Introduction to Quantum Field Theory and Quantum Statistics

1 Introduction to Quantum Field Theory and Quantum Statistics

Michael Bonitz

Insitut f¨urTheoretische Physik und Astrophysik Kiel University

February 9, 2015

preliminary lecture notes, not for distribution 2 Contents

1 Canonical Quantization 11 1.1 Minimal action principle ...... 11 1.1.1 Classical mechanics of a point particle ...... 16 1.1.2 Canonical momentum and Hamilton density of classical fields ...... 17 1.2 Conservation laws in classical field theory ...... 19 1.2.1 Translational invariance. Energy and momentum conservation ...... 21 1.3 Field quantization ...... 24 1.4 Phonons ...... 26 1.4.1 Application of canonical field theory ...... 27 1.4.2 Expansion in terms of eigenfunctions ...... 28 1.4.3 Quantization of the displacement field ...... 30 1.5 Photons ...... 34 1.5.1 Maxwell’s equations. Electromagnetic potentials. Field tensor ...... 34 1.5.2 Lagrange density of the free electromagnetic field . . . . 38 1.5.3 Normal mode expansion of the electromagnetic field ...... 42 1.5.4 Quantization of the electromagnetic field ...... 44 1.6 EMF Quantization in Matter ...... 48 1.6.1 Lagrangian of a classical relativistic particle ...... 48 1.6.2 Relativistic particle coupled to the electromagnetic field ...... 50 1.6.3 Lagrangian of charged particles in an EM field ...... 51 1.6.4 Quantization of the electromagnetic field coupled to charges 54 1.6.5 Quantization of the EM field in a dielectric medium or plasma ...... 56 1.7 Quantization of the Schrödinger field ...... 58 1.8 Quantization of the Klein-Gordon field ...... 58

3 4 CONTENTS

1.9 Coupled equations for the Schr¨odingerand Maxwell ﬁelds . . . . 58 1.10 Problems to Chapter 1 ...... 58

2 Second Quantization 59 2.1 Second quantization in phase space ...... 59 2.1.1 Classical dynamics in terms of point particles ...... 59 2.1.2 Point particles coupled via classical fields ...... 60 2.1.3 Classicle dynamics via particle and Maxwell fields . . . . 61 2.2 Quantum mechanics and first quantization ...... 65 2.3 The ladder operators ...... 66 2.3.1 One-dimensional harmonic oscillator ...... 67 2.3.2 Generalization to several uncoupled oscillators ...... 70 2.4 Interacting Particles ...... 71 2.4.1 One-dimensional chain and its normal modes ...... 71 2.4.2 Quantization of the 1d chain ...... 75 2.4.3 Generalization to arbitrary interaction ...... 77 2.4.4 Quantization of the N-particle system ...... 80 2.5 Continuous systems ...... 81 2.5.1 Continuum limit of 1d chain ...... 81 2.5.2 Equation of motion of the 1d string ...... 82 2.6 Problems to chapter 2 ...... 85

3 Fermions and bosons 87 3.1 Spin statistics theorem ...... 87 3.2 N-particle wave functions ...... 89 3.2.1 Occupation number representation ...... 91 3.2.2 Fock space ...... 92 3.2.3 Many-fermion wave function ...... 92 3.2.4 Many-boson wave function ...... 93 3.2.5 Interacting bosons and fermions ...... 95 3.3 Second quantization for bosons ...... 97 3.3.1 Creation and annihilation operators for bosons ...... 97 3.4 Second quantization for fermions ...... 104 3.4.1 Creation and annihilation operators for fermions . . . . . 104 3.4.2 Matrix elements in Fock space ...... 110 3.4.3 Fock Matrix of the binary interaction ...... 115 3.5 Field operators ...... 118 3.5.1 Deﬁnition of ﬁeld operators ...... 118 3.5.2 Representation of operators ...... 119 3.6 Momentum representation ...... 122 CONTENTS 5

3.6.1 Creation and annihilation operators in momentum space ...... 122 3.6.2 Representation of operators ...... 123 3.7 Problems to Chapter 3 ...... 126

4 Bosons and fermions in equilibrium 127 4.1 Density operator ...... 127 4.2 Path Integral Monte Carlo ...... 127 4.3 Conﬁguration Path Integral ...... 127 4.3.1 Canonical ensemble ...... 128 4.3.2 Evaluation of Correlation energy contributions ...... 138 4.3.3 Electron gas at ﬁnite temperature ...... 138 4.4 Problems to Chapter 4 ...... 138

5 Dynamics of field operators 139 5.1 Equation of motion of the field operators ...... 139 5.2 General Operator Dynamics ...... 143 5.3 Extension to time-dependent hamiltonians ...... 144 5.4 Schrödingerdynamics of the field operators ...... 145 5.5 Dynamics of the density matrix operator ...... 147 5.6 Fluctuations and correlations ...... 151 5.6.1 Fluctuations and correlations ...... 151 5.6.2 Stochastic Mean Field Approximation ...... 154 5.6.3 Iterative Improvement of Stochastic Mean Field Approx- imation ...... 155 5.6.4 Ensemble average of the field operators ...... 155 5.6.5 Field operators and reduced density matrices ...... 156 5.6.6 BBGKY-hierarchy ...... 159 5.7 Problems to Chapter 5 ...... 159

6 Hubbard model 161 6.1 One-dimensional Hubbard chain ...... 161 6.2 Ensemble averages ...... 166

7 Nonequilibrium Green Functions 169 7.1 Introduction ...... 170 7.2 Nonequilibrium Green functions ...... 173 7.2.1 Keldysh Contour ...... 173 7.2.2 One-Particle Nonequilibrium Green Function ...... 182 7.2.3 Matrix Representation of the Green Function ...... 187 7.2.4 Langreth-Wilkins Rules ...... 191 6 CONTENTS

7.2.5 Proporties of the Nonequilibrium Green function . . . . 195 7.3 Kadanoff-Baym-Ansatz ...... 202 7.4 Keldysh-Kadanoff-Baym Equations ...... 210 7.4.1 Derivation of the Martin-Schwinger hierarchy ...... 210 7.4.2 Selfenergy. Keldysh-Kadanoff-Baym equations ...... 218 7.4.3 Equilibrium Limit. Dyson Equation ...... 220 7.5 Many-Body Approximations ...... 221 7.5.1 Requirements for a Conserving Scheme ...... 222 7.5.2 Perturbation Expansions. Born approximation ...... 223 7.5.3 Vertex function ...... 228 7.5.4 Bethe-Salpeter equation ...... 232 7.5.5 Strong coupling. T-matrix selfenergy ...... 232 7.6 Single-time equations ...... 234 7.6.1 The Reconstruction Problem for the One-Particle Green Function ...... 235 7.6.2 The Generalized Kadanoff-Baym Ansatz ...... 239

8 Matter in strong fields 243 8.1 Free particle in a time-dependent field ...... 243 8.1.1 Classical approach ...... 243 8.1.2 Quantum-mechanical approach ...... 245 8.1.3 Alternative analytical methods ...... 252 8.1.4 Computational methods ...... 253 8.2 Atoms in electromagnetic fields ...... 253 8.2.1 Schrödinger equation for atom in EM field ...... 253 8.2.2 Stationary electrical fields. Stark effect...... 254 8.2.3 Ionization processes ...... 255 8.3 Excitation and ionization processes in strong fields ...... 256 8.3.1 Classification of laser intensities ...... 256 8.3.2 Multiphoton processes in weak fields (γ 1) ...... 256 8.3.3 Strong field effects, γ 1 ...... 258 8.3.4 Strong field approximation. Keldysh theory...... 258 8.4 Strong field phenomena ...... 260 8.4.1 Above threshold ionization ...... 260 8.4.2 Higher harmonics generation (HHG) ...... 260 8.5 Problems to Chapter 8 ...... 261

A Perturbation expansion for NEGF 263 A.1 Functional Derivative of a Contour-Ordered Operator Product ...... 263 CONTENTS 7

A.2 Selfenergy ...... 265

A Solutions to problems 269 A.1 Problems from chapter 2 ...... 269 A.2 Problems from chapter 3 ...... 269 A.3 Problems from chapter 4 ...... 271 A.4 Problems from chapter 5 ...... 271 A.5 Problems from chapter 6 ...... 272

Bibliography 293 Chapter 2

Introduction to second quantization

2.1 Second quantization in phase space

2.1.1 Classical dynamics in terms of point particles We consider systems of a large number N of identical particles which interact via pair potentials V and may be subject to an external ﬁeld U. The system is described by the Hamilton function

N N X p2 X X H(p, q) = i + U(r ) + V (r − r ) (2.1) 2m i i j i=1 i=1 1≤i

∂H pi q˙i = = , (2.2) ∂pi m ∂H ∂U X ∂V p˙i = − = − − , (2.3) ∂qi ∂qi ∂qi j6=i

1Generalized equations of motion can, of course, also be obtained for non-Hamiltonian (dissipative) systems

59 60 CHAPTER 2. SECOND QUANTIZATION

which is to be understood as two systems of 3N scalar equations for x1, y1, . . . zN and px1, py1, . . . pzN where ∂/∂q ≡ {∂/∂r1, . . . ∂/∂rN }, and the last equalities are obtained by inserting the hamiltonian (2.1). The system (2.3) is nothing but Newton’s equations containing the forces arising from the gradient of the external potential and the gradient from all pair interactions involving the given particle, i.e. for any particle i = 1 ...N

∂U(r1,... rN ) X ∂V (ri − rj) ˙pi = − − . (2.4) ∂ri ∂ri j6=i Consider, as an example, a system of identical charged particles with charge ei which may be subject to an external electrostatic potential φext and interact eiej with each other via the Coulomb potential Vc = . Then Newton’s equa- |ri−rj | tion (2.4) for particle i contains, on the r.h.s., the gradients of the potential U = eiφext and of the N − 1 Coulomb potentials involving all other particles.

∂eiφext(ri) X ∂ eiej ˙pi = − − , (2.5) ∂ri ∂ri |ri − rj| j6=i

2.1.2 Point particles coupled via classical fields An alternative way of writing Eq. (2.5) is to describe the particle interaction not by all pair interactions but to compute the total electric field, E(r, t), all particles produce in the whole space. The force particle “i” experiences is then just the Lorentz force, eiE, which is minus the gradient of the total electrostatic potential φ which is readily identified from the r.h.s. of Eq. (2.5)

∂φ(r, t) ˙pi = eiE(ri, t) = −ei , (2.6) ∂r r=ri N X ej φ(r, t) = φ (r) + . (2.7) ext |r − r (t)| j=1 j In this case we explicitly know the form of the potential (2.7) but we can also rewrite this in terms of the Poisson equation which is solved by the potential (2.7)

N X ∆φ(r, t) = −4π ejδ[r − rj(t)] = −4πρ(r, t). (2.8) j=1 φ contains the external potential and the potentials induced by all particles at a given space point r at time t. When computing the force on a given particle 2.1. SECOND QUANTIZATION IN PHASE SPACE 61

i the potential has to be taken at r = ri(t) and the contribution of particle “i” to the sum over the particles in Eqs. (2.7, 2.8) has to be excluded (to avoid selfinteraction). Also, on the r.h.s. we have introduced the charge density ρ of the system of N point particles. Considering the formal structure of Eq. (2.6) we notice that the Coulomb forces between discrete particles have been completely eliminated in favor of a space dependent function, the electric field. Obviously, this description is readily generalized to the case of time-dependent external potentials and magnetic fields which yields a coupled set of Newton’s and Maxwell’s equations,

1 ˙p = e E(r , t) + v × B(r , t) , (2.9) i i i c i i 1 ∂B(r, t) divE(r, t) = 4πρ(r, t), ∇ × E(r, t) = − , (2.10) c ∂t 4π 1 ∂E(r, t) divB(r, t) = 0, ∇ × B(r, t) = j(r, t) + , (2.11) c c ∂t where we introduced the current density j which, for a system of point particles PN is given by j(r, t) = j=1 ejvj(t)δ[r − rj(t)]. Charge and current density are determined by the instantaneous phase space trajectories {q(t), p(t)} of all particles. The two sets of equations (2.6,2.8) and (2.9,2.10, 2.11) form closed systems coupling the dynamics of classical charged point particles and a classical electromagnetic field. This coupling occurs, in the particle equation, via the Lorentz force and, in the field equations, via charge and current density. The classical description, therefore, requires knowledge of the N discrete particle trajectories {q(t), p(t)} and of the two continuous vector fields E(r, t), B(r, t). Using the electric and magnetic fields, we may rewrite the Hamilton function (2.1) corresponding to the full system (2.6, 2.10, 2.11)

N N X p2 X 1 Z H = i + U(r ) + d3r E2(r, t) + B2(r, t) (2.12) 2m i 8π i=1 i=1 where the integral contains the energy of the electromagnetic ﬁeld familiar from standard electrodynamics.

2.1.3 Classicle dynamics via particle and Maxwell ﬁelds We have now found two alternative descriptions of the dynamics of interacting point charges: 62 CHAPTER 2. SECOND QUANTIZATION

1. via the system (2.5), involving only discrete point particles, and

2. via the system (2.9, 2.10, 2.11) which gives a hybrid description in which the particles are discrete but the fields (or the particle interaction) continuous. One may ask if there is third form which contains only continuous field-type quantities. This would require to represent also the particles by fields. This is, indeed possible, as we show in this section. In fact, the right hand sides of Maxwell’s equation already do contain (formally) continuous quantities ρ and j representing the particles. However, they contain (via the delta functions) only the particle coordinates. It is, therefore, tempting to consider a symmetric with respect to q and p quantity – the mi- croscopic phase space density which was introduced by Yuri Klimontovich in the 1950s [Kli57]

N N X X N(r, p, t) = δ[r − ri(t)]δ[p − pi(t)] ≡ δ[x − xi(t)] (2.13) i=1 i=1

2 where we introduced the short notations x ≡ {r, p} and xi(t) ≡ {ri(t), pi(t)}. The function N is related to the particle density n(r, t) and it obeys a normalization condition, Z d3p N(r, p, t) = n(r, t), (2.14) Z d6x N(r, p, t) = N(t). (2.15)

If there are no particle sources or sinks, N(t) =const, and there exists a local d conservation law dt N = 0. From this we obtain the equation of motion of N:

N dN(r, p, t) ∂N X ∂ ∂ri = + δ[p − p (t)] δ[r − r (t)] + dt ∂t i ∂r i ∂t i=1 N X ∂ ∂pi δ[r − r (t)] δ[p − p (t)] . i ∂p i ∂t i=1 Using Newton’s equations (2.5) the time derivatives can be computed after which the delta functions allow to replace ri → r and pi → p which can be

2 Note that for a vector y = {y1, y2, y3}, δ[y] ≡ δ[y1]δ[y2]δ[y3], so N contains a product of six scalar delta functions. 2.1. SECOND QUANTIZATION IN PHASE SPACE 63 taken out of the sum. As a result we obtain3 ∂ ∂ 1 ∂ + v + e E(r, t) + v × B(r, t) N(r, p, t) = 0. (2.16) ∂t ∂r c ∂p Thus we have obtained a ﬁeld description of the particles via the function N and eliminated (formally) all discrete particle information. The ﬁeld N also replaces the charge and current density in Maxwell’s equations (2.10, 2.11) via the relations [cf. Eq. (2.14)] Z ρ(r, t) = e d3p N(r, p, t), (2.17) Z j(r, t) = e d3p vN(r, p, t). (2.18)

A particularly simple form is obtained in the absence of a magnetic ﬁeld, in the case of particles interacting by Coulomb potentials, cf. Eqs. (2.6, 2.7) above. Then, we may use the solution of Poisson’s equation, expressing ρ via N Z ρ(r0, t) φ(r, t) = φ (r, t) + dr0 ext |r − r0| Z Z eN(r0, p0, t) = φ (r, t) + dr0 dp0 . (2.19) ext |r − r0| With this expression the electrostatic ﬁeld has been eliminated and the particle dynamics (2.16) becomes a closed equation for N:

∂ ∂ ∂ Z eN(r0, p0, t) ∂ + v − e φ + d6x0 N(r, p, t) = 0 (2.20) ∂t ∂r ∂r ext |r − r0| ∂p

Using the phase space density, all observables of the system can be expressed in terms of ﬁelds. For example, the hamilton function now becomes Z p2 Z H = d6x N(r, p, t) + d6x U(r)N(r, p, t) + 2m 1 Z + d3r E2(r, t) + B2(r, t) , (2.21) 8π We underline that this is an exact equation (as long as a classical description is valid) – no assumptions with respect to the interactions have been

3To simplify the notation, in the following we consider a one-component system with identical charges, e1 = . . . eN = e. An extension to multi-component systems is straightforwardly done by introducing a separate function Na, for each component. 64 CHAPTER 2. SECOND QUANTIZATION made. It is fully equivalent to Newton’s equations (2.6). The discrete nature of the particles has now vanished – it is hidden in the highly singular phase space ﬁeld N. As Newton’s equation, Eq. (2.20) allows for an exact solution of the particle dynamics, once the initial conditions are precisely known. This may be the case for a few particles. In contrast, if our goal is to describe the behavior of a macroscopic particle ensemble any particular initial condition, and the resulting dynamics have to be regarded as random. However, we will be interested in statistically reliable predictions so we need to average over a certain statistical ensemble, e.g. over all possible initial conditions. An ensemble average of N yields directly the single-particle phase space distribution hN(r, p, t)i = f(r, p, t), and Eq. (2.20) turns into an equation for the distribution function f, i.e. a kinetic equation. The use of the phase space density N as starting point for derivation of a kinetic theory of gases and plasmas has been successfully demonstrated by Klimontovich, for details see his text books [Kli75, Kli80].

With Eqs. (2.16) and (2.20) we have realized the third picture of coupled particle electromagnetic field dynamics – in terms of the particle field N and the electric and magnetic fields. While we have concentrated on charged particles and Coulomb interaction, the approach may be equally applied to other interactions, e.g. electrons interaction with lattice vibrations of a solid described by the displacement field, see Sec. 2.5.1. Thus the basis for a classical field theory has been achieved.

Of course, this picture is based on classical physics, i.e. on Newton’s equations for point particles and Maxwell’s equations for the electromagnetic field. No quantum effects appear, neither in the description of the particles nor the field. The classical picture has been questioned only at the end of the 19th century where the experiments on black body radiation could not be correctly explained by Maxwell’s theory of the electromagnetic field. The classical expression for the field energy of an electromagnetic wave which only depends on the field amplitudes E0, B0 and, therefore, is a continuous function, cf. hamiltonian (2.21), did not reproduce the measured spectral energy density. The solution which was found by Max Planck indicated that the field energy cannot depend on the field amplitudes alone. The energy exchange between electromagnetic field and matter is even entirely independent of E0, B0, instead it depends on the frequency ω of the wave. Thus the total energy of an electromagnetic wave of frequency ω is an integer multiple of an elementary energy, Wfield(ω) = N~ω, where ~ is Planck’s constant, i.e. the energy is quantized. 2.2. QUANTUM MECHANICS AND FIRST QUANTIZATION 65

What Planck had discovered in 1900 was the quantization of the electromagnetic field 4. This concept is very different from the quantum mechanical description of the electron dynamics from which it is, therefore, clearly distin- guished by the now common notion of “second quantization”. Interestingly, however, the “first quantization” of the motion of microparticles was introduced only a quarter century later when quantum mechanics was discovered.

2.2 Quantum mechanics and ﬁrst quantization

Let us brieﬂy recall the main ideas of quantum theory. The essence of quantum mechanics or “ﬁrst” quantization is to replace functions by operators, starting from the coordinate and momentum,

r → ˆr = r, ∂ p → pˆ = ~ , i ∂r where the last equalities refer to the coordinate representation. These operators are hermitean, ˆr† = ˆr and pˆ† = pˆ, and do not commute

[ˆxi, pˆj] = i~δi,j, (2.22) which means that coordinate and momentum (the same components) cannot be measured simultaneously. The minimal uncertainty of such a simultaneous measurement is given by the Heisenberg relation

∆ˆx ∆ˆp ≥ ~, (2.23) i i 2 where the standard deviation (“uncertainty”) of an operator Aˆ is definied as s 2 ∆Aˆ = Aˆ − hAî , (2.24) and the average is computed in a given state |ψi, i.e. hAî = hψ|Aˆ|ψi. The general formulation of quantum mechanics describes an arbitrary quantum system in terms of abstract states |ψi that belong to a Hilbert space (Dirac’s

4Planck himself regarded the introduction of the energy quantum ~ω only as a formal mathematical trick and did not question the validity of Maxwell’s field theory. Only half a century later, when field quantization was systematically derived, the coexistence of the concepts of energy quanta and electromagnetic waves became understandable, see Sec. 1.5 66 CHAPTER 2. SECOND QUANTIZATION notation), and operators act on these state returning another Hilbert space state, Aˆ|ψi = |φi. The central quantity of classical mechanics – the hamilton function – retains its functional dependence on coordinate and momentum in quantum mechanics as well (correspondence principle) but becomes an operator depending on operators, H(r, p) → Hˆ (ˆr, pˆ). The classical equations of motion – Hamil- ton’s equations or Newton’s equation (2.4) – are now replaced by a partial differential equation for the Hamilton operator, the Schrödingerequation

∂ i |ψ(t)i = Hˆ |ψ(t)i. (2.25) ~∂t

Stationary properties are governed by the stationary Schr¨odingerequation that follows from the ansatz5

− i Htˆ |ψ(t)i = e ~ |ψi. Hˆ |ψi = E|ψi, (2.26)

The latter is an eigenvalue equation for the Hamilton operator with the eigenfunctions |ψi and corresponding eigenvalues E. “First” quantization is evident in the case of particle motion in a conﬁning potential U(r), such as an oscillator potential: classical bounded motion transforms, in quantum mechanics, into a set of eigenstates |ψni (that are localized as well) that exist only for a sequence of discrete (quantized) energies En. This example is discussed more in detail below.

2.3 The linear harmonic oscillator and the ladder operators

Let us now recall the simplest example of quantum mechanics: one particle in m 2 2 a one-dimensional harmonic potential U(x) = 2 ω x , i.e. in Eq. (2.1), N = 1 and the interaction potentials vanish. We will use this example to introduce the basic idea of “second quantization”. In writing the potential U(x) we switched to the coordinate representation where states |ψni are represented by functions of the coordinate, ψn(x). At the end we will return to the abstract notation in terms of Dirac states.

