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Introduction to Second Quantization Contents 1 Preliminaries Contents 1 Concepts Fondamentaux de la Physique Introduction to Second Quantization Christophe Mora Contents 1 Preliminaries 1 1.1 Single-particle Hilbert space . 1 1.2 Many-particle Hilbert space . 2 2 Basics of second quantization 2 2.1 Occupation number representation . 2 2.2 Creation and annihilation operators . 3 2.3 Bosons . 3 3 Representation of operators 4 3.1 Change of basis and the field operator . 4 3.2 Representation of one-body and two-body operators . 5 Part of the complexity in the many-body problem - systems involving many particles - comes from the indistinguishability of identical particles, fermions or bosons. Calculations in first quantization thus involve the cumbersome (anti-)symmetrization of wavefunctions. Second quantization is an efficient technical tool that describes many-body systems in a compact and intuitive way. 1 Preliminaries Before entering the details of second quantization, it is worth drawing a clear distinction between the single-particle and the many-particle Hilbert spaces. 1.1 Single-particle Hilbert space ^ Consider a single particle described by the hamiltonian h acting on the Hilbert space H1. H1 is generated by the complete set of eigenfunctions jλi (λ = k; σ; ν; : : :) ^ hjλi = "λjλi; with the eigenvalues "λ. The identity operator in H1 is given by the completeness relation 1 P = λ jλihλj. Examples: 1. single particle in free space, h^ = −~2r2=(2m). The eigenfunctions are labeled by the ik·r wavevectors k with (r) = hrjki = ep and the energies " = 2k2=2m. k V k ~ 2. spin 1=2 in a magnetic field, h^ = −BSz. The Hilbert space has dimension 2, generated by the eigenstates j "i and j #i of the spin operator Sz. 1.2 Many-particle Hilbert space 2 The basis of two-particle states, given by the set of (anti-)symmetrized functions, + for bosons and − for fermions, 1 λ,ν(1; 2) = p ['λ(1) 'ν(2) ± 'λ(2) 'ν(1)] ; 2 1 is built out of the single-particle states 'λ(1) = h1jλi. The corresponding Hilbert space is denoted F2. 1.2 Many-particle Hilbert space We first discuss fermions. Following the two-particle case, the set of antisymmetrized Slater determinants 1 X P λ ,...,λ (1;:::;N) = p (−1) 'λ (P1) :::'λ (PN ); (1) 1 N N! 1 N P 2SN where the summation runs over all permutations of f1;:::;Ng, forms the basis2 of the Hilbert space FN . In the bosonic case, the basis is obtained from symmetrized states, i.e. Eq. (1) where (−1)P is replaced by 1. The hamiltonian may describe independent particles in which case N X H^ = h^(i); i=1 where each piece h^(i) acts only on the particle i. Examples: ^ P 2 1. for particles in free space, H = i pi =(2m). ^ PN z 2. for an assembly of N distinguishable spins in a magnetic field, H = −B i=1 Si . The Hilbert space has dimension 2N and symmetrization is not required. ^ P ^(i) ^ ^ Interactions between particles can be added, H = i h + V , where V includes all multi-particle interactions. For example, Coulomb interactions read N N X X e2 1 V^ = : (2) Coulomb 4π" jr − r j i=1 j>i 0 i j 2 Basics of second quantization So far, we have introduced and discussed the many-body problem in the language of first quantization. Second quantization corresponds to a different labelling of the basis of states Eq. (1) together with the introduction of creation and annihilation operators that connect spaces with different numbers of particles. 1 Restricted to (anti-)symmetrized wavefunctions, F2 is a subset of the larger space H1 ⊗ H1. 2A different choice for the set of single-particle states jλi gives, using Eq. (1), a different many-particle basis that nevertheless spans the same Hilbert space FN . 2.1 Occupation number representation 3 2.1 Occupation number representation Since identical particles are indistinguishable, it is not possible, for a state of the form Eq. (1), to ascribe a definite single-particle state to a given particle. Therefore, instead of focusing on the wavefunction of each particle individually, one can reverse the perspective and characterize the states of Eq. (1) by the set of single-particle states fλ1; : : : ; λN g that are occupied by particles, all other single-particle states being empty. 3 In terms of notations, jfnλgi represents j λ1,...,λN i with, for fermions, nλ = 1 for λ = λi, i = 1 :::N, and nλ = 0 otherwise. The state can be written schematically as λ1 λ2 λN jfnλgi = j0 ::: 1 ::: 0 ::: 1 ::: 0 ::: (:::) 1 i; (3) where it is explicitly specified on the right-hand-side which states are occupied and which state are empty. Bosonic states have similar expressions although the occupation numbers nλ can take values larger than 1, for example λ1 λ2 λN jfnλgi = j0 ::: 5 ::: 0 ::: 1 ::: 0 ::: (:::) 7 i; (4) for nλ1 = 5, nλ2 = 1, :::, nλN = 7. 