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Theory of Many-Particle Systems

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Theory of Many-Particle Systems Theory of Many-Particle Systems Lecture notes for P654, Cornell University, spring 2005. c Piet Brouwer, 2005. Permission is granted to print and copy these notes, if kept together with the title page and this copyright notice. Chapter 1 Preliminaries 1.1 Thermal average In the literature, two types of theories are constructed to describe a system of many particles: Theories of the many-particle ground state, and theories that address a thermal average at temperature T . In this course, we'll limit ourselves to finite-temperature thermal equilibria and to nonequilibrium situations that arise from a finite-temperature equilibrium by switch- ing on a time-dependent perturbation in the Hamiltonian. In all cases of interest to us, results at zero temperature can be obtained as the zero temperature limit of the finite-temperature theory. Below, we briefly recall the basic results of equilibrium statistical mechanics. The thermal average of an observable A is defined as A = tr A^ρ,^ (1.1) h i where ρ^ is the density matrix and A^ is the operator corresponding to the observable A. The density matrix ρ^ describes the thermal distribution over the different eigenstates of the system. The symbol tr denotes the \trace" of an operator, the summation over the expectation values of the operator over an orthonormal basis set, tr Aρ = n Aρ n : (1.2) h j j i n X The basis set n can be the collection of many-particle eigenstates of the Hamiltonian H^ , fj ig or any other orthonormal basis set. In the canonical ensemble of statistical mechanics, the trace is taken over all states with N particles and one has 1 ρ^ = e−H^ =T ; Z = tr e−H^ =T ; (1.3) Z 1 2 CHAPTER 1. PRELIMINARIES where H^ is the Hamiltonian. Using the basis of N-particle eigenstates n of the Hamiltonian j i H^ , with eigenvalues E , the thermal average A can then be written as n h i e−En=T n A^ n A = n h j j i: (1.4) h i e−En=T P n In the grand-canonical ensemble, the tracePis taken over all states, irrespective of particle number, and one has 1 ρ = e−(H^ −µN^)=T ; Z = tr e−(H^ −µN^)=T : (1.5) Z Here µ is the chemical potential and N is the particle number operator. Usually we will include the term µN^ into the definition of the Hamiltonian H^ , so that expressions for the − thermal average in the canonical and grand canonical ensembles are formally identical. 1.2 Schr¨odinger, Heisenberg, and interaction picture There exist formally different but mathematically equivalent ways to formulate the quantum mechanical dynamics. These different formulations are called \pictures". For an observable A^, the expectation value in the quantum state is j i A = A^ : (1.6) h j j i The time-dependence of the expectation value of A can be encoded through the time- dependence of the quantum state , or through the time-dependence of the operator A^, or through both. You can find a detailed discussion of the pictures in a textbook on quantum mechanics, e.g., chapter 8 of Quantum Mechanics, by A. Messiah, North-Holland (1961). 1.2.1 Schr¨odinger picture In the Schr¨odinger picture, all time-dependence is encoded in the quantum state (t) ; operators are time independent, except for a possible explicit time dependence.1 Thej time-i evolution of the quantum state (t) is governed by a unitary evolution operator U^(t; t ) j i 0 that relates the quantum state at time t to the quantum state at a reference time t0, (t) = U^(t; t ) (t ) : (1.7) j i 0 j 0 i 1An example of an operator with an explicit time dependence is the velocity operator in the presence of a time-dependent vector potential, 1 e v^ = i~r A(r; t) : m − − c 1.2. SCHRODINGER,¨ HEISENBERG, AND INTERACTION PICTURE 3 Quantum states at two arbitrary times t and t0 are related by the evolution operator U^(t; t0) = y 0 U^(t; t0)U^ (t ; t0). The evolution operator U^ satisfies the group properties U^(t; t) = 1^ and U^(t; t0) = U^(t; t00)U^(t00; t0) for any three times t, t0, and t00, and the unitarity condition U^(t; t0) = U^ y(t0; t): (1.8) The time-dependence of the evolution operator U^(t; t0) is given by the Schr¨odinger equation @U^(t; t ) i~ 0 = H^ U^(t; t ): (1.9) @t 0 For a time-independent Hamiltonian H^ , the evolution operator U^(t; t0) is an exponential function of H^ , U^(t; t0) = exp[ iH^ (t t0)=~]: (1.10) − − If the Hamiltonian H^ is time dependent, there is no such simple result as Eq. (1.10) above, and one has to solve the Schr¨odinger equation (1.9). 