5Here we assume a time-independent hamiltonian. 2.3. THE LADDER OPERATORS 67

2.3.1 One-dimensional harmonic oscillator The stationary properties of the harmonic oscillator follow from the stationary Schr¨odingerequation (2.26) which now becomes, in coordinate representation

pˆ2 mω2 Hˆ (ˆx, pˆ)ψ (x) = + xˆ2 ψ (x) = E ψ (x), (2.27) n 2m 2 n n n

~ d wherex ˆ = x andp ˆ = i dx . We may bring the Hamilton operator to a more symmetric form by introducing the dimensionless coordinate ξ = x/x0 with 1/2 the length scale x0 = [~/mω] , whereas energies will be measured in units of ω. Then we can replace d = 1 d and obtain ~ dx x0 dξ

Hˆ 1 ∂2 = − + ξ2 . (2.28) ~ω 2 ∂ξ2 This quadratic form can be rewritten in terms of a product of two ﬁrst order operators a, a†, the “ladder operators”,

1 ∂ a = √ + ξ , (2.29) 2 ∂ξ 1 ∂ a† = √ − + ξ . (2.30) 2 ∂ξ Indeed, computing the product

1 ∂ 1 ∂ Nˆ = a†a = √ + ξ √ − + ξ (2.31) 2 ∂ξ 2 ∂ξ 1 ∂2 = − + ξ2 − 1 , 2 ∂ξ2 the hamiltonian (2.28) can be written as

Hˆ 1 = Nˆ + . (2.32) ~ω 2 It is obvious from (2.32) that Nˆ commutes with the hamiltonian,

[H,ˆ Nˆ] = 0, (2.33) and thus the two have common eigenstates. This way we have transformed the hamiltonian from a function of the two non-commuting hermitean operators 68 CHAPTER 2. SECOND QUANTIZATION xˆ andp ˆ into a function of the two operators a and a† which are also non- commuting6, but not hermitean, instead they are the hermitean conjugate of each other,

[a, a†] = 1, (2.34) (a)† = a†, (2.35) which is easily veriﬁed. The advantage of the ladder operators is that they allow for a straightforward computation of the energy spectrum of Hˆ , using only the properties (2.31) and (2.34), without need to solve the Schr¨odingerequation, i.e. avoid- 7 ing explicit computation of the eigenfunctions ψn(ξ) . This allows us to return to a representation-independent notation for the eigenstates, ψn → |ni. The only thing we require is that these states are complete and orthonormal, P 0 1 = n |nihn| and hn|n i = δn,n0 . Now, acting with Nˆ on an eigenstate, using Eq. (2.32), we obtain E 1 Nˆ|ni = a†a |ni = n − |ni = n|ni, ∀n, (2.36) ~ω 2 E 1 n = n − , ~ω 2 where the last line relates the eigenvalues of Nˆ andn ˆ that correspond to the common eigenstate |ni. Let us now introduce two new states that are created by the action of the ladder operators,

a|ni = |n˜i, a†|ni = |n¯i, where this action is easily computed. In fact, multiplying Eq. (2.36) from the left by a, we obtain E 1 aa† |n˜i = n − |n˜i. ~ω 2 Using the commutation relation (2.34) this expression becomes E 3 a†a |n˜i = n − |n˜i = (n − 1)|n˜i, ~ω 2 6The appearance of the standard commutator indicates that these operators describe bosonic excitations. 7We will use the previous notation ψ, which means that the normalization is R 2 x0 dξ|ψ(ξ)| =1 2.3. THE LADDER OPERATORS 69 which means the state |n˜i is an eigenstate of Nˆ [and, therefore, of Hˆ ] and has an energy lower than |ni by ~ω whereas the eigenvalue of Nˆ isn ˜ = n−1. Thus, the action of the operator a is to switch from an eigen state with eigenvalue n to one with eigenvalue n − 1. Obviously, this is impossible for the ground state, i.e. when a acts on |0i, so we have to require

|0˜i = a|0i ≡ 0. (2.37)

When we use this result in Eq. (2.36) for n = 0, the l.h.s. is zero with the consequence that the term in parantheses must vanish. This immediately leads to the well-known result for the ground state energy: E0 = ~ω/2, corresponding to the eigenvalue 0 of Nˆ. From this we now obtain the energy spectrum of the excited states: acting with a† from the left on Eq. (2.36) and using the commutation relation (2.34), we obtain E 1 Nˆ |n¯i = n − + 1 |n¯i =n ¯|n¯i. ~ω 2

Thus,n ¯ is again an eigenstate of Nˆ and Hˆ . Further, if the eigenstate |ni has an energy En, cf. Eq. (2.36), thenn ¯ has an energy En + ~ω, whereas the associated eigenvalue of Nˆ isn ¯ = n + 1. Starting from the ground state † and acting repeatedly with a we construct the whole spectrum, En, and may express all eigenfunctions via ψ0:

1 E = ω n + , n = 0, 1, 2,... (2.38) n ~ 2 †n |ni = Cn a |0i. (2.39) 1 Cn = √ , (2.40) n!

† where the normalization constant Cn will be veriﬁed from the properties of a below. The above result shows that the eigenvalue of the operator Nˆ is just the quantum number n of the eigenfunction |ni. In other words, since |ni is obtained by applying a† to the ground state function n times or by “n-fold excitation”, the operator Nˆ is the number operator counting the number of excitations (above the ground state). Therefore, if we are not interested in the analytical details of the eigenstates we may use the operator Nˆ to count the number of excitations “contained” in the system. For this reason, the common notion for the operator a (a†) is “annihilation” (“creation”) operator of an excitation. For an illustration, see Fig. 2.3.1. 70 CHAPTER 2. SECOND QUANTIZATION

Figure 2.1: Left: oscillator potential and energy spectrum. The action of the operators a and a† is illustrated. Right: alternative interpretation: the operators transform between “many-particle” states containing diﬀerent number of elementary excitations.

From the eigenvalue problem of Nˆ, Eq. (2.36) we may also obtain the explicit action of the two operators a and a†. Since the operator a transforms a state into one with quantum number n lower by 1 we have √ a|ni = n|n − 1i, n = 0, 1, 2,... (2.41) where the prefactor may be understood as an ansatz8. The correctness is proven by deriving, from Eq. (2.41), the action of a† and then verifying that we recover the eigenvalue problem of Nˆ, Eq. (2.36). The action of the creation operator is readily obtained using the property (2.35): X X a†|ni = |n¯ihn¯|a†|ni = |n¯i a[hn¯|] |ni n¯ n¯ X √ √ = |n¯i n¯ hn¯ − 1 |ni = n + 1 |n + 1i. (2.42) n¯ Inserting these explicit results for a and a† into Eq. (2.36), we immediately verify the consistency of the choice (2.41). Obviously the oscillator eigenstates |ni are no eigenstates of the creation and annihilation operators 9.

A more detailed discussion is part of the problems, see Sec. 2.6.

2.3.2 Generalization to several uncoupled oscillators The previous results are directly generalized to a three-dimensional harmonic oscillator with frequencies ωi, i = 1, 2, 3, which is described by the hamiltonian

3 ˆ X ˆ H = H(ˆxi, pˆi), (2.43) i=1 which is the sum of three one-dimensional hamiltonians (2.27) with the po- m 2 2 2 2 2 2 tential energy U(x1, x2, x3) = 2 (ω1x1 + ω2x2 + ω3x3). Since [pi, xk] ∼ δk,i 8This expression is valid also for n = 0 where the prefactor assures that application of a to the ground state does not lead to a contradiction. 9A particular case are Glauber states (coherent states) that are a special superposition of the oscillator states which are the eigenstate of the operator a. 2.4. INTERACTING PARTICLES 71 all three hamiltonians commute and have joint eigenfunction (product states). The problem reduces to a superposition of three independent one-dimensional oscillators. Thus we may introduce ladder operators for each component inde- pendently as in the 1d case before, 1 ∂ ai = √ + ξi , (2.44) 2 ∂ξi † 1 ∂ † ai = √ − + ξi , [ai, ak] = δi,k. (2.45) 2 ∂ξi Thus the hamiltonian and its eigenfunctions and eigenvalues can be written as

3 X 1 Hˆ = ω a†a + ~ i i i 2 i=1 ai|0i = 0, i = 1, 2, 3 1 † n1 † n2 † n3 ψn1,n2,n3 = |n1n2n3i = √ (a1) (a2) (a3) |0i (2.46) n1!n2!n3! 3 X 1 E = ω n + . ~ i i 2 i=1 Here |0i ≡ |000i = |0i|0i|0i denotes the ground state and a general state |n1n2n3i = |n1i|n2i|n3i contains ni elementary excitations in direction i, cre- † ated by ni times applying operator ai to the ground state. Finally, we may consider a more general situation of any number M of coupled independent linear oscillators and generalize all results by replacing the dimension 3 → M.

2.4 Generalization to interacting particles. Normal modes

The previous examples of independent linear harmonic oscillators are of course the simplest situations which, however, are of limited interest. In most problems of many-particle physics the interaction between the particles which was neglected so far, is of crucial importance. We now discuss how to apply the formalism of the creation and annihilation operators to interacting systems.

2.4.1 One-dimensional chain and its normal modes We consider the simplest case of an interacting many-particle system: N identical classical particles arranged in a linear chain and interacting with their 72 CHAPTER 2. SECOND QUANTIZATION

Figure 2.2: Illustration of the one-dimensional chain with nearest neighbor interaction. The chain is made inﬁnite by connecting particle N + 1 with particle 1 (periodic boundary conditions). left and right neighbor via springs with constant k.10, see Fig. 2.2.

This is the simplest model of interacting particles because each particle is assumed to be ﬁxed around a certain position xi in space around which it can perform oscillations with the displacement qi and the associated momentum 11 pi. Then the hamiltonian (2.1) becomes

N 2 X pj k 2 H(p, q) = + (q − q ) . (2.47) 2m 2 j j+1 j=1 Applying Hamilton’s equations we obtain the system of equations of motion (2.4) mq¨j = k (qj+1 − 2qj + qj−1) , j = 1 ...N (2.48) which have to be supplemented with boundary and initial conditions. In the following we consider a macroscopic system and will not be interested in effects of the left and right boundary. This can be achieved by using “periodic” boundary conditions, i.e. periodically repeating the system according to qj+N (t) = qj(t) for all j [for solutions for the case of a ﬁnite system, see Problem 5, Sec. 2.6]. We start with looking for particular (real) solutions of the following form12 i(−ωt+jl) qj(t) = e + c.c., (2.49) which, inserted into the equation of motion, yield for any j

2 ∗ il −il ∗ −mω qj + qj = k e − 2 + e qj + qj , (2.50) resulting in the following relation between ω and k (dispersion relation)13: l k ω2(l) = ω2 sin2 , ω2 = 4 . (2.51) 0 2 0 m 10Here we follow the discussion of Huang [Hua98]. 11Such “lattice” models are very popular in theoretical physics because they allow to study many-body eﬀects in the most simple way. Examples include the Ising model, the Anderson model or the Hubbard model of condensed matter physics. 12 0 0 In principle, we could use a prefactor qj = q diﬀerent from one, but by rescaling of q it can always be eliminated. The key is that the amplitudes of all particles are strictly coupled. 13 2 x We use the relation 1 − cos x = 2 sin 2 . 2.4. INTERACTING PARTICLES 73

Figure 2.3: Dispersion of the normal modes, Eq. (2.51), of the 1d chain with periodic boundary conditions.

Here ω0 is just the eigenfrequency of a spring with constant k, and the prefactor 2 arises from the fact that each particle interacts with two neighbors. While 0 the condition (2.51) is independent of the amplitudes qj , i.e. of the initial conditions, we still need to account for the boundary (periodicity) condition. Inserting it into the solution (2.49) gives the following condition for l, indepen- n N dently of ω: l → ln = N 2π, where n = 0, ±1, ±2, · · · ± 2 . Thus there exists a discrete spectrum of N frequencies of modes which can propagate along the chain (we have to exclude n = 0 since this corresponds to a time-independent trivial constant displacement), k nπ N ω2 = 4 sin2 , n = ±1, ±2, · · · ± . (2.52) n m N 2 This spectrum is shown in Fig. 2.3. These N solutions are the complete set of normal modes of the system (2.47), corresponding to its N degrees of freedom. These are collective modes in which all particles participate, all oscillate with the same frequency but with a well deﬁned phase which depends on the particle number. These normal modes are waves running along the chain with a phase 14 velocity cn ∼ ωn/ln. Due to the completeness of the system of normal modes, we can expand any excitation of particle j and the corresponding momentum pj(t) = mq˙j(t) into a supersposition of normal mode contributions (n 6= 0)

N N 2 2 1 X n 1 X 0 i(−ωnt+2π N j) −iωnt qj(t) = √ Qn e = √ e Qn(j) (2.53) N N N N n=− 2 n=− 2 N N 2 2 1 X n 1 X 0 i(−ωnt+2π N j) −iωnt pj(t) = √ Pn e = √ e Pn(j), (2.54) N N N N n=− 2 n=− 2

0 0 where Pn = −imωnQn. Note that the complex conjugate contribution to mode n is contained in the sum (term −n). Also, qj(t) and pj(t) are real functions. ∗ By computing the complex conjugate qj and equating the result to qj we obtain 0 ∗ 0 the conditions (Qn) = Q−n and ω−n = −ω−n. Analogously we obtain for the 0 ∗ 0 momenta (Pn ) = P−n. To make the notation more compact we introduced

14 The actual phase velocity is ωn/kn, where the wave number kn = ln/a involves a length scale a which does not appear in the present discrete model. 74 CHAPTER 2. SECOND QUANTIZATION

~ ~ the N-dimensional complex vectors Qn and Pn with the component j being 0 i2πnj/N 0 i2πnj/N 15 equal to Qn(j) = Qne and Pn(j) = Pn e . One readily proofs that these vectors form an orthogonal system by computing the scalar product

N X n+m ~ ~ 0 0 i2π N j 0 0 QnQm = QnQm e = NQnQmδn,−m. (2.55) j=1

Using this property it is now straightforward to compute the hamilton function in normal mode representation. Consider ﬁrst the momentum contribution, N N N X 1 X2 X2 p2(t) = P~ P~ e−i(ωn+ωm)t, (2.56) j N n m j=1 N N n=− 2 m=− 2 where the sum over j has been absorbed in the scalar product. Using now the orthogonality condition (2.55) we immediately simplify

N X 2 X 0 2 pj (t) = |Pn | . (2.57) j=1 n

Analogously, we compute the potential energy

N N N 2 2 k X 2 k X X U = [q (t) − q (t)] = e−i(ωn+ωm)t 2 j j+1 2N j=1 N N n=− 2 m=− 2 N X n n m m 0 0 i2π N j i2π N (j+1) i2π N j i2π N (j+1) × QnQm e − e e − e . j=1

The sum over j can again be simpliﬁed using the orthogonality condition (2.55), which allows to replace m by −n,

N 1 X 0 0 i2π n j i2π n (j+1) i2π m j i2π m (j+1) Q Q e N − e N e N − e N = N n m j=1 i2π n i2π m ~ ~ = 1 − e N 1 − e N QnQm = 2 2πn 0 0 ωn 0 2 = 2 1 − cos δn,−mQnQm = 4 2 δn,−m|Qn| , N ω0

15See problem 5, Sec. 2.6. 2.4. INTERACTING PARTICLES 75

2 x where we have used Eq. (2.51) and the relation 1−cos x = 2 sin 2 . This yields for the potential energy

k X mω2 U = n 2 k n and for the total hamilton function

N X2 1 m H(P,Q) = |P 0|2 + ω2|Q0 |2 . (2.58) 2m n 2 n n N n=− 2

2.4.2 Quantization of the 1d chain We now quantize the interacting system (2.47) by replacing coordinates and momenta of all particles by operators

(qi, pi) → (ˆqi, pî) , i = 1,...N, † † withq î =q ˆj, pî =p î, [ˆqi, pˆj] = i~δij. (2.59) The Hamilton function (2.47) now becomes an operator of the same functional form (correspondence principle),

N 2 X pˆj k 2 Hˆ (ˆp, qˆ) = + (ˆq − qˆ ) , 2m 2 j j+1 j=1 and we still use the periodic boundary conditionsq ˆN+i =q ˆi. The normal modes of the classical system remain normal modes in the quantum case as 0 0 well, only the amplitudes Qn and Pn become operators

N 1 2 X −iωnt ˆ qˆj(t) = √ e Qn(j) (2.60) N N n=− 2 N 1 2 X −iωnt ˆ pj(t) = √ e Pn(j), (2.61) N N n=− 2

ˆ ˆ0 ˆ ˆ0 ˆ0 where Qn(j) = Qn exp{i2πnj/N}, Pn(j) = Pn exp{i2πnj/N} and Pn = ˆ0 −imωnQn. ˆ0 What remains is to impose the necessary restrictions on the operators Qn ˆ0 and Pn such that they guarantee the properties (2.59). One readily veriﬁes 76 CHAPTER 2. SECOND QUANTIZATION

ˆ0 † ˆ0 ˆ0 † ˆ0 that hermiticity of the operators is fulﬁlled if (Q )n = Q−n,(P )n = P−n and ω−n = −ωn. Next, consider the commutator ofq ˆi andp ˆj and use the normal mode representations (2.60, 2.61),

1 X X 0 0 −i(ω +ω )t i 2π (kn+jm) [ˆq , pˆ ] = [Qˆ , Pˆ ]e n m e N . (2.62) k j N n m n m

ˆ0 ˆ0 A suﬃcient condition for this expression to be equal i~δk,j is evidently [Qn, Pm] = i~δn,−m which is veriﬁed separately for the cases k = j and k 6= j. In other words, the normal mode operators obey the commutation relation h i ˆ0 ˆ0 † Qn, (Pm) = i~δn,m, (2.63) and the hamiltonian becomes, in normal mode representation,

N X2 1 m Hˆ (P,ˆ Qˆ) = |Pˆ0|2 + ω2|Qˆ0 |2 . (2.64) 2m n 2 n n N n=− 2 This is a superposition of N independent linear harmonic oscillators with the frequencies ωn given by Eq. (2.52). Applying the results for the superposition of oscillators, Sec. 2.3.2, we readily can perform the second quantization by p mωn N N deﬁning dimensionless coordinates, ξn = Qn, n = − ,... , n 6= 0, ~ 2 2 and introducing the creation and annihilation operators, 1 ∂ an = √ + ξn , (2.65) 2 ∂ξn † 1 ∂ † an = √ − + ξn , [an, ak] = δn,k. (2.66) 2 ∂ξn Thus the hamiltonian and its eigenfunctions and eigenvalues can be written as

N X2 1 Hˆ = ω a† a + ~ n n n 2 N n=− 2 N N a |0i = 0, n = − ,... n 2 2 1 m1 mN ψ = |m . . . m i = √ a† ... a† |0i m1,...mN 1 N − N N m1! . . . mN ! 2 2 N X2 1 E = ω m + . ~ n n 2 N n=− 2 2.4. INTERACTING PARTICLES 77

Here |0i ≡ |0 ... 0i = |0i ... |0i [N factors] denotes the ground state and a general state |m−N/2 . . . mN/2i = |m−N/2i ... |mN/2i contains mn elementary † excitations of the normal mode n, created by mn times applying operator an to the ground state.

Finally we notice that the commutation relation (2.63) is that of bosons. This result was independent of whether the particles in the chain are fermions or bosons. This is discussed in Problem 6, Sec. 2.6.