2.2 Creation and annihilation operators P The constraint on the number of particles, λ nλ = N, can be released by working in the extended Hilbert space +1 M F = FN ; N=0 called the Fock space. Here, F1 = H1 is the single-particle Hilbert space, F0 contains a unique vacuum state, often noted j0i, in which no particle is present. In the Fock space, creation operators are introduced that raise the number of particles in a given single-particle state by 1. For fermions, it reads λ1 λ2 λN λ1 λ2 λN cy j0 ::: ::: 0 ::: ::: 0 ::: (:::) i = j0 ::: ::: 0 ::: ::: 0 ::: (:::) i; λ1 0 1 1 1 1 1 while particle creation in a single-particle state that is already occupied gives zero, λ1 λ2 λN cy j0 ::: ::: 0 ::: ::: 0 ::: (:::) i = 0: λ2 1 1 1 y The annihilation operator cλ, lowering the number by 1, is the hermitian conjugate of cλ. The full basis of the Fock space F is in fact generated by creation operators applied on the vacuum state, namely jn = 1; : : : ; n = 1i = cy : : : cy j0 >. λ1 λN λ1 λN The antisymmetric properties of the basis states (Slater determinants) jnλ1 : : : nλN i are ensured by the anticommutation relations y fcα; cβg = cαcβ + cβcα = 0; fcα; cβg = δα,β: (5) y The productn ^λ = cλcλ gives the number of fermions occupying the state jλi, y cλcλjfnαgi = nλjfnαgi where nλ = 0 or 1. 3 λ1,...,λN (1;:::;N) = h1;:::;Nj λ1,...,λN i. 2.3 Bosons 4 2.3 Bosons There are only slight differences in the way second quantization works for fermions and for bosons. In the case of bosons, the basis states are symmetrized functions and the number of bosons in a given single-particle state is not restricted. These properties are ensured by the commutation relations y [bα; bβ] = bαbβ − bβbα = 0; [bα; bβ] = δα,β; (6) y 4 with the (annihilation) creation operators (bα) bα. From Eq. (6), one can prove that y p bλjnλi = nλ + 1jnλ + 1i p (7) bλjnλi = nλjnλ − 1i y such thatn ^λ = bλbλ is indeed the number operator,n ^λjnλi = nλjnλi. 3 Representation of operators The complexity associated with wavefunction (anti)symmetrization has been reduced, in the formalism of second quantization, to the surprisingly simple commutation relations, Eq. (5) for fermions and Eq. (6) for bosons. Had the usual operators of the theory complicated expressions in terms of creation/annihilation operators, this would not be very useful. How- ever, as we shall see below, the hamiltonian as well as standard operators do have simple expressions in second quantization. 3.1 Change of basis and the field operator y 1 P Starting with the expression jλi = cλj0i, one can insert the closure relation = λ jλihλj to derive the transformation law for the creation/annihilation operators y X y X cα = hλjαi cλ; cα = hαjλi cλ; (8) λ λ from one basis to another. Hence, the change of basis only requires the calculation of matrix elements hαjλi involving single-particle states. By convention, the field operator Ψ(r) in a continuous problem is associated to the basis of position states jri, X Ψ(r) = hrjλi cλ: (9) λ Using Eq. (5) and Eq. (6), one finds the commutation relation fΨ(r); Ψ(r0)g = 0; fΨ(r); Ψy(r0)g = δ(r − r0); fermions; (10) [Ψ(r); Ψ(r0)] = 0; [Ψ(r); Ψy(r0)] = δ(r − r0); bosons: The total number of particles (fermions or bosons) is then given by Z ^ X y d N = cλcλ = d r ρ^(r) (11) λ 4 The states jnλi are chosen to be normalized to 1. 3.2 Representation of one-body and two-body operators 5 where the local density operatorρ ^(r) = Ψy(r)Ψ(r) has been introduced. Example: The transformation to the Fourier momentum representation reads 1 X ik·r Ψ(r) = p e ck; (12) V k where ck destroys a particle with momentum k. The total number of particle is given by ^ P y N = k ckck. 3.2 Representation of one-body and two-body operators ^(1) PN (1) Single-particle or one-body operators have the form O = i=1 o^ [i] in first quantization, whereo ^(1)[i] is a single-particle operator acting on the ith particle. In the language of second quantization, they take the form ^(1) X (1) y O = hαjo^ jβi cαcβ; (13) α,β (1) R ∗ (1) with the matrix elements hαjo^ jβi = d1 d2 'α(1)h1jo^ j2i'β(2). Examples: ^ P 2 1. The kinetic energy operator T = i pi =(2m), describing independent particles, reads in second quantization X 2k2 T^ = ~ cy c ; (14) 2m k k k i.e. it is diagonal in the momentum basis.
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