1.2.2 Heisenberg picture In the Heisenberg picture, the time-dependence is encoded in operators A^(t), whereas the quantum state is time independent. The Heisenberg picture operator A^(t) is related to the Schr¨odingerj picturei operator A^ as A^(t) = U^(t0; t)A^U^(t; t0); (1.11) where t0 is a reference time. The Heisenberg picture quantum state has no dynamics and is equal to the Schr¨odinger picture quantum state (t ) at the referencej i time t . The j 0 i 0 time evolution of A^(t) then follows from Eq. (1.9) above, dA^(t) i @A^(t) = [H^ (t); A^(t)] + ; (1.12) dt ~ − @t where H^ (t) = U^(t0; t)H^ (t0)U^(t; t0) is the Heisenberg representation of the Hamiltonian and the last term is the Heisenberg picture representation of an eventual explicit time dependence of the observable A. The square brackets [ ; ] denote a commutator. The Heisenberg picture · · − Hamiltonian H^ (t) also satisfies the evolution equation (1.12), dH^ (t) i @H^ @H^ (t) = [H^ (t); H^ (t)] + = : dt ~ − @t @t 4 CHAPTER 1. PRELIMINARIES For a time-independent Hamiltonian, the solution of Eq. (1.12) takes the familiar form ^ ~ ^ ~ A^(t) = eiH(t−t0)= Ae^ −iH(t−t0)= : (1.13) You can easily verify that the Heisenberg and Schr¨odinger pictures are equivalent. Indeed, in both pictures, the expectation values (A)t of the observable A at time t is related to the operators A^(t ) and quantum state (t ) at the reference time t by the same equation, 0 j 0 i 0 (A) = (t) A^ (t) t h j j i = (t ) U^(t ; t)A^U^(t; t ) (t ) h 0 j 0 0 j 0 i = A^(t) : (1.14) h j j i 1.2.3 Interaction picture The interaction picture is a mixture of the Heisenberg and Schr¨odinger pictures: both the quantum state (t) and the operator A^(t) are time dependent. The interaction picture is j i usually used if the Hamiltonian is separated into a time-independent unperturbed part H^0 and a possibly time-dependent perturbation H^1, H^ (t) = H^0 + H^1: (1.15) In the interaction picture, the operators evolve according to the unperturbed Hamiltonian H^0, iH^0(t−t0)=~ −iH^0(t−t0)=~ A^(t) = e A^(t0)e ; (1.16) whereas the quantum state evolves according to a modified evolution operator U^ (t; t ), j i I 0 iH^0(t−t0)=~ U^I (t; t0) = e U^(t; t0): (1.17) You easily verify that this assignment leads to the same time-dependent expectation value (1.14) as the Schr¨odinger and Heisenberg pictures. The evolution operator that relates interaction picture quantum states at two arbitrary times t and t0 is 0 0 iH^0(t−t0)=~ 0 −iH^0(t −t0)=~ U^I (t; t ) = e U^(t; t )e : (1.18) The interaction picture evolution operator also satisfies the group and unitarity properties. A differential equation for the time dependence of the operator A^(t) is readily obtained from the definition (1.16), dA^(t) i @A^(t) = [H^ ; A^(t)] + : (1.19) dt ~ 0 − @t 1.2. SCHRODINGER,¨ HEISENBERG, AND INTERACTION PICTURE 5 The time dependence of the interaction picture evolution operator is governed by the interaction- picture representation of the perturbation H^1 only, @U^ (t; t ) i~ I 0 = H^ (t)U^ (t; t ); (1.20) @t 1 I 0 where iH^0(t−t0)=~ −iH^0(t−t0) H^1(t) = e H^1e : (1.21) The solution of Eq. (1.20) is not a simple exponential function of H^1, as it was in the case of the Schr¨odinger picture. The reason is that the perturbation H^1 is always time dependent 2 in the interaction picture, see Eq. (1.21). (An exception is when H^1 and H^0 commute, but in that case H^1 is not a real perturbation.) However, we can write down a formal solution of Eq. (1.20) in the form of a \time-ordered" exponential. For t > t0, this time-ordered exponential reads ~ t 0 ^ 0 −(i= ) t dt H1(t ) U^I (t; t0) = Tte 0 : (1.22) R In this formal solution, the exponent has to be interpreted as its power series, and in each term the \time-ordering operator" Tt arranges the factors H^ with descending time arguments. 0 For example, for a product of two operators H^1 taken at different times t and t one has H^ (t)H^ (t0) if t > t0, T H^ (t)H^ (t0) = 1 1 (1.23) t 1 1 H^ (t0)H^ (t) if t0 > t.
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