2.4.3 Generalization to arbitrary interaction Of course, the simple 1d chain is a model with a limited range of applicability. A real system of N interacting particles in 1d will be more difficult, at least by three issues: first, the pair interaction potential V may have any form. Second, the interaction, in general, involves not only nearest neighbors, and third, the effect of the full 3d geometry may be relevant. We, therefore, now return to the general 3d system of N classical particles (2.1) with the total potential energy16 N X X Utot(q) = U(ri) + V (ri − rj), (2.67) i=1 1≤i

∂ m¨ri = − Utot(q), i = 1,...N. (2.68) ∂ri Let us consider stationary solutions, where the time derivatives on the l.h.s. vanish. The system will then be in a stationary state “s00 corresponding to (0) (0) (0) a minimum qs of Utot of depth Us = Utot(qs ) [the classical ground state corresponds to the deepest minimum]. In the case of weak excitations from the (0) (0) minimum, q = qs + ξ, with |ξ| << qs , the potential energy can be expanded in a Taylor series17

∂ 1 U (q) = U (0) + U (q = q(0))ξ + ξT H(s)ξ + ... (2.69) tot s ∂q tot s 2 where all ﬁrst derivatives are zero, and we limit ourselves to the second order (harmonic approximation). Here we introduced the 3N × 3N Hesse matrix

16Here we follow the discussion of Ref. [HKL+09] 17 (0) Recall that q, qs and ξ are 3N-dimensional vectors in conﬁguration space. 78 CHAPTER 2. SECOND QUANTIZATION

(s) ∂2 (0) T H = Utot(q = qs ), where xi, xj = x1, y1, . . . zN , and ξ is the trans- ij ∂xi∂xj posed vector (row) of ξ. Thus, for weak excitations, the potential energy (0) change ∆Utot = Utot(q) − Us is reduced to an expression which is quadratic in the displacements ξ, i.e. we are dealing with a system of coupled harmonic oscillators18 We can easily transform this to a system of uncoupled oscillators by di- agonalizing the Hesse matrix which can be achieved by solving the eigenvalue problem (we take the mass out for dimensional reasons)

λnmQn = HQn, n = 1,... 3N. (2.70)

19 Since H is real, symmetric and positive deﬁnite the eigenvalues√ are real and positive corresponding to the normal mode frequencies ωn = λn. Fur- thermore, as a result of the diagonalization, the 3N-dimensional eigenvectors form a complete orthogonal system {Qn} with the scalar product QnQm ≡ P3N i=1 Qn(i)Qm(i) ∼ δm,n which means that any excitation can be expanded into a superposition of the eigenvectors (normal modes),

3N (0) X q(t) = qs + cn(t)Qn. (2.71) n=1

The expansion coeﬃcients cn(t) (scalar functions) are the normal coordinates. Their equation of motion is readily obtained by inserting a Taylor expansion of the gradient of Utot [analogous to (2.69)] into (2.68), ∂U 0 = mq¨ + tot = mq¨ + H· ξ, (2.72) ∂q and, using Eq. (2.71) forq ¨ and eliminating H with the help of (2.70),

3N X 2 0 = m c¨n(t) + cn(t)ωn Qn. (2.73) n=1

Due to the orthogonality of the Qn which are non-zero, the solution of this equation implies that the terms in the parantheses vanish simultaneously for every n, leading to an equation for a harmonic oscillator with the solution

cn(t) = An cos{ωnt + Bn}, n = 1,... 3N, (2.74) 18Strictly speaking, from the 3N degrees of freedom, up to three [depending on the symmetry of U] may correspond to rotations of the whole system (around one of the three coordinate axes, these are center of mass excitations which do not change the particle distance), and the remaining are oscillations. 19 (0) qs corresponds to a mininmum, so the local curvature of Utot is positive in all directions 2.4. INTERACTING PARTICLES 79

where the coeﬃcients An and Bn depend on the initial conditions. Thus, the normal coordinates behave as independent linear 1d harmonic oscillators. In analogy to the coordinates, also the particle momenta, corresponding to some excitation q(t), can be expanded in terms of normal modes by using p(t) = mq˙(t). Using the result for cn(t), Eq. (2.74), we have the following general expansion

3N 3N (0) X X q(t) − qs = An cos{ωnt + Bn}Qn ≡ Qn(t) (2.75) n=1 n=1 3N 3N X X p(t) = An sin{ωnt + Bn}Pn ≡ Pn(t), (2.76) n=1 n=1

where the momentum amplitude vector is Pn = −mωnQn. Finally, we can transform the Hamilton function into normal mode representation, using the harmonic expansion (2.69) of the potential energy

2 N 2 p X p 1 X (s) H(p, q) = + U (q) = U (0) + i + ξT (i)H ξ(j). (2.77) 2m tot s 2m 2 ij i=1 i6=j

Eliminating the Hesse matrix with the help of (2.70) and inserting the expansions (2.75) and (2.76) we obtain

3N 3N (0) X X Pn(t)Pn0 (t) m 2 H(p, q) − U = + ω δ 0 Q (t)Q 0 (t) s 2m 2 n n,n n n n=1 n0=1 3N X P 2(t) m = n + ω2Q2 (t) ≡ H(P,Q), (2.78) 2m 2 n n n=1 where, in the last line, the orthogonality of the eigenvectors has been used. Thus we have succeeded to diagonalize the hamiltonian of the N-particle system with arbitrary interaction. Assuming weak excitations from a stationary state the hamiltonian can be written as a superposition of 3N normal modes. This means, we can again apply the results from the case of uncoupled harmonic oscillators, Sec. 2.3.2, and immediately perform the “ﬁrst” and “second” quantization. 80 CHAPTER 2. SECOND QUANTIZATION

2.4.4 Quantization of the N-particle system For the ﬁrst quantization we have to replace the normal mode coordinates and momenta by operators, ˆ ˆ Qn(t) → Qn(t) = An cos{ωnt + Bn}Qn ˆ ˆ Pn(t) → Pn(t) = An sin{ωnt + Bn}Pn, (2.79) leaving the time-dependence of the classical system unchanged. Further we have to make sure that the standard commutation relations are fulﬁlled, i.e. ˆ ˆ [Qn, Pm] = i~δn,m. This should follow from the commutation relations of the original particle coordinates and momenta, [xiα, pjβ] = i~δi,jδα,β, where α, β = 1, 2, 3 and i, j = 1,...N, see Problem 7, Sec. 2.6. Then, the Hamilton operator becomes, in normal mode representation

3N ( ) X Pˆ2(t) m Hˆ (P,ˆ Qˆ) = n + ω2Qˆ2 (t) , (2.80) 2m 2 n n n=1 which allows us to directly introduce the creation and annihilation operators p mωn ˆ by introducing ξn = Qn, n = 1,... 3N) ~ 1 ∂ an = √ + ξn , (2.81) 2 ∂ξn † 1 ∂ † an = √ − + ξn , [an, ak] = δn,k. (2.82) 2 ∂ξn Thus the hamiltonian and its eigenfunctions and eigenvalues can be written as

3N X 1 Hˆ = ω a† a + ~ n n n 2 n=1 an|0i = 0, n = 1,... 3N 1 † n1 † n3N ψn1,...n3N = |n1 . . . n3N i = √ (a1) ... (a3N ) |0i n1! . . . n3N ! 3N X 1 E = ω n + . ~ n n 2 n=1

Here |0i ≡ |0 ... 0i = |0i ... |0i [3N factors] denotes the ground state and a general state |n1 . . . n3N i = |n1i ... |n3N i contains nn elementary excitations of † the normal mode n, created by nn times applying operator an to the ground state. 2.5. CONTINUOUS SYSTEMS 81

In summary, in ﬁnding the normal modes of the interacting N-particle system the description is reduced to a superposition of independent contributions from 3N degrees of freedom. Depending on the system dimensionality, these include (for a three-dimsional system) 3 translations of the center of mass and 3 rotations of the system as a whole around the coordinate axes. The remaining normal modes correspond to excitations where the particle distances change. Due to the stability of the stationary state with respect to weak excitations, these relative excitations are harmonic oscillations which have been quantized. In other words, we have 3N −6 phonon modes associated with the corresponding creation and annihilation operators and energy quanta. The frequencies of the modes are determined by the local curvature of the total potential energy (the diagonal elements of the Hesse matrix).

Additional discussions of the present concept are subject of problems 7 and 8, cf. Sec. 2.6.

2.5 Continuous systems

2.5.1 Continuum limit of 1d chain

So far we have considered discrete systems containing N point particles. If the number of particles grows and their spacing becomes small we will eventually reach a continuous system – the 1d chain becomes a 1d string. We start with assigning particle i a coordinate xj = ja where j = 0,...N, a is the constant interparticle distance and the total length of the system is l = Na, see Fig.

We again consider a macroscopic system which is now periodically repeated after length l, i.e. points x = 0 and x = l are identical20. In the discrete system we have an equally spaced distribution of masses m of point particles with a linear mass density ρ = m/a. The interaction between the masses is characterized by an elastic tension σ = κa where we relabeled the spring constant by κ. The continuum limit is now performed by simulataneously increasing the particle number and reducing a but requiring that the density

20Thus we have formally introduced N + 1 lattice points but only N are diﬀerent. 82 CHAPTER 2. SECOND QUANTIZATION and the tension remain unchanged,

a, m −→ 0 N, κ −→ ∞ l, ρ, σ = const.

We now consider the central quantity, the displacement of the individual particles qi(t) which now transforms into a continuous displacement ﬁeld q(x, t). Further, with the continuum limit, diﬀerences become derivatives and the sum over the particles is replaced by an integral according to

qj(t) −→ q(x, t) ∂q q − q −→ a j+1 j ∂x X 1 Z l −→ dx. a j 0

Instead of the Hamilton function (2.47) we now consider the Lagrange function which is the diﬀerence of kinetic and potential energy, L = T − V , which in the continuum limit transforms to

N X nm 2 κ 2o L(q, q˙) = (q ˙ ) − (q − q ) 2 j 2 j j+1 j=1 ( ) 1 Z l ∂q(x, t)2 ∂q(x, t)2 −→ dx ρ − σ (2.83) 2 0 ∂t ∂x

The advantage of using the Lagrange function which now is a functional of the displacement ﬁeld, L = L[q(x, t)], is that there exists a very general method of ﬁnding the corresponding equations of motion – the minimal action principle.

2.5.2 Equation of motion of the 1d string We now deﬁne the one-dimensional Lagrange density L

Z l L = dx L[q ˙(x, t), q0(x, t)], (2.84) 0 where Eq. (2.83) shows that Lagrange density of the spring depends only on two ﬁelds – the time derivativeq ˙ and space derivative q0 of the displacement 2.5. CONTINUOUS SYSTEMS 83

Figure 2.4: Illustration of the minimal action principle: the physical equation of motion corresponds to the tractory q(x, t) which minimizes the action, Eq. (2.85) at ﬁxed initial and ﬁnal points (ti, 0) and (tf , l).

field. The action is defined as the time integral of the Lagrange function between a fixed initial time ti and final time tf

Z tf Z tf Z l S = dtL = dt dx L[q ˙(x, t), q0(x, t)]. (2.85) ti ti 0 The equation of motion of the 1d string follows from minimizing the action with respect to the independent variables of L [this “minimal action principle” will be discussed in detail in Chapter 1, Sec. 1.1], for illustration, see Fig. 2.4,

Z tf Z l δL δL 0 0 = δS = dt dx δq˙ + 0 δq ti 0 δq˙ δq Z tf Z l = dt dx {ρq˙ δq˙ − σq0 δq0} . (2.86) ti 0

∂ 0 We now change the order of diﬀerentiation and variation, δq˙ = ∂t δq and δq = ∂ ∂x δq and perform partial integrations with respect to t in the ﬁrst term and x in the second term of (2.86)

Z tf Z l 0 = − dt dx {ρq¨ − σq00} δq, (2.87) ti 0 where the boundary values vanish because one requires that the variation δq(x, t) are zero at the border of the integration region, δq(0, t) = δq(l, t) ≡ 0. Since this equation has to be fulﬁlled for any ﬂuctuation δq(x, t) the term in the parantheses has to vanish which yields the equation of motion of the 1d string

∂2q(x, t) ∂2q(x, t) rσ r κ − c2 = 0, with c = = a . (2.88) ∂t2 ∂x2 ρ m

This is a linear wave equation for the displacement ﬁeld, and we introduced the phase velocity, i.e. the sound speed c. The solution of this equation can be written as i(kx−ωt) q(x, t) = q0e + c.c., (2.89) 84 CHAPTER 2. SECOND QUANTIZATION

Figure 2.5: Dispersion of the normal modes of the discrete 1D chain and of the associated continuous system – the 1D string. The dispersions agree for small k up to a kmax=π/a. which, inserted into Eq. (2.88), yields the dispersion relation

ω(k) = c · k, (2.90) i.e., the displacement of the string performs a wave motion with linear dispersion – we observe an acoustic wave where the wave number k is continuous. It is now interesting to compare this result with the behavior of the original discrete N−particle system. There the oscillation frequencies ωn were given 21 by Eq. (2.52), and the wave numbers are discrete kn = 2πn/Na with n = ±1, · · · ± N/2, and the maximum wave number is kmax = π/a. Obviously, the discrete system does not have a linear dispersion, but we may consider the small k limit and expand the sin to ﬁrst order:

κ πn2 c2 ak 2 ω2 ≈ 4 = 4 n = ck , (2.91) n m N a2 2 n i.e. for small k the discrete system has exactly the same dispersion as the continuous system. The comparison with the discrete system also gives a hint at the existence of an upper limit for the wave number in the continuous system. In fact, k cannot be larger than π/amin where amin is the minimal distance of neighboring particles in the “continuous medium”. The two dispersions are shown in Fig. 2.5. One may, of course, ask whether a continuum model has its own right of existence, without being a limit of a discrete system. In other words, this would correspond to a system with an infinite particle number and, correspondingly, an infinite number M of normal modes. While we have not yet discussed how to quantize continuum systems it is immediately clear that there should be problems if the number of modes is unlimited. In fact, the total energy contains a zero point contribution for each mode which, with M going to infinity, will diverge. This problem does not occur for any realistic system because the particle number is always finite (though, possibly large). But a pure continuum model will be only physically relevant if such divergencies are avoided. The solution is found by co-called “renormalization” procedures where a maximum k-value (a cut-off) is introduced. This maybe not easy to derive for any specific field theory, however, based on the information from discrete systems,

21 The wave number follows from the mode numbers ln by dividing by a 2.6. PROBLEMS TO CHAPTER 2 85 such a cut-oﬀ can always be motivated by choosing a physically relevant particle number, as we have seen in this chapter.

Thus we have succeeded to perform the continuum limit of the 1d chain – the 1d string and derive and solve its equation of motion. The solution is a continuum of acoustic waves which are the normal modes of the medium which replace the discrete normal modes of the linear chain. Now the question remains how to perform a quantization of the continuous system, how to introduce creation and annihilation operators. To this end we have to develop a more general formalism which is called canonical quatization and which will be discussed in the next chapter.

2.6 Problems to chapter 2

1. Calculate the explicit form of the ground state wave function by using Eq. (2.37). √ 2. Show that the matrix elements of a† are given by n + 1|a†|n = n + 1, where n = 0, 1,... , and are zero otherwise. √ 3. Show that the matrix elements of a are given by hn − 1|a|ni = n, where n = 0, 1,... , and are zero otherwise.

4. Proof relation (2.40).

5. Proof the orthogonality relation (2.55).

6. The commutation relation (2.63) which was derived to satisfy the commutation relations of coordinates and momenta is that of bosons. This result was independent of whether the particles in the chain are fermions or bosons. Discuss this seeming contradiction. ˆ ˆ 7. Proof the commutation relation [Qn, Pm] = i~δn,m. 8. Apply the concept of the eigenvalue problem of the Hesse matrix to the solution of the normal modes of the 1d chain. Rederive the normal mode representation of the hamiltonian and check if the time dependencies vanish. 86 CHAPTER 2. SECOND QUANTIZATION Chapter 3

Fermions and bosons

We now turn to the quantum statistical description of many-particle systems. The indistinguishability of microparticles leads to a number of far-reaching consequences for the behavior of particle ensembles. Among them are the symmetry properties of the wave function. As we will see there exist only two diﬀerent symmetries leading to either Bose or Fermi-Dirac statistics. Consider a single nonrelativistic quantum particle described by the hamiltonian hˆ. The stationary eigenvalue problem is given by the Schr¨odingerequation ˆ h|φii = i|φii, i = 1, 2,... (3.1) where the eigenvalues of the hamiltonian are ordered, 1 < 2 < 3 ... . The associated single-particle orbitals φi form a complete orthonormal set of states in the single-particle Hilbert space1

hφi|φji = δi,j, ∞ X |φiihφi| = 1. (3.2) i=1

3.1 Spin statistics theorem

We now consider the quantum mechanical state |Ψi of N identical particles which is characterized by a set of N quantum numbers j1, j2, ..., jN , meaning that particle i is in single-particle state |φji i. The states |Ψi are elements of the N-particle Hilbert space which we deﬁne as the direct product of single-particle

1The eigenvalues are assumed non-degenerate. Also, the extension to the case of a continuous basis is straightforward.

87 88 CHAPTER 3. FERMIONS AND BOSONS

Figure 3.1: Example of the occupation of single-particle orbitals by 3 particles. Exchange of identical particles (right) cannot change the measurable physical properties, such as the occupation probability.

Hilbert spaces, HN = H1 ⊗ H1 ⊗ H1 ⊗ ... (N factors), and are eigenstates of the total hamiltonian Hˆ , ˆ H|Ψ{j}i = E{j}|Ψ{j}i, {j} = {j1, j2,... } (3.3) The explicit structure of the N−particle states is not important now and will be discussed later2. Since the particles are assumed indistinguishable it is clear that all physical observables cannot depend upon which of the particles occupies which single particle state, as long as all occupied orbitals, i.e. the set j, remain unchainged. In other words, exchanging two particles k and l (exchanging their orbitals, jk ↔ jl) in the state |Ψi may not change the probability density, cf. Fig. 3.1. The mathematical formulation of this statement is based on the permutation operator Pkl with the action

Pkl|Ψ{j}i = Pkl|Ψj1,...,jk,...jl,...,jN i = 0 = |Ψj1,...,jl,...jk,...,jN i ≡ |Ψ{j}i, ∀k, l = 1,...N, (3.4) where we have to require

0 0 hΨ{j}|Ψ{j}i = hΨ{j}|Ψ{j}i. (3.5) ˆ ˆ ˆ Indistinguishability of particles requires PklH = H and [Pkl, H] = 0, i.e. Pkl ˆ and H have common eigenstates. This means Pkl obeys the eigenvalue problem

0 Pkl|Ψ{j}i = λkl|Ψ{j}i = |Ψ{j}i. (3.6)

2In this section we assume that the particles do not interact with each other. The generalization to interacting particles will be discussed in Sec. 3.2.5. 3.2. N-PARTICLE WAVE FUNCTIONS 89

† Obviously, Pkl = Pkl, so the eigenvalue λkl is real. Then, from Eqs. (3.5) and (3.6) immediately follows

2 2 λkl = λ = 1, ∀k, l = 1,...N, (3.7) with the two possible solutions: λ = 1 and λ = −1. From Eq. (3.6) it follows that, for λ = 1, the wave function |Ψi is symmetric under particle exchange whereas, for λ = −1, it changes sign (i.e., it is “anti-symmetric”). This result was obtained for an arbitrary pair of particles and we may expect that it is straightforwardly extended to systems with more than two particles. Experience shows that, in nature, there exist only two classes of microparticles – one which has a totally symmetric wave function with respect to exchange of any particle pair whereas, for the other, the wave function is antisymmetric. The first case describes particles with Bose-Einstein statistics (“bosons”) and the second, particles obeying Fermi-Dirac statistics (“fermions”)3. The one-to-one correspondence of (anti-)symmetric states with bosons (fermions) is the content of the spin-statistics theorem. It was first proven by Fierz [Fie39] and Pauli [Pau40] within relativistic quantum field theory. Require- ments include 1.) Lorentz invariance and relativistic causality, 2.) positivity of the energies of all particles and 3.) positive definiteness of the norm of all states.

3.2 Symmetric and antisymmetric N-particle wave functions

We now explicitly construct the N-particle wave function of a system of many fermions or bosons. For two particles occupying the orbitals |φj1 i and |φj2 i, respectively, there are two possible wave functions: |Ψj1,j2 i and |Ψj2,j1 i which follow from one another by applying the permutation operator P12. Since both wave functions represent the same physical state it is reasonable to eliminate this ambiguity by constructing a new wave function as a suitable linear combination of the two,

± |Ψj1,j2 i = C12 {|Ψj1,j2 i + A12P12|Ψj1,j2 i} , (3.8) with an arbitrary complex coeﬃcient A12. Using the eigenvalue property of the permutation operator, Eq. (3.6), we require that this wave function has

3Fictitious systems with mixed statistics have been investigated by various authors, e.g. [MG64, MG65] and obey “parastatistics”. For a text book discussion, see Ref. [Sch], p. 6. 90 CHAPTER 3. FERMIONS AND BOSONS the proper symmetry,

± ± P12|Ψj1,j2 i = ±|Ψj1,j2 i . (3.9) The explicit form of the coeﬃcients in Eq. (3.8) is obtained by acting on ± this equation with the permutation operator and equating this to ±|Ψj1,j2 i , 2 ˆ according to Eq. (3.9), and using P12 = 1,

± 2 P12|Ψj1,j2 i = C12 |Ψj2,j1 i + A12P12|Ψj1,j2 i =

= C12 {±A12|Ψj2,j1 i ± |Ψj1,j2 i} , which leads to the requirement A = λ, whereas normalization of |Ψ i± √ 12 j1,j2 yields C12 = 1/ 2. The ﬁnal result is

± 1 ± |Ψj ,j i = √ {|Ψj ,j i ± P12|Ψj ,j i} ≡ Λ |Ψj ,j i (3.10) 1 2 2 1 2 1 2 12 1 2 where, ± 1 Λ = √ {1 ± P12}, (3.11) 12 2 denotes the (anti-)symmetrization operator of two particles which is a linear combination of the identity operator and the pair permutation operator. The extension of this result to 3 fermions or bosons is straightforward. For 3 particles (1, 2, 3) there exist 6 = 3! permutations: three pair permutations, (2, 1, 3), (3, 2, 1), (1, 3, 2), that are obtained by acting with the permuation operators P12,P13,P23, respectively on the initial conﬁguration. Further, there are two permutations involving all three particles, i.e. (3, 1, 2), (2, 3, 1), which are obtained by applying the operators P13P12 and P23P12, respectively. Thus, the three-particle (anti-)symmetrization operator has the form

± 1 Λ = √ {1 ± P12 ± P13 ± P23 + P13P12 + P23P12}, (3.12) 123 3! where we took into account the necessary sign change in the case of fermions resulting for any pair permutation. This result is generalized to N particles where there exists a total of N! permutations, according to4

± ± |Ψ{j}i = Λ1...N |Ψ{j}i, (3.13)

4This result applies only to fermions. For bosons the prefactor has to be corrected, cf. Eq. (3.25). 3.2. N-PARTICLE WAVE FUNCTIONS 91 with the deﬁnition of the (anti-)symmetrization operator of N particles,

1 X Λ± = √ sign(P )Pˆ (3.14) 1...N N! P SN where the sum is over all possible permutations Pˆ which are elements of the Np permutation group SN . Each permutation P has the parity, sign(P ) = (±1) , ˆ which is equal to the number Np of successive pair permuations into which P can be decomposed (cf. the example N = 3 above). Below we will construct ± the (anti-)symmetric state |Ψ{j}i explicitly. But before this we consider an alternative and very eﬃcient notation which is based on the occupation number formalism. ± The properties of the (anti-)symmetrization operators Λ1...N are analyzed in Problem 1, see Sec. 3.7

3.2.1 Occupation number representation

The original N-particle state |Ψ{j}i contained clear information about which particle occupies which state. Of course this is unphysical, as it is in conﬂict with the indistinguishability of particles. With the construction of the sym- ± metric or anti-symmetric N-particle state |Ψ{j}i this information about the identity of particles is eliminated, an the only information which remained is how many particles np occupy single-particle orbital |φpi. We thus may use a ± diﬀerent notation for the state |Ψ{j}i in terms of the occupation numbers np of the single-particle orbitals,

± |Ψ{j}i = |n1n2 ... i ≡ |{n}i, np = 0, 1, 2, . . . , p = 1, 2,... (3.15)

Here {n} denotes the total set of occupation numbers of all single-particle orbitals. Since this is the complete information about the N-particle system these states form a complete system which is orthonormal by construction of the (anti-)symmetrization operators,

0 h{n}|{n }i = δ 0 = δ 0 δ 0 ... {n},{n } n1,n1 n2,n2 X |{n}ih{n}| = 1. (3.16) {n}

The nice feature of this representation is that it is equally applicable to fermions and bosons. The only diﬀerence between the two lies in the possible values of the occupation numbers, as we will see in the next two sections. 92 CHAPTER 3. FERMIONS AND BOSONS

3.2.2 Fock space

In Sec. 3.1 we have introduced the N-particle Hilbert space HN . In the following we will need either totally symmetric or totally anti-symmetric states + − which form the sub-spaces HN and HN of the Hilbert space. Furthermore, below we will develop the formalism of second quantization by defining creation and annihilation operators acting on symmetric or anti-symmetric states. Obviously, the action of these operators will give rise to a state with N + 1 or N − 1 particles. Thus, we have to introduce, in addition, a more general space containing states with different particle numbers: We define the symmetric (anti-symmetric) Fock space F ± as the direct sum of symmetric ± (anti-symmetric) Hilbert spaces HN with particle numbers N = 0, 1, 2,... ,

+ + + F = H0 ∪ H1 ∪ H2 ∪ ..., − − − F = H0 ∪ H1 ∪ H2 ∪ .... (3.17)

Here, we included the vacuum state |0i = |0, 0,... i which is the state without particles which belongs to both Fock spaces.

3.2.3 Many-fermion wave function Let us return to the case of two particles, Eq. (3.10), and consider the case j1 = j2. Due to the minus sign in front of P12, we immediately conclude that − |Ψj1,j1 i ≡ 0. This state is not normalizable and thus cannot be physically realized. In other words, two fermions cannot occupy the same single-particle orbital – this is the Pauli principle which has far-reaching consequences for the behavior of fermions. We now construct the explicit form of the anti-symmetric wave function. This is particularly simple if the particles are non-interacting. Then, the total hamiltonian is additive5, N ˆ X ˆ H = hi, (3.18) i=1 ˆ ˆ and all hamiltonians commute, [hi, hj] = 0, for all i and j. Then all particles have common eigenstates, and the total wave function (prior to anti- symmetrization) has the form of a product

|Ψ{j}i = |Ψj1,j2,...jN i = |φj1 (1)i|φj2 (2)i ... |φjN (N)i

5This is an example of an observable of single-particle type which will be discussed more in detail in Sec. 3.3.1. 3.2. N-PARTICLE WAVE FUNCTIONS 93 where the argument of the orbitals denotes the number (index) of the particle that occupies this orbital. As we have just seen, for fermions, all orbitals have to be diﬀerent. Now, the anti-symmetrization of this state can be performed − immediately, by applying the operator Λ1...N given by Eq. (3.14). For two particles, we obtain

− 1 |Ψj ,j i = √ {|φj (1)i|φj (2)i − |φj (2)i|φj (1)i} = 1 2 2! 1 2 1 2 = |0, 0,..., 1,..., 1,... i. (3.19) In the last line, we used the occupation number representation, which has everywhere zeroes (unoccupied orbitals) except for the two orbitals with numbers j1 and j2. Obviously, the combination of orbitals in the ﬁrst line can be written as a determinant which allows for a compact notation of the general wave function of N fermions as a Slater determinant,

|φj1 (1)i |φj1 (2)i ... |φj1 (N)i

− 1 |φj2 (1)i |φj2 (2)i ... |φj2 (N)i |Ψj1,j2,...j i = √ = N N! ......

...... = |0, 0,..., 1,..., 1,..., 1,..., 1,... i. (3.20) In the last line, the 1’s are at the positions of the occupied orbitals. This becomes obvious if the system is in the ground state, then the N energetically lowest orbitals are occupied, j1 = 1, j2 = 2, . . . jN = N, and the state has the simple notation |1, 1,... 1, 0, 0 ... i with N subsequent 1’s. Obviously, the anti-symmetric wave function is normalized to one. As discussed in Sec. 3.2.1, the (anti-)symmetric states form an orthonormal basis in Fock space. For fermions, the restriction of the occupation numbers leads to a slight modiﬁcation of the completeness relation which we, therefore, repeat: 0 h{n}|{n }i = δ 0 δ 0 ..., n1,n1 n2,n2 1 1 X X ... |{n}ih{n}| = 1. (3.21)

n1=0 n2=0

3.2.4 Many-boson wave function The case of bosons is analyzed analogously. Considering again the two-particle case

+ 1 |Ψj ,j i = √ {|φj (1)i|φj (2)i + |φj (2)i|φj (1)i} = 1 2 2! 1 2 1 2 = |0, 0,..., 1,..., 1,... i, (3.22) 94 CHAPTER 3. FERMIONS AND BOSONS the main diﬀerence to the fermions is the plus sign. Thus, this wave function is not represented by a determinant, but this combination of products with positive sign is called a permanent. The plus sign in the wave function (3.22) has the immediate consequence that the situation j1 = j2 now leads to a physical state, i.e., for bosons, there is no restriction on the occupation numbers, except for their normalization

∞ X np = N, np = 0, 1, 2,...N, ∀p. (3.23) p=1

Thus, the two single-particle orbitals |φj1 i and |φj2 i occuring in Eq. (3.22) can accomodate an arbitrary number of bosons. If, for example, the two particles are both in the state |φji, the symmetric wave function becomes

+ |Ψj,ji = |0, 0,..., 2,... i = 1 = C(nj)√ |φj(1)i|φj(2)i + |φj(2)i|φj(1)i , (3.24) 2! where the coeﬃcient C(nj) is introduced to assure the normalization condition + + hΨj,j|Ψj,ji = 1. Since the two terms in (3.24) are identical√ the normalization 2 gives 1 = 4|C(nj)| /2, i.e. we obtain C(nj = 2) = 1/ 2. Repeating this analysis for a state with an arbitrary occupation number n , there will be n ! j √ j identical terms, and we obtain the general result C(nj) = 1/ nj. Finally, if P∞ there are several states with occupation numbers n1, n2,... with p=1 np = N, −1/2 the normalization constant becomes C(n1, n2, ...) = (n1! n2! ... ) . Thus, for + the case of bosons action of the symmetrization operator Λ1...N , Eq. (3.14), on the state |Ψj1,j2,...jN i will not yield a normalized state. A normalized symmetric state is obtained by the following prescription,

+ 1 + |Ψj1,j2,...jN i = √ Λ1...N |Ψj1,j2,...jN i (3.25) n1!n2!...

1 X Λ+ = √ P.ˆ (3.26) 1...N N! P SN

−1/2 Hence the symmetric state contains the prefactor (N!n1!n2!...) in front of the permanent. An example of the wave function of N bosons is

k + X |Ψj1,j2,...jN i = |n1n2 . . . nk, 0, 0,... i, np = N, (3.27) p=1 3.2. N-PARTICLE WAVE FUNCTIONS 95

where np 6= 0, for all p ≤ k, whereas all orbitals with the number p > k are empty. In particular, the energetically lowest state of N non-interacting bosons (ground state) is the state where all particles occupy the lowest orbital |φ1i, + i.e. |Ψj1,j2,...jN iGS = |N0 ... 0i. This eﬀect of a macroscopic population which is possible only for particles with Bose statistics is called Bose-Einstein con- densation. Note, however, that in the case of interaction between the particles, a permanent constructed from the free single-particle orbitals will not be an eigenstate of the system. In that case, in a Bose condensate a ﬁnite fraction of particles will occupy excited orbitals (“condensate depletion”). The construction of the N-particle state for interacting bosons and fermions is subject of the next section.

3.2.5 Interacting bosons and fermions So far we have assumed that there is no interaction between the particles and the total hamiltonian is a sum of single-particle hamiltonians. In the case of interactions, N ˆ X ˆ ˆ H = hi + Hint, (3.28) i=1 and the N-particle wave function will, in general, deviate from a product of single-particle orbitals. Moreover, there is no reason why interacting particles should occupy single-particle orbitals |φpi of a non-interacting system. The solution to this problem is based on the fact that the (anti-)symmetric ± states, |Ψ{j}i = |{n}i, form a complete orthonormal set in the N-particle Hilbert space, cf. Eq. (3.16). This means, any symmetric or antisymmetric state can be represented as a superposition of N-particle permanents or determinants, respectively,

± X ± |Ψ{j}i = C{n}|{n}i (3.29) {n},N=const

The eﬀect of the interaction between the particles on the ground state wave function is to “add” contributions from determinants (permanents) involving higher lying orbitals to the ideal wave function, i.e. the interacting ground state includes contributions from (non-interacting) excited states. For weak interaction, we may expect that energetically low-lying orbitals will give the dominating contribution to the wave function. For example, for two fermions, the dominating states in the expansion (3.29) will be |1, 1, 0,... i, |1, 0, 1,... i, |1, 0, 0, 1 ... i, |0, 1, 1,, 0 ... i and so on. The computation of the ground state of an interacting many-particle system is thus transformed into the computation 96 CHAPTER 3. FERMIONS AND BOSONS

± of the expansion coeﬃcients C{n}. This is the basis of the exact diagonalization method or conﬁguration interaction (CI). It is obvious that, if we would have obtained the eigenfunctions of the interacting hamiltonian, it would be represented by a diagonal matrix in this basis whith the eigenvalues populating the diagonal.6

This approach of computing the N-particle state via a superposition of permanents or determinants can be extended beyond the ground state properties. Indeed, extensions to thermodynamic equilibrium (mixed ensemble where the superpositions carry weights proportional to Boltzmann factors, e.g. [SBF+11]) and also nonequilibrium versions of CI (time-dependent CI, TDCI) that use pure states are meanwhile well established. In the latter, ± the coeﬃcients become time-dependent, C{n}(t), whereas the orbitals remain ﬁxed. We will consider the extension of the occupation number formalism to the thermodynamic properties of interacting bosons and fermions in Chapter 4. Further, nonequilibrium many-particle systems will be considered in Chapter 7 where we will develop an alternative approach based on nonequilibrium Green functions.

The main problem of CI-type methods is the exponential scaling with the number of particles which dramatically limits the class of solvable problems. Therefore, in recent years a large variety of approximate methods has been developed. Here we mention multiconfiguration (MC) approaches such as MC Hartree or MC Hartree-Fock which exist also in time-dependent variants (MCTDH and MCTDHF), e.g. [MMC90] and are now frequently applied to interacting Bose and Fermi systems. In this method not only the coefficients C±(t) are optimized but also the orbitals are adapted in a time-dependent fashion. The main advantage is the reduction of the basis size, as sompared to CI. A recent time-dependent application to the photoionization of helium can be found in Ref. [HB11]. Another very general approach consists in subdivid- ing the N-particle state in various classes with different properties. This has been termed “Generalized Active Space” (or restricted active space) approach and is very promising due to its generality [HB12, HB13]. An overview on first results is given in Ref. [HHB14].

6This N-particle state can be constructed from interacting single-particle orbitals as well. These are called “natural orbitals” and are the eigenvalues of the reduced one-particle density matrix. For a discussion see [SvL13]. 3.3. SECOND QUANTIZATION FOR BOSONS 97 3.3 Second quantization for bosons

We have seen in Chapter 1 for the example of the harmonic oscillator that an elegant approach to quantum many-particle systems is given by the method of second quantization. Using properly defined creation and annihilation operators, the hamiltonian of various N-particle systems was diagonalized. The examples studied in Chapter 1 did not explicitly include an interaction contribution to the hamiltonian – a simplification which will now be dropped. We will now consider the full hamiltonian (3.28) and transform it into second quantization representation. While this hamiltonian will, in general, not be diagonal, nevertheless the use of creation and annihilation operators yields a quite efficient approach to the many-particle problem.

3.3.1 Creation and annihilation operators for bosons † We now introduce the creation operatora ˆi acting on states from the symmetric Fock space F +, cf. Sec. 3.2.2. It has the property to increase the occupation number ni of single-particle orbital |φii by one. In analogy to the harmonic oscillator, Sec. 2.3 we use the following deﬁnition

† √ aˆi |n1n2 . . . ni ... i = ni + 1 |n1n2 . . . ni + 1 ... i (3.30)

While in case of coupled harmonic oscillators this operator created an additional excitation in oscillator “i”, now its action leads to a state with an additional particle in orbital “i”. The associated annihilation operatora î of orbital |φii is now constructed as the hermitean adjoint (we use this as its def- † † † inition) ofa î , i.e. [âi ] =a î, and its action can be deduced from the definition (3.30),

X 0 0 aî|n1n2 . . . ni ... i = |{n }ih{n }|aî|n1n2 . . . ni ... i {n0} X 0 † 0 0 ∗ = |{n }ihn1n2 . . . ni ... |aî |n1 . . . ni ... i = {n0} X p 0 i 0 = n + 1 δ 0 δ 0 |{n }i = i {n},{n } ni,ni+1 {n0} √ = ni |n1n2 . . . ni − 1 ... i, (3.31) yielding the same explicit definition that is familiar from the harmonic os- 7 † cillator : the adjoint ofa î is indeed an annihilation operator reducing the 7See our results for coupled harmonic oscillators in section 2.3.2. 98 CHAPTER 3. FERMIONS AND BOSONS

occupation of orbital |φii by one. In the third line of Eq. (3.31) we introduced the modiﬁed Kronecker symbol in which the occupation number of orbital i is missing,

i δ 0 = δ 0 . . . δ 0 δ 0 .... (3.32) {n},{n } n1,n1 ni−1,ni−1 ni+1,ni+1 ik δ 0 = δ 0 . . . δ 0 δ 0 . . . .δ 0 δ 0 .... (3.33) {n},{n } n1,n1 ni−1,ni−1 ni+1,ni+1 nk−1,nk−1 nk+1,nk+1

In the second line, this deﬁnition is extended to two missing orbitals. We now need to verify the proper bosonic commutation relations, which are given by the Theorem: The creation and annihilation operators deﬁned by Eqs. (3.30, 3.31) obey the relations

† † [âi, aˆk] = [âi , aˆk] = 0, ∀i, k, (3.34) h † i aî, aˆk = δi,k. (3.35)

Proof of relation (3.35): Consider ﬁrst the case i 6= k and evaluate the commutator acting on an arbitrary state

h † i √ aˆi, aˆk |{n}i =a ˆi nk + 1| . . . ni, . . . nk + 1 ... i † √ − aˆk ni| . . . ni − 1, . . . nk ... i = 0

Consider now the case i = k: Then

h † i aˆk, aˆk |{n}i = (nk + 1)|{n}i − nk|{n}i = |{n}i, which proves the statement since no restrictions with respect to i and k were made. Analogously one proves the relations (3.34), see problem 18. We now consider the second quantization representation of important operators.

Construction of the N-particle state As for the harmonic oscillator or any quantized ﬁeld, an arbitrary many- particle state can be constructed from the vacuum state by repeatedly applying

8From this property we may also conclude that the ladder operators of the harmonic oscillator have bosonic nature, i.e. the elementary exciations of the oscillator (oscillation quanta or phonons) are bosons. 3.3. SECOND QUANTIZATION FOR BOSONS 99 the creation operator(s). For example, single and two-particle states with the proper normalization are obtained via

|1i =a ˆ†|0i, † |0, 0 ... 1, 0,... i =a ˆi |0i, 1 2 |0, 0 ... 2, 0,... i = √ aˆ† |0i, 2! i † † |0, 0 ... 1, 0,... 1, 0,... i =a ˆi aˆj|0i, i 6= j, where, in the second (third) line, the 1 (2) stands on position i, whereas in the last line the 1’s are at positions i and j. This is readily generalized to an arbitrary symmetric N-particle state according to9.

n n 1 † 1 † 2 |n1, n2,... i = √ aˆ1 aˆ2 ... |0i (3.36) n1!n2! ...

Particle number operators The operator † nî =a î aî (3.37) is the occupation number operator for orbital i because, for ni ≥ 1,

† †√ aî aî|{n}i =a î ni|n1 . . . ni − 1 ... i = ni|{n}i,

† whereas, for ni = 0,a î aî|{n}i = 0. Thus, the symmetric state |{n}i is an eigenstate ofn î with the eigenvalue coinciding with the occupation number ni of this state. In other words: alln î have common eigenfunctions with the hamiltonian and commute, [ˆni,H] = 0. The total particle number operator is defined as

∞ ∞ ˆ X X † N = nî = aî aî, (3.38) i=1 i=1 because its action yields the total number of particles in the system: Nˆ|{n}i = P∞ ˆ i=1 ni|{n}i = N|{n}i. Thus also N commutes with the hamiltonian and has the same eigenfunctions.

9The origin of the prefactors was discussed in Sec. 3.2.4 and is also analogous to the case of the harmonic oscillator Sec. 2.3. 100 CHAPTER 3. FERMIONS AND BOSONS

Single-particle operators Consider now a general single-particle operator deﬁned as

N ˆ X ˆ B1 = bα, (3.39) α=1 ˆ where bα acts only on the variables associated with particle with number “α”. We will now transform this operator into second quantization representation. To this end we deﬁne the matrix element with respect to the single-particle orbitals ˆ bij = hi|b|ji, (3.40) and the generalized projection operator10

N ˆ X Πij = |iiαhj|α, (3.41) α=1 where |iiα denotes the orbital i occupied by particle α.

Theorem: The second quantization representation of a single-particle operator is given by ∞ ∞ ˆ X ˆ X † B1 = bij Πij = bij aˆi aˆj (3.42) i,j=1 i,j=1 Proof: We ﬁrst expand ˆb, for an arbitrary particle α into a basis of single-particle orbitals |ii = |φii,

N ∞ ∞ ˆ X X X ˆ B1 = bij|iiαhj|α = bijΠij, (3.43) α=1 i,j=1 i,j=1

10For i = j this deﬁnition contains the standard projection operator on state |ii, i.e. |iihi|, whereas for i 6= j this operator projects onto a transition, i.e. transfers an arbitrary particle from state |ji to state |ii. 3.3. SECOND QUANTIZATION FOR BOSONS 101

ˆ We now express Πij in terms of creation and annihilation operators by an- ˆ alyzing its action on a symmetric state (3.25), taking into account that Πij + 11 commutes with the symmetrization operator Λ1...N , Eq. (3.26) , N 1 X Πˆ |{n}i = √ Λ+ |ii hj| · |j i|j i ... |j i. (3.44) ij n !n ! ... 1...N α α 1 2 N 1 2 α=1 The product state is constructed from all orbitals including the orbitals |ii and |ji. Among the N factors there are, in general, ni factors |ii and nj factors |ji. Let us consider two cases. 1), j 6= i: Since the single-particle orbitals form an orthonormal basis, hj|ji = 1, multiplication with hj|α reduces the number of occurences of orbital j in the product state by one, whereas multiplication with |iiα increases the number of orbitals i by one. The occurence of nj such orbitals in the product state gives rise to an overall factor of nj. Finally, the properly symmetrized state which follows from |{n}i by increasing ni by one and decreasing nj by one will be denoted by i {n}j = |n1, n2 . . . ni + 1 . . . nj − 1 ... i

1 + = p Λ1...N · |j1i|j2i ... |jN i. (3.45) n1! ... (ni + 1)! ... (nj − 1)! ... It contains the same particle number N as the state |{n}i but, due to the differ- + √ √ ent orbital occupations, the prefactor in front of Λ1...N differs by nj/ ni + 1 compared to the one in Eq. (3.44) which we, therefore, can rewrite as √ ˆ ni + 1 i Πij|{n}i = nj √ {n}j nj † =a î aˆj|{n}i. (3.46)

2), j = i: The same derivation now leads again to a number ni of factors, whereas the square roots in Eq. (3.46) compensate, and we obtain ˆ Πjj|{n}i = nj |{n}i † =a ˆjaˆj|{n}i. (3.47) Thus, the results (3.46) and (3.47) can be combined to the operator identity

N X † |iiαhj|α =a ˆi aˆj (3.48) α=1

11 From the definition (3.41) it is obvious that Πˆ ij is totally symmetric in all particle indices. 102 CHAPTER 3. FERMIONS AND BOSONS which, together with the definition (3.45), proves the theorem12. For the special case that the orbitals are eigenfunctions of an operator, ˆ bα|φii = bi|φii—such as the single-particle hamiltonian, the corresponding matrix is diagonal, bij = biδij, and the representation (3.42) simplifies to

∞ ∞ ˆ X † X B1 = bi aî aî = bi nî, (3.49) i=1 i=1

ˆ where bi are the eigenvalues of b. Equation (3.49) naturally generalizes the familiar spectral representation of quantum mechanical operators to the case of many-body systems with arbitrary variable particle number.

Two-particle operators A two-particle operator is of the form

N 1 X Bˆ = ˆb , (3.50) 2 2! α,β α6=β=1

ˆ where bα,β acts only on particles α and β. An example is the operator of ˆ the pair interaction, bα,β → w(|rα − rβ|). We introduce again matrix elements, now with respect to two-particle states composed as products of single-particle orbitals, which belong to the two-particle Hilbert space H2 = H1 ⊗ H1, ˆ bijkl = hij|b|kli, (3.51)

Theorem: The second quantization representation of a two-particle operator is given by ∞ 1 X Bˆ = b aˆ†aˆ†aˆ aˆ (3.52) 2 2! ijkl i j l k i,j,k,l=1

Proof: We expand ˆb for an arbitrary pair α, β into a basis of two-particle orbitals |iji = |φii|φji,

12See problem 2. 3.3. SECOND QUANTIZATION FOR BOSONS 103 leading to the following result for the total two-particle operator,

∞ N 1 X X Bˆ = b |ii |ji hk| hl| . (3.53) 2 2! ijkl α β α β i,j,k,l=1 α6=β=1 The second sum is readily transformed, using the property (3.48) of the sigle- particle states. We ﬁrst extend the summation over the particles to include α = β,

General many-particle operators The above results are directly extended to more general operators involving K particles out of N N 1 X Bˆ = ˆb , (3.54) K K! α1,...αK α16=α26=...αK =1 and which have the second quantization representation

∞ ˆ 1 X † † BK = bj ...j m ...m aˆ ... aˆ aˆm ....aˆm (3.55) K! 1 k 1 k j1 jk k 1 j1...jkm1...mk=1 where we used the general matrix elements with respect to k-particle product ˆ states, bj1...jkm1...mk = hj1 . . . jk|b|m1 . . . mki. Note again the inverse ordering of the annihilation operators. Obviously, the result (3.55) includes the previous examples of single and two-particle operators as special cases.

13Note that the order of the creation operators and of the annihilators, respectively, is arbitrary. In Eq. (3.52) we have chosen an ascending order of the creators (same order as the indices of the matrix element) and a descending order of the annihilators, since this leads to an expression which is the same for Bose and Fermi statistics, cf. Sec. 3.4.1 104 CHAPTER 3. FERMIONS AND BOSONS 3.4 Second quantization for fermions

We now turn to particles with half-integer spin, i.e. fermions, which are described by anti-symmetric wave functions and obey the Pauli principle, cf. Sec. 3.2.3.

3.4.1 Creation and annihilation operators for fermions As for bosons we wish to introduce creation and annihilation operators that should again allow to construct any many-body state out of the vacuum state according to [cf. Eq. (3.36)]

n n − † 1 † 2 |n1, n2,... i = Λ1...N |i1 . . . iN i = aˆ1 aˆ2 ... |0i. ni = 0, 1, (3.56)

Due to the Pauli principle we expect that there will be no additional prefactors resulting from multiple occupations of orbitals, as in the case of bosons14. So far we do not know how these operators look like explicitly. Their deﬁnition has to make sure that the N-particle states have the correct anti-symmetry and that application of any creator more than once will return zero. To solve this problem, consider the example of two fermions which can occupy the orbitals k or l. The two-particle state has the symmetry |kli = −|lki, upon particle exchange. The anti-symmetrized state is constructed of the product state of particle 1 in state k and particle 2 in state l and has the properties

− † † − † † Λ1...N |kli =a ˆl aˆk|0i = |11i = −Λ1...N |lki = −aˆkaˆl |0i, (3.57) i.e., it changes sign upon exchange of the particles (third equality). This indicates that the state depends on the order in which the orbitals are ﬁlled, i.e., on the order of action of the two creation operators. One possible choice is used in the above equation and immediately implies that

† † † † † † aˆkaˆl +a ˆl aˆk = [ˆak, aˆl ]+ = 0, ∀k, l, (3.58) where we have introduced the anti-commutator15. In the special case, k = l, 2 † we immediately obtain aˆk = 0, for an arbitrary state, in agreement with the Pauli principle. Calculating the hermitean adjoint of Eq. (3.58) we obtain that the anti-commutator of two annihilators vanishes as well,

[ˆak, aˆl]+ = 0, ∀k, l. (3.59)

14The prefactors are always equal to unity because 1! = 1 15This was introduced by P. Jordan and E. Wigner in 1927. 3.4. SECOND QUANTIZATION FOR FERMIONS 105

Figure 3.2: Illustration of the phase factor α in the fermionic creation and annihiliation operators. The single-particle orbitals are assumed to be in a deﬁnite order (e.g. with respect to the energy eigenvalues). The position of the particle that is added to (removed from) orbital φk is characterized by the number αk of particles occupying orbitals to the left, i.e. with energies smaller than k.

We expect that this property holds for any two orbitals k, l and for any N- particle state that involves these orbitals. Now we can introduce an explicit deﬁnition of the fermionic creation operator which has all these properties. The operator creating a fermion in orbital k of a general many-body state is deﬁned as16

† αk X aˆk| . . . , nk,... i = (1 − nk)(−1) | . . . , nk + 1,... i, αk = nl (3.60) l

X 0 0 aˆk| . . . , nk,... i = |{n }ih{n }|aˆk| . . . , nk,... i = {n0} X 0 † 0 ∗ = |{n }ih{n}|aˆk|{n }i {n0} X 0 α0 k 0 = (1 − n )(−1) k δ 0 δ 0 |{n }i k {n },{n} nk,nk+1 {n0}

αk = (2 − nk)(−1) | . . . , nk − 1,... i αk ≡ nk(−1) | . . . , nk − 1,... i

0 where, in the third line, we used deﬁnition (3.32). Also, αk = αk because the sum involves only occupation numbers that are not altered. Note that the factor 2 − nk = 1, for nk = 1. However, for nk = 0 the present result is not

16 There can be other conventions which diﬀer from ours by the choice of the exponent αk which, however, is irrelevant for physical observables. 106 CHAPTER 3. FERMIONS AND BOSONS correct, as it should return zero. To this end, in the last line we have added the factor nk that takes care of this case. At the same time this factor does not alter the result for nk = 1. Thus, the factor 2−nk can be skipped entirely, and we obtain the expression for the fermionic annihilation operator of a particle in orbital k

αk aˆk| . . . , nk,... i = nk(−1) | . . . , nk − 1,... i (3.61)

Using the deﬁnitions (3.60) and (3.61) one readily proves the anti-commutation relations given by the Theorem: The creation and annihilation operators deﬁned by Eqs. (3.60) and (3.61) obey the relations

† † [âi, aˆk]+ = [âi , aˆk]+ = 0, ∀i, k, (3.62) h † i aî, aˆk = δi,k. (3.63) + Proof of relation (3.62): Consider, the case of two annihilators and the action on an arbitrary anti- symmetric state

[âi, aˆk]+|{n}i = (âiaˆk +a ˆkaî) |{n}i, (3.64) and consider first case i = k. Inserting the definition (3.61), we obtain

2 (ˆak) |{n}i ∼ nkaˆk|n1 . . . nk − 1 ... i = 0, and thus the anti-commutator vanishes as well. Consider now the case17 i < k:

P nl aˆiaˆk|{n}i =a ˆink(−1) l

Now we compute the result of the action of the exchanged operator pair

P nl aˆkaˆi|{n}i =a ˆkni(−1) l

The only difference compared to the first result is in the additional −1 in the second exponent. It arises because, upon action ofa ˆk aftera î, the number of particles to the left of k is already reduced by one. Thus, the two expressions differ just by a minus sign, which proves vanishing of the anti-commutator.

17This covers the general case of i 6= k, since i and k are arbitrary. 3.4. SECOND QUANTIZATION FOR FERMIONS 107

The proof of relation (3.63) proceeds analogously and is subject of Problem 3, cf. Sec. ??. Thus we have proved all anti-commutation relations for the fermionic operators and confirmed that the definitions (3.60) and (3.61) obey all properties required for fermionic field operators. We can now proceed to use thes operators to bring arbitrary quantum-mechanical operators into second quantized form in terms of fermionic orbitals.

Particle number operators As in the case of bosons, the operator

† nî =a î aî (3.65) is the occupation number operator for orbital i because, for ni = 0, 1,

† † αi aî aî|{n}i =a î (−1) |n1 . . . ni − 1 ... i = ni[1 − (ni − 1)]|{n}i, where the prefactor equals ni, for ni = 1 and zero otherwise. Thus, the anti- symmetric state |{n}i is an eigenstate ofn î with the eigenvalue coinciding with 18 the occupation number ni of this state . The total particle number operator is defined as

∞ ∞ ˆ X X † N = nî = aî aî, (3.66) i=1 i=1

ˆ P∞ because its action yields the total particle number: N|{n}i = i=1 ni|{n}i = N|{n}i.

Single-particle operators Consider now again a single-particle operator

N ˆ X ˆ B1 = bα, (3.67) α=1 and let us ﬁnd its second quantization representation.

18This result, together with the anti-commutation relations for the operators a and a† proves the consistency of the deﬁnitions (3.60) and (3.61). 108 CHAPTER 3. FERMIONS AND BOSONS

Theorem: The second quantization representation of a single-particle operator is given by

∞ ˆ X † B1 = bij aˆi aˆj (3.68) i,j=1

Proof: As for bosons, cf. Eq. (3.43), we have

N ∞ ∞ ˆ X X X ˆ B1 = bij|iiαhj|α = bijΠij, (3.69) α=1 i,j=1 i,j=1

ˆ ˆ † where Πij was deﬁned by (3.41), and it remains to show that Πij =a ˆi aˆj, for ˆ fermions as well. To this end we consider action of Πij on an anti-symmetric ˆ state, taking into accont that Πij commutes with the anti-symmetrization op- − erator Λ1...N , Eq. (3.14),

N ˆ 1 X X Πij|{n}i = √ sign(π)|iiαhj|α · |j1iπ(1)|j2iπ(2) ... |jN iπ(N). (3.70) N! α=1 πSN

If the product state does not contain the orbital |ji expression (3.70) vanishes, due to the orthogonality of the orbitals. Otherwise, let jk = j. Then hj|jki = 1, and the orbital |jki will be replaced by |ii, unless the state |ii is already present, then we again obtain zero due to the Pauli principle, i.e.

ˆ i Πij|{n}i ∼ (1 − ni)nj {n}j , (3.71) where we used the notation (3.45). What remains is to ﬁgure out the sign change due to the removal of a particle from the i-th orbital and creation of one in the k-th orbital. To this end we ﬁrst “move” the (empty) orbital |ji P past all orbitals to the left occupied by αj = p

19 Note that, if i > j, the occupation numbers occuring in αi have changed by one compared to those in αj. 3.4. SECOND QUANTIZATION FOR FERMIONS 109 account the deﬁnitions (3.60) and (3.61) we obtain20

ˆ αi+αj i † Πij|{n}i = (−1) (1 − ni)nj {n}j =a ˆi aˆj|{n}i (3.72) which, together with the equation (3.69), proves the theorem. Thus, the second quantization representation of single-particle operators is the same for bosons and fermions.

Two-particle operators As for bosons, we now derive the second quantization representation of a two- ˆ particle operator B2. Theorem: The second quantization representation of a two-particle operator is given by ∞ 1 X Bˆ = b aˆ†aˆ†aˆ aˆ (3.73) 2 2! ijkl i j l k i,j,k,l=1 Proof: ˆ As for bosons, we expand B into a basis of two-particle orbitals |iji = |φii|φji,

∞ N 1 X X Bˆ = b |ii |ji hk| hl| , (3.74) 2 2! ijkl α β α β i,j,k,l=1 α6=β=1 and transform the second sum

20One readily veriﬁes that this result applies also to the case j = i. Then the prefactor is just [1 − (nj − 1)]nj = nj, and αi = αj, resulting in a plus sign ˆ † Πjj|{n}i = nj |{n}i =a ˆjaˆj|{n}i. 110 CHAPTER 3. FERMIONS AND BOSONS

General many-particle operators The above results are directly extended to a general K-particle operator, K ≤ N, which was deﬁned in Eq. (3.54). Its second quantization representation is found to be

∞ ˆ 1 X † † BK = bj ...j m ...m aˆ ... aˆ aˆm ....aˆm (3.75) K! 1 k 1 k j1 jk k 1 j1...jkm1...mk=1 where we used the general matrix elements with respect to k-particle product ˆ states, bj1...jkm1...mk = hj1 . . . jk|b|m1 . . . mki. Note again the inverse ordering of the annihilation operators which exactly agrees with the expression for a bosonic system. Obviously, the result (3.75) includes the previous examples of single and two-particle operators as special cases.

3.4.2 Matrix elements in Fock space We now further extend the analysis of the anti-symmetric Fock space. A convenient orthonormal basis for a system of N fermions are the anti-symmetric states |{n}i, cf. Eq. (3.21). Then operators are completely deﬁned by their action on these states and by their matrix elements. For fermions the occupation number representation can be cast into a simple spinor formulation which we consider next.

Spinor representation of single-particle states The fact that the fermionic occupation numbers have only two possible values is very similar to the two spin projections of spin 1/2 particles and allows for a very intuitive description in terms of spinors. Thus, an empty or singly occupied orbital can be written as a column

 1 |0i → − empty state, (3.76) 0 0 |1i → − occupied state, (3.77) 1 and, analogously for the “bra”-states,

Spinor representation of operators In the spinor representation each second quantization operator becomes a 2×2 matrix, A A Aˆ → 00 01 , (3.80) A10 A11 ˆ where Aαβ = hα|A|βi and α, β = 0, 1. The particle number operator has the following action 1 nˆ = 0, (3.81) 0 0 0 nˆ = 1 , (3.82) 1 1 and is, therefore, given by a diagonal matrix in this spinor representation with its eigenvalues on the diagonal,21 0 0 nˆ → hn|nˆ|n0i = n01ˆ = , (3.83) 0 1 and one readily conﬁrms that this is consistent with the action of the operator given by Eqs. (3.81) and (3.82).

Spinor representation of aˆ and aˆ† Using the deﬁnitions (3.60) and (3.61) we readily obtain the matrix elements of the creation and annihilation operator. We again consider the matrices with respect to single-particle states |φki and take into account that, for fermions, nk is either 0 or 1. As a result, we obtain D † 0 E 0 0 † nk aˆ n = = δn ,1δn ,n0 +1 ≡ A , (3.84) k k 1 0 k k k k 0 0 1 hnk |aˆk| n i = = δn ,0δn ,n0 −1 ≡ Ak, (3.85) k 0 0 k k k

† where the matrix ofa ˆk is the transposed of that ofa ˆk and we introduced the short-hand notation A for the matrix δ δ 0 in the space of single-particle k nk,0 nk,1 22 orbitals |φki . 21The first [second] row corresponds to the case hn| = h0| [hn| = h1|], whereas the first [second] column corresponds to |ni = |0i [|ni = |1i]. 22 † We summarize the main properties of the matrices Ak and Ak which are a consequence † of the properties ofa ˆk anda ˆk and can be verified by direct matrix multiplication: 112 CHAPTER 3. FERMIONS AND BOSONS

Matrix elements of aˆ and aˆ† in Fock space It is now easy to extend this to matrix elements with respect to anti-symmetric N-particle states. These matrices will have the same structure as (3.84) and (3.85), with respect to orbital k, and be diagonal with respect to all other orbitals. In addition, there will be a sign factor depending on the position of orbital k in the N-particle state, cf. deﬁnitions (3.60) and (3.61),

D E † 0 αk k † {n} aˆ {n } = (−1) δ 0 A (3.86) k {n},{n } k

0 where the original prefactor 1 − nk has been transformed into an additional Kronecker delta for nk. The matrix of the annihilation operator is

0 αk k h{n} |aˆk| {n }i = (−1) δ{n},{n0} Ak (3.87)

Matrix elements of one-particle operators in Fock space To compute the matrix elements of one-particle operators, Eq. (3.43), we need ˆ the matrix of the projector Πkl. Using the results (3.87) for the annihiliator and (3.86) for the creator successively we obtain, for the case k 6= l,

D E X D E {n} aˆ†aˆ {n0} = {n} aˆ† {n¯} h{n¯} |aˆ | {n0}i = l k l k {n¯} 0 α X α¯l k l = (−1) k (−1) δ δ 0 δ 0 δ δ δ n¯k,0 {n¯},{n } n¯k,nk−1 nl,1 {n¯},{n} n¯l+1,nl {n¯} 0 α kl X α¯l = (−1) k δ δ 0 (−1) δ δ δ 0 δ 0 δ nl,1 {n},{n } n¯k,0 n¯k,nk n¯l,nl n¯k,nk−1 n¯l+1,nl n,¯ n¯k

αk0l kl † X 0 X = (−1) δ{n},{n0}Al Ak, αk0l = nm + nm, (3.88) m

2 2 † 1. Ak = Ak = 0.

0 0 2. A† A = = n 1ˆ , – a diagonal matrix with the eigenvalues ofn ˆ on the k k 0 1 k k k diagonal, cf. Eq. (3.83). 1 0 3. A A† = = (1 − n )1ˆ = 1 − A† A , i.e. A† and A anti-commute. k k 0 0 k k k k k k

† † † 4. For diﬀerent single-particle spaces, k 6= l,[Ak, Al]+ = [Ak, Al ]+ = [Ak, Al]+ = 0. 3.4. SECOND QUANTIZATION FOR FERMIONS 113 which is a diagonal matrix in all orbitals except k and l whereas, with respect to orbital k, it has the structure of the matrix (3.86) and, for orbital l, the form of matrix (3.87). Note that the occupation numbers entering the exponent αk0l are restricted by the Kronecker symbols. For the case k = l we recover the matrix of the particle number operator which is completely diagonal23

D † 0 E 0 {n} aˆ aˆ {n } = h{n} |nˆ | {n }i = n δ 0 . (3.89) k k k k {n},{n } With the results (3.88) and (3.89) we readily obtain the matrix representation of a single-particle operator, deﬁned by Eq. (3.67),

∞ D E X D E {n} Bˆ {n0} = b {n} aˆ†aˆ {n0} (3.90) 1 lk l k k,l=1

diag First, for a diagonal operator B , blk = bkδkl, the result is simply

∞ N D diag 0 E X X {n} Bˆ {n } = δ 0 b n = δ 0 b . (3.91) 1 {n},{n } k k {n},{n } nk k=1 k=1 where, in the last equality, we have simpliﬁed the summation by including only the occupied orbitals which have the numbers n1, n2 . . . nN . For the general case of a non-diagonal operator it follows from (3.90)24

( N D 0 E X {n} Bˆ {n } = δ 0 b + 1 {n},{n } nknk k=1 N ) X + (−1)k+l+γkl b δnknl A† A , . (3.92) nlnk {n},{n0} nl nk k6=l=1 where γkl = 1, for k < l, and 0, otherwise. 23This result is contained in expression (3.88). Indeed, in the special case k = l we obtain P 0 kl k αk0l → m

24The non-diagonal matrix elements are transformed to summation over occupied orbitals as

∞ N X D † 0 E X † 0 blk {n} aˆ aˆk {n } = bn n {n} aˆ aˆn {n } , l l k nl k k6=l=1 k6=l=1 where it remains to carry out the action oft the two operators. Note that the sign of the result is diﬀerent for nl < nk and nl > nk. 114 CHAPTER 3. FERMIONS AND BOSONS

Matrix elements of two-particle operators in Fock space To compute the matrix elements of two-particle operators, Eq. (3.73), we need the matrix elements of four-operator products, which we transform, using the anti-commutation relations (3.63) according to

† † † † † aî aˆjaˆlaˆk = −aî aˆlaˆjaˆk + δjl aî aˆk. (3.93)

Next, transform the matrix element of the first term on the right, D E D ED E † † 0 X † † 0 {n} aî aˆlaˆjaˆk {n } = {n} aî aˆl {n¯} {n¯} aˆjaˆk {n } = {n¯}

X αi¯l il α¯jk0 jk = (−1) δ δ δ δ δ × (−1) δ 0 δ δ 0 δ δ 0 , {n},{n¯} ni,1 n¯i,0 nl,0 n¯l,1 {n¯},{n } n¯j ,1 nj ,0 n¯k,0 nk,1 {n¯}

P P 0 where α¯jk0 = p

D † † E iljk † † {n} aˆ aˆ aˆ aˆ {n0} = (−1)αilj0k0 δ A A A A (3.94) i l j k {n},{n0} i l j k X X X 0 X 0 with αilj0k0 = np + np + np + np. p

This is a general result which also contains the cases of equal index pairs. Then, proceeding as in footnote 23, we obtain the results for the special cases.

αj0k0 jk † i=l: (−1) δ{n},{n0}niAjAk

αil il † j=k: (−1) δ{n},{n0}njAi Al

† αik0 ik l=j: (−1) δ{n},{n0}(1 − nl)Ai Ak i=l, j=k: δ{n},{n0}ninj

αlj0 lj † αil il † k=i: (−1) δ{n},{n0}niAlAj + (−1) δ{n},{n0}δijAi Al

25We ﬁrst rewrite

X il jk X iljk δ δ 0 = δ 0 δn ,n¯ δn ,n¯ δ 0 δ 0 , {n},{n¯p} {n¯p},{n } {n},{n } j j k k n¯i,ni n¯l,nl {n¯p} n¯in¯ln¯j n¯k

Taking into account the other Kronecker deltas we can combine pairs and perform the P remaining four summations, δ 0 δ = δ 0 and so on. n¯i n¯i,ni n¯i,0 ni,0 3.4. SECOND QUANTIZATION FOR FERMIONS 115

3.4.3 Fock Matrix of the binary interaction Of particular importance is the occupation number matrix representation of the interaction potential. This is an example of a two-particle quantity the properties of which we discussed in section 3.4.2. But for this special case, we can make further progress26. Starting point is the pair interaction

N 1 X Vˆ = wˆ(α, β), (3.95) 2 α6=β=1 with the second quantization representation (3.73) 1 X Vˆ = w aˆ†aˆ†aˆ aˆ , (3.96) 2 ijkl i j l k ijkl where the matrix elements are deﬁned as Z 3 3 ∗ ∗ wijkl = d xd y φi (x)φj (y)w(x, y)φk(x)φl(y), (3.97) and have the following symmetries

wijkl = = wjilk, (3.98) ∗ wijkl = wklij, (3.99) where property (3.99) follows from the symmetry of the potential w(x, y) = w(y, x). This allows us to eliminate double counting of pairs from the sum in Eq. (3.96)27

∞ ∞ ˆ X X − † † V = wijklaˆjaˆi aˆkaˆl, (3.100) 1≤i

26M. Heimsoth contributed to this section. 27We summarize the main steps: First, using the anti-commutation relations of the annihilators and perfoming an index change, we transform (the contribution k = l vanishes),

X X X − wijklaˆlaˆk = (wijkl − wijlk)ˆalaˆk = wijklaˆlaˆk. kl k

X † † X † † (wijkl − wijlk)ˆai aˆjaˆlaˆk = (wijkl − wjikl − wijlk + wjilk)ˆai aˆjaˆlaˆk = ij,k

∞ ∞ ˆ 0 X X − † † 0 h{n}|V |{n }i = wijklh{n}|aˆjaˆi aˆkaˆl|{n }i. (3.102) 1≤i

Each of the two vectors contains N particles (the interaction does not change the particle number), i.e. exactly N occupied orbitals which are all diﬀerent. So the sums over i, j and k, l, in fact, run over two (possibly diﬀerent) sets of N 0 0 0 28 orbitals with the numbers (m1, m2 . . . mN ) and (m1, m2 . . . mN ), respectively,

h{n}|Vˆ |{n0}i → h{m}|Vˆ |{m0}i = N N X X − † † 0 = w 0 0 h{m}|aˆm aˆm aˆm0 aˆm0 |{m }i. (3.103) mimj mkml j i k l 1≤i

Using the deﬁnitions of the creation and annihilation operators, Eqs. (3.60), (3.61), and taking advantage of the operator order in (3.103)29, the operators can be evaluated, with the result

N N 0 X X i+j+k+l − 0 ˆ 0 0 h{m}|V |{m }i = (−1) w 0 0 h{m}|mi,mj |{m }im ,m , mimj mkml k l 1≤i

N − 2 particles, and similarly for h{m}|mi,mj . The scalar product of the two anti-symmetric N −2-particle states in (3.104) is non-zero only if the two states contain N − 2 identical orbitals. To simplify the analysis, in Eq. (3.104) we have moved the missing orbitals to positions one and two in the states thereby accumulating the total sign factor contained in this expression. Thus, the remaining orbitals are not only identical but they also have identical numbers, 0 0 i.e. m3 = m3, m4 = m4,... . 28by |{m}i = |{m}i(|{n}i) we will denote the subset of N occupied orbitals contained in the state |{n}i. For example, a three-particle state |{n}i = |1, 0, 0, 1, 1i transforms into |m1m2m3i where the mi point to the original orbitals with numbers m1 = 1, m2 = 4, m3 = 5. Note that the matrix h{n}|Vˆ |{n0}i is diagonal in all orbitals missing simultaneously in h{m}| and |{m0}i. 29Since i < j and k < l, the signs produced by the ﬁrst and second operators are independent of each other. 3.4. SECOND QUANTIZATION FOR FERMIONS 117

Finally, expression (3.104) will be only non-zero if the missing orbitals fall in one of three cases30:

1. The two states are identical, {n} ≡ {n0} and, consequently {m} ≡ {m0}. Then Eq. (3.104) yields

N 0 X − h{n}|Vˆ |{n }i = δ 0 w . (3.105) {n},{n } mimj mimj 1≤i

2. The two states are identical except for one orbital: the orbital mp with number p is present in state h{m}| but is missing in state |{m0}i which, instead, contains an orbital mr with number r missing in h{m}|. Then the scalar product of the two N −2 particle states is nonzero only if both these states are annihilated and Eq. (3.104) yields31

N−1 0 ˆ 0 mpmr † X p+r − h{n}|V |{n }i = δ 0 A Am0 (−1) · Θ(p, r, i) w 0 {n},{n } mp r mimpmimr 1≤i,i6=p,r (3.106) 0 Here Θ(p, r, i) = −1, if either mp < mi or mr < mi, otherwise Θ(p, r, i) = 0 r 1. This case describes single-particle excitations where |{n }i = |{n}pi.

3. The two states are identical except for two orbitals with the numbers mp 0 0 0 and mq in h{m}| and mr and ms in |{m }i, respectively. Without loss of 0 0 generality we can use mp < mq and mr < ms. Then Eq. (3.104) yields

0 0 ˆ 0 mpmqmrms † † p+q+r+s − h{n}|V |{n }i = δ 0 A Am0 A Am0 (−1) w 0 0 {n},{n } mp r mq s mpmqmrms (3.107) 0 rs This case describes two-particle excitations where |{n }i = |{n}pqi. These results are known as Slater-Condon rules and were obtained by those two authors in 1929 and 1930 [Sla29, Con30] and are of prime importance for wave function based many-body methods such as configuration interaction (CI) and Multiconfiguration Hartree-Fock (MCHF) and their time-dependent extensions, e.g. [HHB14]. Similarly this representation is used in configuration path integral Monte Carlo simulations of strongly correlated fermions, e.g. [SBF+11] and references therein.

30Thereby we return to the full vectors (including the empty orbitals) and restore the delta functions. 31To obtain the correct sign we move the orbitals p and r to the last place in the product in state h{n}| and in |{n0}i, respectively and count the number of transpositions (diﬀerence). 118 CHAPTER 3. FERMIONS AND BOSONS 3.5 Coordinate representation of second quantization operators. Field operators

So far we have considered the creation and annihilation operators in an arbitrary basis of single-particle states. The coordinate and momentum representations are of particular importance and will be considered in the following. As before, an advantage of the present second quantization approach is that these considerations are entirely analogous for fermions and bosons and can be performed at once for both, the only diﬀerence being the details of the commutation (anticommutation) rules of the respective creation and annihilation operators. Here we start with the coordinate representation whereas the momentum representation will be introduced below, in Sec. 3.6.

3.5.1 Deﬁnition of ﬁeld operators

We now introduce operators that create or annihilate a particle at a given space point rather than in given orbital φi(r). To this end we consider the superposition in terms of the functions φi(r) where the coeﬃcients are the creation and annihilation operators,

∞ ˆ X ψ(x) = φi(x)ˆai, (3.108) i=1 ∞ ˆ† X ∗ † ψ (x) = φi (x)ˆai . (3.109) i=1

Here x = (r, σ), i.e. φi(x) is an eigenstate of the operator ˆr, and the φi(x) form a complete orthonormal set. Obviously, these operators have the desired property to create (annihilate) a particle at space point r in spin state σ. From the (anti-)symmetrization properties of the operators a and a† we immediately obtain h i ψˆ(x), ψˆ(x0) = 0, (3.110) ∓ h i ψˆ†(x), ψˆ†(x0) = 0, (3.111) ∓ h i ψˆ(x), ψˆ†(x0) = δ(x − x0). (3.112) ∓ 3.5. FIELD OPERATORS 119

Figure 3.3: Illustration of the relation of the field operators to the second quantization operators defined on a general basis {φi(x)}. The field operator ψˆ†(x) creates a particle at space point x (in spin state |σi) to which all single- particle orbitals φi contribute. The orbitals are vertically shifted for clarity.

These relations are straightforwardly proven by direct insertion of the definitions (3.108) and (3.109). We demonstrate this for the last expression.

∞ h ˆ ˆ† 0 i X ∗ 0 h †i ψ(x), ψ (x ) = φi(x)φj (x ) aˆi, aˆj = ∓ ∓ i,j=1 ∞ X ∗ 0 0 0 = φi(x)φi (x ) = δ(x − x ) = δ(r − r )δσ,σ0 , i=1 where, in the last line, we used the representation of the delta function in terms of a complete set of functions.

3.5.2 Representation of operators We now transform operators into second quantization representation using the ﬁeld operators, taking advantage of the identical expressions for bosons and fermions.

Single-particle operators The general second-quantization representation was given by [cf. Secs. 3.3, 3.4] ∞ ˆ X ˆ † B1 = hi|b|jiai aj. (3.113) i,j=1 120 CHAPTER 3. FERMIONS AND BOSONS

We now transform the matrix element to coordinate representation: Z ˆ 0 ∗ ˆ 0 0 hi|b|ji = dxdx φi (x)hx|b|x iφj(x ), (3.114) and obtain for the operator, taking into account the deﬁnitions (3.108) and (3.109), ∞ Z ˆ X 0 † ∗ ˆ 0 0 B1 = dxdx ai φi (x)hx|b|x iφj(x )aj = i,j=1 Z = dxdx0 ψˆ†(x)hx|ˆb|x0iψˆ(x0). (3.115)

For the important case that the matrix is diagonal, hx|ˆb|x0i = ˆb(x)δ(x − x0), the ﬁnal expression simpliﬁes to Z ˆ ˆ† ˆ ˆ B1 = dx ψ (x)b(x)ψ(x) (3.116)

Consider a few important examples. The ﬁrst is again the density operator. In ﬁrst quantization the operator of the particle density for N particles follows from quantizing the classical result for point particles,

N X nˆ(x) = δ(x − xα), (3.117) α=1

32 and the expectation value in a certain N-particle state Ψ(x1, x2, . . . xN ) is

N X hnˆi(x) = h Ψ| δ(x − xα)|Ψ i α=1 Z = N d2d3 . . . dN|Ψ(1, 2,...N)|2 = n(r, σ), (3.118) which is the single-particle spin density of a (in general correlated) N-particle system. The second quantization representation of the density operator follows from our above result (3.116) by replacing ˆb → δ(x − x0), i.e. Z nˆ(x) = dx0ψˆ†(x0)δ(x − x0)ψˆ(x0) = ψˆ†(x)ψˆ(x), (3.119)

32This is the example of an (anti-)symmetrized pure state which is easily extended to mixed states. 3.5. FIELD OPERATORS 121 and the operator of the total density is the sum (integral) over all coordinate- spin states Z Z Nˆ = dx nˆ(x) = dx ψˆ†(x)ψˆ(x), (3.120) naturally extending the previous results for a discrete basis to continuous states. The second example is the kinetic energy operator which is also diagonal and has the second-quantized representation Z 1 Tˆ = dx ψˆ†(x) − ∇2 ψˆ(x). (3.121) 2 The third example is the second quantization representation of the single- particle potential v(r) given by Z Vˆ = dx ψˆ†(x)v(r)ψˆ(x). (3.122)

Two-particle operators In similar manner we obtain the ﬁeld operator representation of a general two-particle operator

∞ 1 X Bˆ = hij|ˆb|klia†a†a a . (3.123) 2 2 i j l k i,j,k,l=1 We now transform the matrix element to coordinate representation: Z ˆ ∗ ∗ ˆ hij|b|kli = dx1dx2dx3dx4φi (x1)φj (x2)hx1x2|b|x3x4iφl(x3)φk(x4), (3.124) and, assuming that the matrix is diagonal, ˆ ˆ hx1x2|b|x3x4i = b(x1, x2)δ(x1 − x3)δ(x2 − x4), we obtain, after inserting this result into (3.123),

∞ 1 X Z Bˆ = dx dx a†φ∗(x )a†φ∗(x )ˆb(x , x )φ (x )a φ (x )a . 2 2 1 2 i i 1 j j 2 1 2 l 1 l k 2 k i,j,k,l=1 Using again the defintion of the field operators the final result for a diagonal two-particle operator in coordinate representation is

1 Z Bˆ = dx dx ψˆ†(x )ψˆ†(x )ˆb(x , x )ψˆ(x )ψˆ(x ) (3.125) 2 2 1 2 1 2 1 2 2 1 122 CHAPTER 3. FERMIONS AND BOSONS

Note again the inverse ordering of the destruction operators which makes this result universally applicable to fermions and bosons. The most important example of this representation is the operator of the two-particle interaction, ˆ ˆ W , which is obtained by replacing b(x1, x2) → w(x1 − x2).

3.6 Momentum representation of second quantization operators

We now consider the momentum representation of the creation and annihilation operators. This is useful for translationally invariant systems such as the electron gas or the jellium model, since the eigenfunctions of the momentum operator, 1 hx|φ i = φ (x) = eik·rδ , x = (r, σ), (3.126) k,s k,s V1/2 s,σ are eigenfunctions of the translation operator. Here we use periodic boundary conditions to represent an infinite system by a finite box of length L and volume 3 V = L , so the wave numbers have discrete values kx = 2πnx/L, . . . kz = 2πnz/L with nx, ny, nz being integer numbers. The eigenfunctions (3.126) form a complete orthonormal set, where the orthonormality condition reads Z 1 3 i(k0−k)r X hφ |φ 0 0 i = d r e δ δ 0 = δ 0 δ 0 , (3.127) k,s k ,s V s,σ s ,σ k,k s,s V σ where we took into account that the integral over the finite volume V equals zero for k 6= k0 and V otherwise.

3.6.1 Creation and annihilation operators in momentum space The creation and annihilation operators on the Fock space of N-particle states constructed from the orbitals (3.126) are obtained by inverting the definition of the field operators (3.108) written with respect to the momentum-spin states (3.126) ˆ X ψ(x) = φk0,σ0 (x)ak0,σ0 . k0σ0 ∗ Multiplication by φk,σ(x) and integrating over x yields, with the help of condition (3.127), Z Z ∗ ˆ 1 3 −ik·r ˆ ak,σ = dx φk0σ0 ψ(x) = 1/2 d r e ψ(r, σ), (3.128) V V 3.6. MOMENTUM REPRESENTATION 123 and, similarly for the creation operator, Z † 1 3 ik·r ˆ† ak,σ = 1/2 d r e ψ (r, σ). (3.129) V V Relations (3.128) and (3.129) are nothing but the Fourier transforms of the field operators. These operators obey the same (anti-)commutation relations as the field operators, which is a consequence of the linear superpositions (3.128), (3.129), cf. the proof of Eq. (3.112).

3.6.2 Representation of operators We again construct the second quantization representation of the relevant operators, now in terms of creation and annihilation operators in momentum space.

Single-particle operators For a single-particle operator we have, according to our general result, Eq. (3.69), and denoting x = (r, s), x0 = (r0, s0),

ˆ X X † ˆ 0 0 B1 = akσhkσ|b|k σ i ak0σ0 kσ k0σ0 Z X X 0 † ˆ 0 0 0 0 = dx dx akσhkσ|xihx|b|x ihx |k σ i ak0σ0 kσ k0σ0 Z 1 X X 0 † −ikr 0 ik0r0 = dx dx a e hx|ˆb|x ie a 0 0 δ δ 0 0 , (3.130) V kσ k σ σ,s σ ,s kσ k0σ0 where, in the last line, we inserted complete sets of momentum eigenstates (3.126). If the momentum matrix elements of the operator ˆb are known, the ﬁrst line can be used directly. Otherwise, the matrix element is obtained from the the known coordinate space result in the last line. For an operator that commutes with the momentum operator and, thus, is given by a diagonal matrix one integration (and spin summation) can be performed. We demonstrate this for the example of the kinetic energy operator. 2 2 ˆ 0 ~ ∇ 0 Then hx|b|x i → − 2m δ(x − x ), and we obtain, using the property (3.127), Z 2 02 1 X X 3 † −ikr ~ k ik0r Tˆ = d r a e e a 0 V kσ 2m k σ kσ k0 V X 2k2 = ~ a† a . (3.131) 2m kσ kσ kσ 124 CHAPTER 3. FERMIONS AND BOSONS

In similar fashion we obtain for the single-particle potential, upon replacing hx|ˆb|x0i → v(r)δ(x − x0), Z X X † 1 3 −ikr ik0r Vˆ = a a 0 d r e v(r) e kσ k σ V kσ k0 V X X † = v˜k−k0 akσ ak0σ, (3.132) kσ k0 where we introduced the Fourier transform of the single-particle potential, −1 R 3 −iqr v˜q = V d r v(r)e . Finally, the operator of the single-particle density becomes, in momentum space by Fourier transformation, X X 1 Z nˆ = nˆ = d3r ψ† (r)ψ (r) e−iqr q qσ V σ σ σ σ V Z 1 X † 1 3 i(k−k0)r −iqr = a 0 a d r e e V k σ kσ V σkk0 V 1 X = a† a . (3.133) V k−q,σ kσ σk

This shows that the Fourier component of the density operator,n ˆq, describes a density ﬂuctuation corresponding to a transition of a particle from state |φkσi to state |φk−q,σi, for arbitrary k. With this result we may rewrite the single-particle potential (3.132) as ˆ X V = V v˜q nˆ−q. (3.134) q

Two-particle operators We now turn to two-particle operators. Rewriting the general result (3.73) for a spin-momentum basis, we obtain

1 X X † † 0 0 0 0 Bˆ = a a hk σ k σ |ˆb|k σ k σ i a 0 0 a 0 0 (3.135) 2 k1σ1 k2σ2 1 1 2 2 1 1 2 2 k2σ2 k1σ1 2! 0 0 0 0 k1σ1k2σ2 k1σ1k2σ2 We now apply this result to the interaction potential where the matrix element 0 0 0 0 in momentum representation was computed before, hk1σ1k2σ2|wˆ|k1σ1k2σ2i = 0 w˜(k − k )δ 0 0 δ 0 δ 0 , andw ˜ denotes the Fourier transform of the 1 1 k1+k2−k1−k2 σ1,σ1 σ2,σ2 pair interaction, and the interaction does not change the spin of the involved particles, see problem 6, Sec. 3.7. Inserting this into Eq. (3.135) and intro- 0 0 ducing the momentum transfer q = k1 − k1 = k2 − k2, we obtain ˆ 1 X X † † W = w˜(q)a a ak −q,σ ak +q,σ , (3.136) 2! k1σ1 k2σ2 2 2 1 1 k1k2q σ1σ2 3.6. MOMENTUM REPRESENTATION 125

In similar manner other two-particle quantities are computed. With this result we can write down the second quantization representation in spin- momentum space of a generic hamiltonian that contains kinetic energy, an external potential and a pair interaction contribution. From the expressions (3.131, 3.132, 3.136) we obtain

2 2 X ~ k † X † Hˆ = a a + v˜ 0 a a 0 2m kσ kσ k−k kσ k σ kσ kk0σ 1 X X † † + w˜(q)a a ak −q,σ ak +q,σ . (3.137) 2! k1σ1 k2σ2 2 2 1 1 k1k2q σ1σ2

This result is a central starting point for many investigations in condensed matter physics, quantum plasmas or nuclear matter. After investigating the basic properties of the method of second quantization we now turn to more advanced topics. One of them is the extension of the analysis to systems at a finite temperature, i.e. in a mixed ensemble. This will be the subject of Chapter 4. After this we turn to the time evolution of the field operators following an external perturbation. This will be studied in detail for the case of single-time observables, in Chapter 5. A second route to nonequilibrium dynamics is to use field operator products that depend on two times which leads to the theory of nonequilibrium Green functions which we discuss in Chapter 7.

Application to relativistic quantum systems

The momentum representation is conveniently extended to relativistic many- particle systems. In fact, since the Dirac equation of a free particle has plane wave solutions, we may use the same single-particle orbitals as in the nonrelativistic case. With this, the matrix elements of the single-particle potential and of the interaction potential remain unchanged (if magnetic corrections to the interaction are neglected). The only change is in the kinetic energy contribution, due to the relativistic modiﬁcation of the single-particle dispersion, 2 2 √ ~ k 2 2 2 2 4 k = 2m → ~ k c + m c , where m is the rest mass. In the ultra-relativistic limit, k = ~ck. Otherwise the hamiltonian (3.137) remains valid. Of course, this is true only as long as pair creation processes are negligible. Otherwise we would need√ to extend the description by introducing the negative 2 2 2 2 4 energy branch k− = − ~ k c + m c and the corresponding second set of plane wave states. In all matrix elements we would need to include a second index (+, −) referring to the energy band. 126 CHAPTER 3. FERMIONS AND BOSONS 3.7 Problems to Chapter 3

± ± 1. Express Λ123 via Λ12, cf. Eqs. (3.11) and (3.12).

± 2. Generalize the previous result to ﬁnd a decomposition of Λ1...N into lower order operators.

3. Prove the bosonic commutation relations (3.34).

4. Prove the anti-commuation relation (3.63) between fermionic creation and annhiliation operators.

5. Discuss what happened to the sum over α in the derivation of Eq. (3.48).

6. Compute the momentum matrix element of the pair interaction. Chapter 5

Dynamics of the creation and annihilation operators

After considering the description of a many-particle system in thermodynamic equilibrium we now extend the formalism of second quantization to nonequilibrium. We obtain the equations of motion for the second quantization operators where we consider fermions and bosons in a common approach. The only dis- tinction will enter through the diﬀerent (anti-)commutation properties of the respective operators.

5.1 Equation of motion of the ﬁeld operators

We start by considering the dynamics of the ﬁeld operators. Their time- dependent form is obtained by transforming to the Heisenberg representation of quantum mechanics according to1.

ˆ † ˆ ψH (x, t) = U (t, t0)ψ(x)U(t, t0) (5.1) where ψˆ(x) is the (time-independent) field operator in the Schrödingerpicture, ˆ i.e. the value of the Heisenberg operator ψH (x, t) at a chosen initial time t0. Furthermore, U(t, t0) is the N-particle Schrödingertime evolution operator that obeys

0 ˆ 0 i~∂tU(t, t ) − H(t)U(t, t ) = 0, (5.2) U(t, t) = 1ˆ, (5.3)

1A critical discussion ot the Heisenberg representation of the ﬁeld operators is given in Sec. 5.4.

139 140 CHAPTER 5. DYNAMICS OF FIELD OPERATORS where Hˆ is the full N-particle hamiltonian. We also give the adjoint equation, † 0 † 0 ˆ 0 0 ˆ 0 = −i~∂tU (t, t ) − U (t, t )H(t) = −i~∂tU(t , t) − U(t , t)H(t) 0 0 ˆ 0 = −i~∂t0 U(t, t ) − U(t, t )H(t ), (5.4) where we used [U(t, t0)]† = U(t0, t) and, in the last line, renamed the time arguments t ↔ t0 and understand H to act to the left. The time evolution of the ﬁeld operators is governed by the hamiltonian for which we use a general expression containing kinetic energy, potential energy and pair interaction energy which we write in second quantization (x = (r, σ), see chapter 3)

Z 2 Hˆ = Tˆ + Uˆ + Wˆ = dx0 ψˆ†(x0) − ~ ∇02 + v(r0) ψˆ(x0) + 2m 1 Z Z = dx0 dx00 ψˆ†(x0)ψˆ†(x00)w(r0, r00)ψˆ(x00)ψˆ(x0). (5.5) 2 The evolution equation of the field operators is given by Heisenberg’s equation (see problem 5.1, Sec. 5.7) 2 ˆ ˆ ˆ † ˆ ˆ i~∂tψH (x, t) = −[HH , ψH (x, t)] = −U (t, t0)[H, ψ(x)]U(t, t0). (5.7) We now evaluate the commutator which is the sum of three commutators involving T,ˆ Uˆ and Wˆ , respectively. This will lead to commutators of different combinations of field operators which we will simplify using the commutation (anticommutation) relations for bosonic (fermionic) operators. We start the derivation by noting the following properties of commutators,

[AˆB,ˆ Cˆ] = Aˆ[B,ˆ Cˆ] + [A,ˆ Cˆ]B,ˆ (5.8) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [AB, C] = A[B, C]∓ ± [A, C]∓B, (5.9) ˆ† 0 ˆ 0 [ψ (x ), ψ(x)]∓ = ∓δ(x − x ), (5.10) where the ﬁrst two are veriﬁed by direct evaluation of the left and right-hand sides (see problem 5.2, Sec. 5.7) and the third follows from the standard (anti- )commutation relations3.

2The derivation starts from the r.h.s. of Heisenberg’s equation that involves two Heisen- berg operators † † ˆ −[U (t, t0)HUˆ (t, t0),U (t, t0)ψ(x)U(t, t0)], (5.6)

† and uses the property U(t, t0)U (t, t0) = 1. 3In the second and third line the upper (lower) sign refers to bosons (fermions), i.e. to the commutator (anti-commutator). 5.1. EQUATION OF MOTION OF THE FIELD OPERATORS 141

Consider ﬁrst the commutator with the external potential which is simpli- ﬁed with the help of Eq. (5.9), Z [V,ˆ ψˆ(x)] = dx0 [ψˆ†(x0)v(r0)ψˆ(x0), ψˆ(x)] = Z 0 0 n ˆ† 0 ˆ 0 ˆ ˆ† 0 ˆ ˆ 0 o dx v(r ) ψ (x )[ψ(x ), ψ(x)]∓ ± [ψ (x ), ψ(x)]∓ψ(x )

= −v(r)ψˆ(x), (5.11) where we took into account that the ﬁrst commutator vanishes and the second is evaluated according to Eq. (5.10). The same derivation applies to the kinetic energy term with the result 2 [T,ˆ ψˆ(x)] = − − ~ ∇2 ψˆ(x). (5.12) 2m Finally, we transform the commutator with the interaction energy using relation (5.8), Z Z 2[W,ˆ ψˆ(x)] = dx0 dx00 [ψˆ†(x0)ψˆ†(x00)w(r0, r00)ψˆ(x00)ψˆ(x0), ψˆ(x)] = Z Z = dx0 dx00 w(r0, r00) ψˆ†(x0)ψˆ†(x00)[ψˆ(x00)ψˆ(x0), ψˆ(x)] + + [ψˆ†(x0)ψˆ†(x00), ψˆ(x)]ψˆ(x00)ψˆ(x0) . (5.13)

The first commutator vanishes as it involves only annihilation operators whereas the second is transformed, using Eqs. (5.9) and (5.10), ˆ† 0 ˆ† 00 ˆ ˆ† 0 ˆ† 00 ˆ ˆ† 0 ˆ ˆ† 00 [ψ (x )ψ (x ), ψ(x)] = ψ (x )[ψ (x ), ψ(x)]∓ ± [ψ (x ), ψ(x)]∓ψ (x ) = ∓ψˆ†(x0)δ(x00 − x) − δ(x0 − x)ψˆ†(x00), (5.14) and the second term in the integral (5.13) becomes [ψˆ†(x0)ψˆ†(x00), ψˆ(x)]ψˆ(x00)ψˆ(x0) = −2δ(x0 − x)ψˆ†(x00)ψˆ(x00)ψˆ(x0), where the first term in Eq. (5.14) is transformed by exchanging x0 ↔ x00, after which it becomes identically equal to the second one4. With this the final result for the commutator becomes Z [W,ˆ ψˆ(x)] = − dx00 w(r, r00)ψˆ†(x00)ψˆ(x00)ψˆ(x). (5.15)

4The derivation assumes w(r0, r00) = w(r00, r0), and the sign change, in the case of fermions, arises from exchanging the order of the two annihilation operators. 142 CHAPTER 5. DYNAMICS OF FIELD OPERATORS

Inserting the results for the three commutators into Eq. (5.7) and applying the time evolution operators (assuming ∂H/∂t = 0 it follows U †v(r) = v(r)U †; for the general case, see Sec. 5.4 and problem 2) we obtain the equation of motion of the ﬁeld operator,

 2 i ∂ ψˆ (x, t) = − ~ ∇2 + v(r) + Uˆ ind(x, t) ψˆ (x, t) (5.16) ~ t H 2m H H Z ˆ ind 00 00 ˆ† 00 ˆ 00 UH (x, t) = dx w(r, r )ψH (x , t)ψH (x , t). (5.17)

eff Thus the field operator is subject to an effective single-particle potentialv ˆH = ˆ ind v(r) + UH . This is an exact result, valid both for fermions and bosons. Remarkably, this equation which was derived from the Heisenberg equation (5.7) has the form of a one-particle time-dependent Schrödingerequation, just as for the wave function, and it shares the same basic properties. First, the equation for the creation operator is obtained by hermitean conjugation of Eq. (5.16):

 2 i ∂ ψˆ† (x, t) = −ψˆ† (x, t) − ~ ∇2 + v(r) + Uˆ ind(x, t) (5.18) ~ t H H 2m H

ˆ ind where the operators ∇ and UH act on the ﬁeld operator to the left, and we ˆ ind † ˆ ind took into account that (UH ) = UH which is a consequence of the fact that ˆ† ˆ ˆ ind the density operator,n ˆH = ψH ψH , appearing in UH , is hermitean. Theorem: As the Schr¨odingerequation in quantum mechanics, Eq. (5.16) is associated with an operator continuity equation describing local particle number conservation,

ˆ ∂tnˆH (x, t) + ∇jH (x, t) = 0 (5.19) n o nˆ = ψˆ† ψˆ , ˆj (x, t) = ~ ψˆ† ∇ψˆ − ∇ψˆ† ψˆ (5.20) H H H H 2im H H H H

While in the continuity equation for the single-particle wave function of standard quantum mechanics the quantities n and j describe the probability density and probability current density, here the analogous quantities refer to an N-particle system5.

5While the probability density is normalized to 1 (the particle number equals one), here the integral ofn ˆ over the volume yields the total particle number N. 5.2. GENERAL OPERATOR DYNAMICS 143

Proof: We compute the time-derivative of the density operator and use the equations of motion (5.16), (5.18), dropping the arguments x, t

† ˙ ˙ † n † † o nˆ˙ = ψˆ ψˆ + ψˆ ψˆ = − ~ ψˆ ∇2ψˆ − ∇2ψˆ ψˆ = H H H H H 2im H H H H n o = − ~ ∇ ψˆ† ∇ψˆ − ∇ψˆ† ψˆ , 2im H H H H and the expression in the brackets is just the current density operator (5.20)6. The key difference between the familiar one-particle Schrödingerequation and Eq. (5.16) for the field operator is the appearance of an effective potential ˆ eff ˆ ind UH = v + UH instead of the external potential v. This “induced” potential includes the whole many-body problem. It has exactly the form of a mean field (Hartree) potential that is created by all particles, as in the case of the quantum Vlasov equation (Hartree equation)7. Thus this equation is the simplest formulation of the nonequilibrium many-body problem for fermions and bosons in its full generality. This simple form arises from the nature of the creation and annihilation operators that are well adapted to this problem. Unfortunately, a direct solution of Eq. (5.16) is impossible due to its operator character. The standard procedure is, therefore, to introduce suitable expectation values. This will be considered in Sec. 5.6. An independent approach that is based on a stochastic approach to this equation will be discussed in Sec. 5.4. But before that we generalize the equation of motion for the field operators to a general basis and derive the equations of motion for the creation and annihilation operators.

5.2 Dynamics of the creation and annihilation operators in an arbitrary representation

After considering the dynamics of the second quantization operators in coordinate representation, we now generalize this result to an arbitrary basis of single-particle states {|ii}. The N-particle states belong to the Fock space and are again written in occupation number representation |n1n2 ... i, cf. Chapter 3. The creation and annihilation operators associated to orbital i are ai and † ai and obey the standard (anti-)commutation relations.

6In the derivation we took into account that the terms with the potentials cancel. 7Interestingly, the same general structure of an exact mean-ﬁeld type form of the many- body problem was obtained before in Ch. 2 for classical systems when we discussed the phase space density N(r, p, t), cf. Eq. (2.20). 144 CHAPTER 5. DYNAMICS OF FIELD OPERATORS

We start with the same hamiltonian as before, Eq. (5.5), for which we use the general second quantization representation,

Hˆ = Tˆ + Uˆ + W,ˆ (5.21) ∞ ∞ ˆ ˆ X † X † T + U = ai (tij + vij) aj = hij ai aj, (5.22) i,j=1 i,j=1 ∞ 1 X Wˆ = a†a† w a a . (5.23) 2 i j ijkl l k i,j,k,l=1 Proceeding as in Sec. 5.1 we introduce Heisenberg operators, † 8 aî(t) = U (t, t0)âiU(t, t0), and consider the Heisenberg equation of motion † ˆ i~∂tai(t) = −U (t, t0)[H, ai]U(t, t0), ai(t0) = ai. (5.24) Evaluating the three commutators in Eq. (5.24) we finally obtain (see problem 5.4, Sec. 5.7)

X X † i~∂tai(t) = (til + vil)al(t) + wimlnam(t)an(t)al(t) (5.25) l lmn where all operators are now Heisenberg operators. This is the generalization of the coordinate space result (5.16) to a general basis representation. All the results discussed before (induced potential, adjoint equation, continuity equation etc.) remain valid. Again we may introduce an eﬀective potential and rewrite the equation of motion in the form of an eﬀective single-particle problem

X eﬀ i~∂tai(t) = til +v ˆH,il(t) al(t), (5.26) l eﬀ X vˆH,il(t) = vil + wimlnnˆmn(t). (5.27) mn 5.3 Extension to time-dependent hamiltonians

So far we have assumed that the hamiltonian does not depend explicitly on time. This was used when applying the time evolution on the ﬁnal step of the derivation. We now remove this restriction and generalize our results to the case of a time-dependent single-particle potential, such as an external electromagnetic ﬁeld. Then the only term that changes is the the one involving the

8 As before, we assume ∂tH = 0. For the general case see Sec. 5.4 5.4. SCHRODINGER¨ DYNAMICS OF THE FIELD OPERATORS 145 potential Uˆ(t), cf. Eq. (5.22), and the contribution to the r.h.s. of Eq. (5.24) becomes

† ˆ † X −U (t, t0)[U, ai]U(t, t0) = U (t, t0) vil(t) al U(t, t0) (5.28) l X † X = U (t, t0)vil(t)U(t, t0) al(t) = vˆH,il(t) al(t). l l where, in the last line we inserted a unity operator between vil and al. Note that, for the general case of a time-depdendent potential v(t) does not commute with the time evolution operator, v(t)U(t, t0) 6= U(t, t0)v(t), and the result, therefore, contains the Heisenberg operator vH,il(t). Thus, our previous result, Eq. (5.27) is generalized to

X eﬀ i~∂tai(t) = til +v ˆH,il(t) al(t), (5.29) l eﬀ X vˆH,il(t) =v ˆH,il(t) + wimlnnˆmn(t), (5.30) mn

† where the time dependence ofv ˆH,il(t) = U (t, t0)vil(t)U(t, t0) is due, both, to the explicit time dependence of the potential v and the two time evolution operators.

5.4 Schr¨odingerdynamics of the creation and annihilation operators

In the previous sections we used the Heisenberg picture for the creation and annihilation operators9. While this is common practice in many text books, this approach has to be critically assessed. The problem is that the “standard” † Heisenberg operatora ˆH (t) = U (tt0)a ˆ U(tt0) has, strictly speaking, no clear mathematical meaning if the Hamilton operator (and, similarly, the evolution operator U) refers to a fixed particle number N. Suppose we act witha ˆH (t) on an arbitrary state |ψN i of the N-particle Hilbert state HN . Then we will, obviously, understand U as an N-particle time evolution operator associated ˆ with the N-particle hamiltonian HN . The action of U produces again a state from HN . Acting now witha ˆ produces a state from the Hilbert space HN−1. † ˆ The final action of U , which is again associated with HN , is then, however, ill-defined. Thus, the use of the standard Heisenberg picture for the operators

9This aspect and the following results have been worked out together with S. Hermanns and C. Hinz. 146 CHAPTER 5. DYNAMICS OF FIELD OPERATORS

† ˆ aˆ anda ˆ is only possible if HN and U do not refer to a fixed N but are defined in Fock space10. Alternatively, if the Hamiltonian is defined in Hilbert space HN , the dynamics of creation and annihilation operators should be formulated in the Schrödingerpicture where no such problem occurs since it involves only a single evolution operator UN . Let us now consider how this is accomplished and compare the results with those of the previous sections of this chapter. We start with an arbitrary complete set of single-particle states |ii for which † the operatorsa î anda î are defined as before. With these operators we can again produce the second quantization respresentation of arbitrary operators, in particular, for the generic N-particle hamiltonian Hˆ , cf. Eq. (5.21). We proceed by constructing properly (anti-)symmetrized N-particle states |{n}i and defining the N-particle evolution operator U, as before, via Eq. (5.16). Now we define the time-dependent annihilation and creation operators that † evolve from the operatorsa î anda î , leaving out the hats

ai(t) = aiU(t, t0) (5.35)

† † † ai (t) = U (t, t0)ai (5.36)

10One possible way to define Heisenberg-type operators with hamiltonians for a fixed N is to work with different Hilbert spaces:

H † ak (t) = UN−1(t, t0)akUN (t, t0). (5.31) Then, the equation of motion is given by

H † † i∂tak (t) = i∂tUN−1akUN + UN−1aki∂tUN † † = −UN−1HN−1akUN + UN−1akHN UN H H H H = −HN−1(t)ak (t) + ak (t)HN (t) (5.32) † ˆ where, in the third line, we inserted unity according to UN−1UN−1 = 1. The density matrix operator is then given by

H †,H H † † † † nkl(t) = ak (t)al (t) = UN akUN−1UN−1alUN = UN nklUN , (5.33) which is a proper Heisenberg operator in N-particle Hilbert space that evolves according to the Heisenberg equation of motion

H H H i~∂t nkl(t) = [nkl(t),Hkl (t)]. (5.34) Thus this modified Heisenberg dynamics of the creation and annihilation operators lead to the same equations of motion for the density matrix operator. Furthermore, it is clear that this modified Heisenberg dynamics will approach the standard definition in the macroscopic limit, N → ∞ when N − 1 → N, and the r.h.s. of Eq. (5.32) approaches the commutator H H [ak (t),HN (t)]. 5.5. DYNAMICS OF THE DENSITY MATRIX OPERATOR 147 where the second line follows by hermitean adjungation of the first one. These definitions mean that the annihilation and creation operators behave like wave functions of first quantization evolving according to the time-dependent N−particle Schrödingerequation,

i~∂tai(t) = aiH(t)U(t, t0), (5.37) † † † −i~∂tai (t) = U (t, t0)H(t)ai . (5.38)

These equations are well deﬁned when U is an N-particle operator11. Let us now see how the corresponding density matrix operator looks like and what its properties are. One immediately ﬁnds

† † † H nji(t) = aj(t)ai(t) = U (t, t0)ajaiU(t, t0) = nji (t), (5.39) † nji(t) = nij(t). (5.40)

The first line shows that the density matrix operator defined with Eqs. (5.35) and (5.36) is a proper Heisenberg operator and its equation of motion is given by the Heisenberg equation (5.34). This way we have at our disposal two independent dynamical equations of the creation and annihilation operators – a Schrödingerequation and a Heisenberg equation. Both have a different applicability range: the first corresponds to Hamiltonians (and time evolution operators) acting in the N-particle Hilbert space, whereas the second assumes operators defined in Fock space with a variable particle number. Both approaches have their advantages and disadvantages for numerical applications as we discuss below.

5.5 Dynamics of the density matrix operator nˆnm(t)

We now consider the dynamics of the operatorn ˆnm(t). This operator is directly related to observable quantities in quantum many-body systems in nonequilibrium and thus of prime importance. Sincen ˆnm(t) is a Heisenberg operator the ambiguity in the dynamics of the ﬁeld operators – Heisenberg vs. Schr¨odinger dynamics – does not play a role here. Both representaions lead to the same results for the density matrix operator. We start from the Heisenberg equation

† ˆ i~∂tnij(t) = −U (t, t0)[Hs(t), nij]U(t, t0), nij(t0) = nij. (5.41)

11This is not a restriction. N can be chosen arbitrary, only U has to be chosen correspondingly. 148 CHAPTER 5. DYNAMICS OF FIELD OPERATORS and evaluate the three commutators in Eq. (5.41), using again the relations † (5.8), (5.9) and [ai , aj]∓ = ∓δi,j,

ˆ ˆ X n † † † † o [T + U, nij] = hkl(t) ak[al, ai aj] + [ak, ai aj]al = kl X n † † † † o = hkl(t) ∓ak[ai , al]∓aj − ai [aj, ak]∓al kl X n † † o = hkl(t) akajδil − ai alδjk kl X n † † o = hki(t) akaj − hjk(t) ai ak (5.42) k X ∗ ∗ ∗ ∗ = − nik hkj(t) − hik(t) nkj = ˆnh (t) − h (t)ˆn, (5.43) k where, in the last expression, we introduced standard matrix notation. The commutator with the interaction energy is transformed similarly,

ˆ † X n † † † † † † o 2[W , ai aj] = wklmn akal [anam, ai aj] + [akal , ai aj]anam = klmn X n † † † † † † o = wklmn −akal [ai , anam]aj − ai [aj, akal ]anam = klmn X n † † † † † o = wklmn akal (anδmi ± amδni) aj − ai al δkj ± akδlj anam klmn X n † † ∗ † † o = −2 wjnklai analak − winklakal anaj . (5.44) kln In the third line, the first terms in the parantheses are equal to the second ones – this is shown by exchanging (k, m) ↔ (l, n) and using the symmetries aman = ±anam and wklmn = wlknm. What remains now is to apply the time evolution operators. There are two ways to proceed. The first is to apply the evolution operators only to the outermost field operators. To this end we use the results for the commutators in the form (5.42) and (5.44) and combine them as follows

† † X eff i~∂tnij(t) = U (t, t0)ai hjk (t)akU(t, t0) − k X † † eff − U (t, t0)akhki (t)ajU(t, t0) (5.45) k eff ˆ ind ˆ ind X † hjk = hjk(t) + Wjk , Wjk = wjnklanal, (5.46) ln 5.5. DYNAMICS OF THE DENSITY MATRIX OPERATOR 149 where we introduced the same operators of the induced potential and effective single-particle potential as before, cf. Sec. 5.3. Note, however, that here the induced potential is still time-independent. This is our first result. It is particularly useful when we consider computation of suitabel averages. Sup- pose we are interested in the average dynamics of the density matrix operator, i.e. the dnamics of the density matrix nij, in a given time-independent N-particle state |Ψi. When the density matrix operator is averaged with |Ψi, we can, for each term, combine the pair of time evolution operators, hΨ|U(t0, t) ...U(t, t0)|Ψi = hΨ(t)| ... |Ψ(t)i with the state vectors to yield time-dependent N-particle states, where the dots denote a time-independent operator, except for the intrinsic time-dependence of hˆ(t). This brings us back to a Schrödinger-type description of the time evolution which is the basis for Configuration Interaction approaches. The second approach consists in reordering the field operators in the interaction term such that this term can be expressed via density matrix operators. Eventually we will also try to achieve a compact matrix equation, as was done for the single-particle terms in Eq. (5.43). To this end we transform the first four-operator product, using nnlak = aknnl − δknal

† † ai analak = niknnl − δnknil

= ± (nilnnk − δlnnik) 1 1 = (n n ± n n ) − (δ n ± δ n ) . (5.47) 2 ik nl il nk 2 nk il ln ik

The ﬁrst two lines correspond to the two options to pair creation and annihilation operators which are both equivalent. Therefore, below we will use the (anti-)symmetrized form given in the third line. Analogously, the second four-operator product becomes

1 1 a† a†a a = (n n ± n n ) − (δ n ± δ n ) . (5.48) k l n j 2 ln kj kn lj 2 nk lj ln kj

The next step is to apply the two time evolution operators which simply leads to the replacement of all density matrix operators be time-dependent (Heisen- berg) operators. Finally, we take into account the induced potential and trans- 150 CHAPTER 5. DYNAMICS OF FIELD OPERATORS form, using Eq. (5.47),

X X wjnkl a† Wˆ inda = {(n n ± n n ) − (δ n ± δ n )} . i jk k 2 ik nl il nk nk il ln ik k kln ∗ ∗ X X wkljn X X wkljn = n n ± n n ik 2 nl il 2 nk k ln l kn ∗ ∗ X X wlkjn ± wkljn − n δ ik ln 2 k ln ∗ ∗ X X wkljn ± wlkjn = n {n ∓ δ } ik 2 nl ln k ln = ˆnUˆ ±, ∗ ∗ X wkljn ± wlkjn with the deﬁnition Uˆ ± = {nˆ ∓ δ } , kj 2 nl ln ln X wjnkl ± wjnlk = {nˆ ∓ δ } , (5.49) 2 nl ln ln where, in the two terms containing nil, we exchanged the summation indices l ↔ k. In Eq. (5.49) we introduced the operator of the (anti-)symmetrized induced potential. Similarly, the second term becomes

X X wklin a† Wˆ inda = {(n n ± n n ) − (δ n ± δ n )} . k ki j 2 ln kj kn lj nk lj ln kj k kln X X wklin ± wlkin = {n ∓ δ } n . 2 ln ln kj k ln = Uˆ ±nˆ. (5.50)

One readily veriﬁes the potential U ± is exactly the one introduced in Eq. (5.49).12

12Starting from the deﬁnition (5.49) we readily transform to the expression (5.50) by, ﬁrst, exchanging the summation indices n, l and then using, in the second term, the property of the matrix elements of the interaction, wklni = wlkin

X wknil ± wknli U ± = {nˆ ∓ δ } ik 2 nl ln ln X wklin ± wklni X wklin ± wlkin = {nˆ ∓ δ } = {nˆ ∓ δ } . 2 ln ln 2 ln ln ln ln 5.6. FLUCTUATIONS AND CORRELATIONS 151

Collecting all the results, we obtain, after applying the time evolution operators,

h n oi h i ∗ ˆ ± ˆ± i~∂tˆn(t) = ˆn(t), hH(t) + UH(t) = ˆn(t), hH(t) (5.51) where all operators are now Heisenberg operators, in particular, the induced potential operator now contains Heisenberg creation and annihilation operators. The term in the curly brackets can again be understood as the operator ˆ± ∗ ˆ ± of an eﬀective (Hartree-Fock-type) potential, hH(t) = hH(t) + UH(t).

5.6 Ensemble average of the Heisenberg equation. Fluctuations and correlations

Despite the formal simplicity of the dynamical equation (5.51) which has the form of a Hartree-Fock equation, the operator nature of the entering ﬁeld operators prohibits a direct access to observable physical quantities. There are (at least) four solutions:

A. Computation of pure state averages using N-particle states and time prop- agation of these states (expansion coeﬃcients) in CI-manner, as discussed in the context of Eq. (5.45).

B. Application of the ﬁeld operators to suitable many-body states. Propaga- tion of individual random trajaectories with subsequent ensemble averaging. An example is the stochastic mean ﬁeld approach of Ayik, Lacroix and others which is discussed in Sec. 5.6.2.

C. Performing a suitable statistical average over ﬁeld operators yielding results in a mixed ensemble. This approach results in a hierarchy of equations for reduced s-particle density matrices (BBGKY-hierarchy or for correlation functions of ﬂuctuations) and will be considered in Sec. 5.6.1.

D. Computation of statistical averages of ﬁeld operator products taken at different times. This leads to the theory of nonequilibrium Green functions and will be discussed in Chapter 7.

5.6.1 Fluctuations and correlations We now perform a statistical average of the operator equation (5.51). We will denote averages of operators by symbols without hat and ﬂuctuations 152 CHAPTER 5. DYNAMICS OF FIELD OPERATORS

(deviations from the average) by the symbol δ, i.e.

hnˆnmi ≡ nnm. (5.52)

δnˆnm ≡ nˆnm − nnm. (5.53) Since averaging is a linear operation, its application to the operator equation of motion (5.51) does not change the equation, except for terms containing products of density matrix operators. For arbitrary operators (or random variables), the average of a product can be written as hAˆBˆi = AB + hδAδˆ Bˆi. We now apply these results to the operator equation (5.51): Dh iE ± ˆ ± ± i∂tn(t) − n(t), hH(t) = δˆn(t), δUH(t) ≡ I (t) (5.54)

Here the l.h.s. contains all (ensemble averaged) mean field terms and con- stitutes a standard time-depdendent Hartree-Fock (TDHF) equation for the density matrix. The r.h.s., in contrast, contains all terms going beyond TDHF. By definition, these are correlation contributions. Here we see that these correlation terms have a one to one correspondence with fluctuations of operator pairs13. We have also introduced the short notation I± for the collision integral. The solution of this inhomogeneous linear (formally) equation, together with the initial condition, n(t0) = n0, is straightforward and given in terms of Hartree-Fock propagators U HF (these are two-dimensional matrices and everywhere matrix multiplication is implied)

HF† HF n(t) = U (t, t0) n0 U (t, t0) + nI (t), (5.55) Z t 1 HF† ± HF nI (t) = dt¯U (t, t¯) I (t¯) U (t, t¯), (5.56) i t0 HF ± HF HF i∂tU (t, t0) = hH(t)U (t, t0), U (t, t) = 1. (5.57) We now can make further progress in evaluating the collision term I± by directly computing the fluctuations δnˆ and, from it, also the fluctuation of the effective potential, X wjnkl ± wjnlk δUˆ ± = δnˆ . (5.58) kj 2 nl ln Indeed the equation of motion of δnˆ follows immediately by taking the difference of Eqs. (5.51) and (5.54) [we suppress the time arguments]

∗ ∗ i∂t (nˆ − n) = [nˆ, hH] − [n, hH] + h i Dh iE ˆ ± ± ˆ ± + nˆ, UH − n, UH − δnˆ, δUH

13This correspondence between correlations and ﬂuctuations is well known from the kinetic theory of classical plasmas and was established by Kadomtsev, Klimontovich and others. 5.6. FLUCTUATIONS AND CORRELATIONS 153

Using the linearity in the density matrix, this can be rewritten as

h i Dh iE ∗ ˆ ± ˆ ± ˆ± i∂tδnˆ − [δnˆ, hH] = δnˆ, δUH − δnˆ, δUH ≡ J (5.59)

The term on the right can be understood as a higher order collision integral h i ˆ± ˆ ± or as the ﬂuctuation of the correlator of the ﬂuctuations, J = δ δnˆ, δUH .

Equation (5.59), together with the initial condition δˆn(t0) = δˆn0, is solved like Eq. (5.54), but using ideal propagators instead of Hartree-Fock propagators:

id† id δˆn(t) = U (t, t0) δnˆ0 U (t, t0) + δnˆJ (t), (5.60) Z t 1 id† ˆ± id δnˆJ (t) = dt¯U (t, t¯) J (t¯) U (t, t¯), (5.61) i t0 id ∗ id id i∂tU (t, t0) = hH(t)U (t, t0), U (t, t) = 1. (5.62)

There are several ways how to proceed. One is to evaluate the r.h.s. of Eq.(5.59) by using again the equation of motion (5.59), multiply by δUˆ ± to derive the equation of motion for the product of fluctuations δnˆδUˆ ± and for their commutator. It is easy to see that this equation, on the r.h.s., will contain products of three operator fluctuations. This shows that a hierarchy of equations for the fluctuations emerges which, in fact, is analogous to the BBGKY-hierarchy for the reduced density operators. As an alternative, we can use the solution δn(t), Eq. (5.60), to compute ˆ ind (α) the commutator [δnˆ(t)δU (t)], for a given initial condition δn0 . This yields a single random realization of the collision integral I±(α) in Eq. (5.54). Re- peating this for a representative set of initial conditions we can compute the expectation value by averaging over an ensemble of initial conditions,

M Dh iE 1 X Dh iE I±(t) = δnˆ(t), δUˆ ±(t) = lim δnˆ(α)(t), δUˆ ±(α)(t) , (5.63) M→∞ M α=1 where (α) denotes the possible realizations that occur with probability pα, P where α pα = 1. This set (α, pα) speciﬁes the ensemble. With this, the r.h.s. of Eq. (5.54) is known and this equation can be solved. Two problems remain. The ﬁrst is how to specify a physically adequate ensemble and the second, how to solve for δnˆ, considering the complicated structure of the collision integral Jˆ±. A very simple and successful approach has recently been proposed by Ayik and co-workers [Ayi08, Lac13] and will be considered in the next section. 154 CHAPTER 5. DYNAMICS OF FIELD OPERATORS

5.6.2 Stochastic Mean Field Approximation One problem in treating the fluctuations of the density matrix operator and of the mean field potential is their time dependence. A first simplifying at- tempt to understand the general physics is, therefore, to neglect this dependence entirely. This can be done by approximating the collision intergral I± in ± ± Eq. (5.54) by a local function according to I (t) → I (t0)δ(t−t0). This means only the initial fluctuations are taken into account. With this the solution for the density matrix, Eq. (5.56) becomes 1 n (t) = U HF†(t, t ) I±(t ) U HF(t, t ), I i 0 0 0 and the total solution for the density matrix is given by 1 Dh iE n(t) = U HF†(t, t ) n + δˆn(t ), δUˆ ±(t ) U HF(t, t ). 0 0 i 0 H 0 0 This means that the evolution of the density matrix n(t) is given by a pure Hartree-Fock dynamics. However, the evolution does not startfrom the initial value of the density matrix, n(t0) but from a value that is shifted by the second term in the parantheses. If we forget for a moment the angular brackets we would have random realizations of initial values. Ayik had the idea [Ayi08] to replace the complicated commutator by a semiclassical ensemble of initial density fluctuations with given mean and variance such that the term in (α) parantheses becomes n0 +∆n0 , for a given realization (α). The corresponding dynamics, starting from this initial state is given by

(α) HF† n (α)o HF nˆ (t) = U(α) (t, t0) n0 + ∆n0 U(α) (t, t0), and the full result is then given by the ensemble average, i.e. by the sum over all realizations, for all times,

M 1 X n(t) = lim nˆ(α)(t). (5.64) M→∞ M α=1 Since only mean field trajectories are involved and the result is obtained from a stochastic sampling over relizations this approach has been called Stochas- tic Mean Field (SMF). Thereby, the incorporation of fluctuations reproduces (part of) the correlations in the system. The most attractive feature is the conceptional simplicity and computational efficiency: THDF propagations are very simple and fast, and sampling of the initial states is very efficiently realized with Monte Carlo methods. Finally, this sampling an be performed in parallel on a large number of computer cores. 5.6. FLUCTUATIONS AND CORRELATIONS 155

Although the SMF concept is very crude since it restricts the ﬂuctuations to those of the initial state, neglecting decay of initial ﬂuctaions and buildup of correlations due to collisions, this method shows remarkable results. Tests for simple Hubbard clusters have shown that the results are not only better than pure time-dpendent Hartree-Fock (TDHF) but also more accurate than NEGF results using selfenergies in second order Born approximation [LHHB14].

5.6.3 Iterative Improvement of Stochastic Mean Field Approximation Here we suggest a further improvement of SMF that allows to include the time- dependence of the fluctuations in an iterative manner. A crucial observation is that SMF does not only yield the one-body expectation value n(t) but a full set of random trajectories. This means, we can also compute fluctuations of the density, correlation functions, higher order density matrices and so on. Using the entire ensemble of trajectories {n(α1)} we can immediately compute the ensemble of fluctuations {δn(α1)} and {δU±(α1)}, cf. Eq. (5.58). Using Eq. (5.63) we can directly evaluate the collision integral and thus compute the evolution of the density matrix. A further improvement would be to avoid the computation of the ensemble average in Eq. (5.63) but instead, again, to propagate random realizations. Then we would again have access to an ensemble of improved trajectories {n(α2)} which can, again, be used as input into the collision integral. This way an iterative procedure can be designed. It remains to be checked how good convergence is. The open problem is to work out an approximation how to compute the commutator of {δn(α)} and {δU±(α)} that enters the collision integral.

5.6.4 Ensemble average of the ﬁeld operators Suppose our many-body system is in a mixed state characterized by some time-independent probability distribution. In the most general case this is the

N-particle density operator ρN . Then we can compute averages, h... iρN = Tr . . . ρN ,

ˆ hψ(r, σ, t)iρN = ψ(r, σ, t) (5.65) ˆ† ∗ hψ (r, σ, t)iρN = ψ (r, σ, t), (5.66) which are already regular functions of coordinate, spin and time. 156 CHAPTER 5. DYNAMICS OF FIELD OPERATORS

5.6.5 Field operators and reduced density matrices ˆ Consider an arbitrary s-particle operator, Bs. To compute its expectation values we have two options. First is to use the (anti-) symmetrized reduced ± s-particle density operator Fs that are deﬁned via the full N-particle density operator according to [Bon] ± ± Fs = FsΛ1...s (5.67) N! N! F = Tr ρ , Tr F = . (5.68) s (N − s)! s+1...N N 1...s s (N − s)! Here we introduced the (anti-)symmetrization operator deﬁned in Eq. (3.14) in Sec. 3. Note that Fs is an ensemble averaged quantity since it is computed from ρN that incorporates a mixed state description with given probabilities of the individual N-particle states. ˆ Using the reduced density operator the expectation value of Bs is computed according to 1 hBˆ i = Tr Bˆ F ± (5.69) s ρN s! 1...s s s ∞ ∞ 1 X X = hi . . . i |F ±|k . . . k ihk . . . k |ˆb |i . . . i i, s! 1 s s 1 s 1 s 1...s 1 s i1...is=1 k1...ks=1 where, in the second line, the trace is performed using a complete set of s- particle states |i1 . . . isi which we can always construct as products of single- particle orbitals. This expression is equivalent14 to averaging over the full N- particle density operator since the trace over the variables of particles s+1 ...N is trivially performed. The second approach to compute this expecation value is to transform the ˆ operator Bs into second quantization representation, cf. Eq. (3.54), ∞ ∞ 1 X X Bˆ = hk . . . k |ˆb |i . . . i ia† . . . a† a . . . a , s s! 1 s 1...s 1 s i1 is ks k1 i1...is=1 k1...ks=1 where the sums run over the complete set of single-particle orbitals. This is still an operator expression. In order to obtain its expectation value in the relevant statistical ensemble, we average this expression with the density operator ρN , taking into account that the matrix element of ˆb is a regular c-function, ∞ ∞ 1 X X hBˆ i = hk . . . k |ˆb |i . . . is iha† . . . a† a . . . a i . s ρN s! 1 s 1...s 1 s i1 is ks k1 ρN i1...is=1 k1...ks=1 (5.70)

14 ± Of course, it is equivalent only if the density operator Fs is known exactly. 5.6. FLUCTUATIONS AND CORRELATIONS 157

Now, comparing the two results, Eq. (5.69) and (5.70), we can establish the connection between the creation and annihiliation operators and the reduced density operators:

hi . . . i |F ±|k . . . k i = ha† . . . a† a . . . a i . (5.71) 1 s s 1 s i1 is ks k1 ρN

This is an important result as it establishes the connection between quantum kinetic theory (reduced density operators) and second quantization and allows us to construct the reduced density operators directly from the second quantization operators. Note that the appearance of the factor ns is a consequence of our deﬁnition of the reduced density operators which obey the normalization s TrFs = V that is convenient for macroscopic systems. From this we obtain the normalization of the expression (5.71):

∞ X N! Tr Fˆ± = ha† . . . a† a . . . a i = . (5.72) 1...s s i1 is is i1 ρN (N − s)! i1...is=1

The deﬁnition (5.71) contains all relevant cases. Let us discuss some of them explicitly. i) The single-particle reduced density operator is obtained from setting, in Eq. (5.71), s → 1:

± † hi|F1 |ki = hai akiρN ≡ nik ≡ n, (5.73)

and its coordinate representation is obtained by using the ﬁeld operators, instead of a and a†, and a basis of coordinate-spin states (x = r, σ),

± 0 0 † 0 0 0 nhrσ|F1 |r σ i = hψ (r, σ)ψ(r σ )iρN ≡ n(x, x ). (5.74)

For completeness, we give the normalization condition of the single- particle density matrix in coordinate representation which follows directly from the general relation (5.72), Z Z X ± X † N = dr hrσ|nF1 |rσi = dr hψ (r, σ)ψ(rσ)iρN . (5.75) σ σ ii) The diagonal elements of the single-particle density operator (5.73) yield the ensemble averaged occupations of the single-particle orbitals |ii,

† hai aiiρN ≡ nii = ni , (5.76) 158 CHAPTER 5. DYNAMICS OF FIELD OPERATORS

whereas the diagonal elements in the coordinate representation yield the local spin density

† hψ (r, σ)ψ(rσ)iρN ≡ n(x, x) = nσ(r). (5.77)

In contrast, the oﬀ-diagonal elements of expression (5.73) describe the statistical probability of transitions between orbital |ki and |ii. Similarly, the oﬀ-diagonal elements of the coordinate-space expression (5.74) are related to the probability of a particle undergoing a transition from spin orbital |r0σ0i to |rσi. iii) The second important case of (5.71) is the two-particle reduced density operator (s = 2),

hi i |F ±|k k i = ha† a† a a i ≡ n(2) ≡ n(2), (5.78) 1 2 2 1 2 i1 i2 k2 k1 ρN i1,i2;k1,k2 whereas its coordinate representation is,

± 0 0 0 0 hr1σ1r2σ2|F2 |r1σ1r2σ2i = (5.79) † † 0 0 0 0 (2) 0 0 hψ (r1, σ1)ψ (r2, σ2)ψ(r2σ2)ψ(r1σ1)iρN ≡ n (x1, x2; x1, x2).

The two-particle density matrix is normalized according to [cf. Eq. (5.72)]

∞ X Tr n(2) = ha† . . . a† a . . . a i = N(N − 1). 12 i1 is is i1 ρN i1i2=1 iv) All the above results are directly extended to time-dependent situations. We simply have to introduce the Heisenberg operators in standard manner,

† ai → aHi(t) ≡ U (t, t0)aiU(t, t0), † ψ(x) → ψH (x, t) ≡ U (t, t0)ψ(x)U(t, t0),

and so on. This will give rise to the time-dependent reduced density op- ± (2) erators Fs (t), nij(t), nij (t), time-dependent densities ni(t) and nσ(r, t) etc. Thereby, the underlying dynamics of the Heisenberg operators was computed above: the ﬁeld operators obey Eqs. (5.16) and (5.18) and the general annihilation operator obeys Eq. (5.25).

As we just discussed, the time-dynamics of a many-body system can be obtained from the time evolution of the second quantization operators in the 5.7. PROBLEMS TO CHAPTER 5 159

Heisenberg picture. Alternatively, the dynamics of the reduced density op- ± erators Fs (t) is known: it is given by the BBGKY-hierarchy which follows from the equation of motion of the N-particle density operator ρN – the von ± Neumann equation. Since we found a one-to-one relation between the Fs (t) and the second quantization operators, this BBGKY-hierarchy has to follow † from the dynamics of ai and ai . This will be discussed below.

5.6.6 BBGKY-hierarchy 5.7 Problems to Chapter 5

Problem 5.1 Derive the equation

ˆ † ˆ ˆ i∂tψH (x, t) = −U (t, t0)[H, ψ(x)]U(t, t0). (5.80)

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Problem 5.2 Prove the identity (5.9): [AB, C] = A[B, C]∓ ± [A, C]∓B Problem 5.3 Discuss the derivation of Eq. (5.16) for the case of a time- dependent hamiltonian. Consider a time-dependent single-particle potential, v(r, t).

Problem 5.4 Derive the general equation of motion (5.25) for the creation and annihilation operators for the case of a time-independent external potential. 268 CHAPTER 5. DYNAMICS OF FIELD OPERATORS Appendix A

Solutions to problems

A.1 Problems from chapter 2

Problem 1: A simple equation for ψ0 is readily obtained by inserting the deﬁnition of a into Eq. (2.37),

0 0 = ψ0(ξ) + ξψ0(ξ), (A.1)

−ξ2/2 with the solution ψ0(ξ) = C0e , where C0 follows from the normal- R ∞ 2 1/2 −1/2 ization x0 −∞ dξψ0 = 1, with the result C0 = (π /x0) , where the phase is arbitrary and chosen to be zero.

Problem 2: Proof: Using hψ|a† = a|ψi and Eq. (2.40), direct computation yields † 1 n+1 † † n ψn+1|a |ψn = ψ0|a a (a ) |ψ0 . pn!(n + 1)! √ The ﬁnal result n + 1 is obtained by induction, starting with n = 0.

Problem 3: This problem reduces to the previous one by applying hermitean conjugation † ∗ √ hψn−1|a|ψni = ψn|a |ψn−1 = n

A.2 Problems from chapter 3

Problem 1:

Problem 2:

Problem 3:

269 270 APPENDIX A. SOLUTIONS TO PROBLEMS

† Problem 4: The Proof of relation (3.63), i.e. [ˆai, aˆk]+ = 0, for fermionic creation and annihilation operators proceeds as follows: Consider, this relation explicitly, in its action on a Fock state:

† † † [âi, aˆk]+|{n}i = aîaˆk +a ˆkaî |{n}i, i) case i = k. Inserting the definitions (3.60) and (3.61) we obtain for the first term

† αi aîaî | . . . ni ... i =a î(1 − ni)(−1) | . . . ni + 1 ... i (A.2) 2αi = (ni + 1)(1 − ni)(−1) | . . . ni ... i = δni,0| . . . ni ... i. Analogously, the second term becomes

† † αi aî aî| . . . ni ... i =a î ni(−1) | . . . ni − 1 ... i (A.3) 2αi = (2 − ni)ni(−1) | . . . ni ... i = δni,1| . . . ni ... i.

Since δni,1 + δni,0 = 1, this proves relation (3.63), for i = k. ii) case i 6= k. Without loss of generality we consider i < k and obtain, as before,

† αk aˆiaˆk| . . . ni . . . nk ... i =a ˆi(1 − nk)(−1) | . . . ni . . . nk + 1 ... i αi+αk = ni(1 − nk)(−1) | . . . ni − 1 . . . nk + 1 ... i. For the second term of the anti-commutator, it follows

† † αi aˆkaˆi| . . . ni . . . nk ... i =a ˆkni(−1) | . . . ni − 1 . . . nk ... i αi+(αk−1) = (1 − nk)ni(−1) | . . . ni − 1 . . . nk + 1 ... i † = −aˆiaˆk| . . . ni . . . nk ... i, which proves relation (3.63) for i 6= k. Note the sign change in the P 0 0 last line that results from the exponent αk − 1 = l

Introducing the distance of the two particles, r = r1 − r2, we rewrite the 0 0 0 exponents according to r1(k1 − k1) + r2(k2 − k2) = r(k1 − k1) + r2(k1 − 0 0 k1 + k2 − k2) and obtain Z Z 0 0 1 −ir(k −k0 ) −ir (k −k0 +k −k0 ) hk k |wˆ|k k i = dr w(r)e 1 1 dr e 2 1 1 2 2 . 1 2 1 2 V2 2

The second integral yields Vδ 0 0 and the ﬁrst is expressed by the k1+k2,k1+k2 Fourier transform of the potentialw ˜(q) = V−1 R dr w(r) e−iqr, with the ﬁnal result

0 0 0 hk k |wˆ|k k i =w ˜(k − k )δ 0 0 . 1 2 1 2 1 1 k1+k2,k1+k2

A.3 Problems from chapter 4

Problem 1:

A.4 Problems from chapter 5

Problem 1: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Problem 2: The commutator identiy [AB, C] = A[B, C]∓±[A, C]∓B is proven by dirct computation. The left hand side equals AˆBˆCˆ − CˆAˆBˆ, whereas the right hand side becomes n o AˆBˆCˆ ∓ AˆCˆBˆ ± AˆCˆBˆ ∓ CˆAˆBˆ .

The terms AˆCˆBˆ exactly cancel and we recover the expression on the left hand side.

Problem 2:

Problem 3: The derivation for a time-dependent single-particle potential proceeds as follows.

Problem 4: The solution of the equation of motion for the annihilion operator with respect to a general single-particle basis starts from the Heisenberg equation

† ˆ i~∂tai(t) = −U (t, t0)[H, ai]U(t, t0), ai(t0) = ai. (A.4) 272 APPENDIX A. SOLUTIONS TO PROBLEMS

We now evaluate the three commutators in Eq. (5.24) using the hamiltonian in second quantization (5.21, 5.22, 5.23) as well as the relations † (5.8), (5.9) and [ai , aj]∓ = ∓δi,j,

ˆ ˆ X n † † o X [T + U, ai] = (tkl + vkl ak[al, ai]∓ ± [ak, ai]∓al = − (til + vil)al. k,l l

The commutator with the interaction energy is transformed similarly,

ˆ X n † † † † o 2[W , ai] = wklmn akal [anam, ai] + [akal , ai]anam = klmn X n † † † †o = wklmn ak[al , ai]∓ ± [ak, ai]∓al anam = klmn X n † † o = wklmn ∓akδl,i − al δk,i anam = klmn X † X † = −2 wilmnal anam = −2 wimlnamanal. lmn lmn In the third line the ﬁrst term is equal to the second one – this is shown by exchanging (k, m) ↔ (l, n) and using the symmetries aman = ±anam and wklmn = wlknm. In the last line we exchanged l ↔ m. Collecting all the results, we obtain from Eq. (A.4), after applying the time evolution operators,

X X † i~∂tai(t) = (til + vil)al(t) + wimlnam(t)an(t)al(t), l lmn where all operators are now Heisenberg operators. This is the desired result (5.25). which generalizes the coordinate space result (5.16) to a general basis representation.

A.5 Problems from chapter 6

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