Quantum Field Theories for Quantum Many-Particle Systems; or "Second Quantization"

Transcription

1 Quantum Field Theories for Quantum Many-Particle Systems; or "Second Quantization" Outline 1) Bosons and Fermions 2) N-particle wave functions ("first quantization") 3) The method of quantized fields ("second quantization") Motivation Quantum field theory can be used as a mathematical technique for describing many-particle systems with identical particles. There are many applications: atomic physics many electron atoms nuclear physics many protons and neutrons condensed matter physics many atoms in quantum statistical mechanics; superfluids, metals, etc. This topic is not in Mandl and Shaw There are whole books devoted to this topic. I'm using Huang, Appendix A. 1

2 Consider a system with N identical particles. In the first-quantized theory, the wave function is Ψ( r 1 r 2 r 3 r N ; t ) where r i = the coordinate vector of particle i. The goal is to solve the time evolution, Or, equivalently, find the energy eigenstates, 2

3 The method of quantized fields /1/ In the second quantized theory we define a quantized field for the system of particles. The field obeys a quantum postulate, which is called the equal-time commutation relations: Boson field [ ψ(r), ψ (r') ] = δ 3 (r r') [ ψ(r), ψ(r') ] = 0 [ ψ (r), ψ (r') ] = 0 Fermion field { ψ(r), ψ (r') } = δ 3 (r r') { ψ(r), ψ(r') } = 0 { ψ (r), ψ (r') } = 0 [ A, B ] = AB BA { A, B } = AB + BA /2/ And we define the Hamiltonian operator H and the number operator N op. 3

4 The Hamiltonian in the second quantized theory is H = K + Ω K = ħ 2 /(2m) d 3 r ψ (r) 2 ψ(r) Ω = ½ d 3 r 1 d 3 r 2 ψ (r 1 ) ψ (r 2 ) v 12 ψ(r 2 ) ψ(r 1 ) where again v 12 = v(r 1, r 2 ). The Number Operator in the second quantized theory is N op = d 3 r ψ (r) ψ(r) and it is a constant in the time evolution. 4

5 The Big Theorem The second quantized theory is equivalent in all predictions to the first quantized theory. To prove this is not so easy, because the two theories look so different. H 1Q = H 2Q = The structures of the two theories are different. The trick is to prove that the predictions are the same. 5

6 1 The wave function (as defined from the 2nd quantized theory) has the correct exchange symmetry or antisymmetry. Ψ ( r 1 r 2 r N ; t ) 1/SQRT(N!) 0 ψ(r 1 ) ψ(r 2 ) ψ(r N ) t,n Consider the coordinate exchange, Ψ ( r 1 r j r i r N ; t ). For bosons, all the ψ(r k ) factors commute; so Ψ (...r j r i ) = Ψ ( r i r j ). For fermions, all the ψ(r k ) factors anticommute; so Ψ (...r j r i ) = Ψ ( r i r j ) ; it takes 2n+1 anticommutations to switch the order. 6

7 2 The time-dependent Schroedinger equation has the correct form. We define Ψ ( r 1 r 2 r N ; t ) = 1/SQRT(N!) 0 ψ(r 1 ) ψ(r 2 ) ψ(r N ) t,n Now calculate i ħ ( / t) Ψ = = 1/SQRT(N!) 0 ψ(r 1 ) ψ(r 2 ) ψ(r N ) i ħ ( / t) t,n = 1/SQRT(N!) 0 ψ(r 1 ) ψ(r 2 ) ψ(r N ) H t,n = H t,n Commute H to the left, commuting H past the ψ(r 1 ) factors. Note 0 H = 0 ( just the adjoint of H 0 = 0 ) 7

8 i ħ ( / t) Ψ = = 1/SQRT(N!) 0 ψ(r 1 ) ψ(r 2 ) ψ(r N ) H t,n Pull H over to the left, repeatedly commuting H past the ψ(r j ) factors. Note 0 H = 0 ( equiv. to H 0 = 0 ). Also, ψ(r j ) H = H ψ(r j ) + [ ψ(r j ), H ] where [ ψ(r), H ] = ħ 2 /(2m) 2 ψ(r) + X(r) ψ(r) and X(r) = d 3 r' ψ (r') v(r',r) ψ(r'). For each commutation we pick up a term ħ 2 /(2m) j 2 ψ(r j ) and a term X(r j ) ψ(r j ) ( ψ ψψ) Now pull the X(r) operators to the left; Note 0 X(r) = 0. Also, where Finally, ψ(r j ) X(r')= X(r') ψ(r j ) + [ ψ(r j ), X(r') ] [ ψ(r j ), X(r') ] = v(r j,r') ψ(r') ψ(r j ) i ħ ( / t) Ψ = ħ 2 /(2m) ( N 2 )Ψ + i<j v(r i, r j ) Ψ [QED] 8

9 3 Expansion in single-particle states Introduce a complete set of single-particle states { u α (r) ; α = } Now expand the quantized field in these functions, ψ(r) = u α (r) c α and ψ (r) = u α *(r) c α Commutators [ ψ(r), ψ (r') ] = δ 3 (r r') [ c α, c β ] = δ(α,β) That is, for bosons, [ c α, c β ] = δ(α,β) and [ c α, c β ] = 0 ; for fermions, { c α, c β } = δ(α,β) and { c α, c β } = 0. More precisely, α = a set of quantum numbers for a single particle; u α *(r) u β (r) d 3 r = δ(α,β) c α = u α *(r) ψ(r) d 3 r We've seen this before! Something new: anticommutation for fermions. 9

10 Non-relativistic quantum fields for identical particles ψ(r) = u α (r) c α and ψ (r) = u α *(r) c α where c α is the annihilation operator for a particle with quantum numbers α and c α is the creation operator. The Pauli exclusion principle for fermions, ( c α ) 2 = 0. Proof: { c α, c β } = 0 = c α c β + c β c α ; so for β = α, c α c α = 0. 10

11 4 Matrix elements agree. 11

12 12

13 One more thing We defined the fermionic operators c k and c k by anticommutators. Can we still interpret them as annihilation and creation operators? Read Mandl and Shaw Section 4.1. Review the bosonic annihilation and creation operators [ a k, a k' ] = δ(k,k') [ a k, a k' ] = 0 and [ a k, a k' ] = 0 N op = a k a k Therefore [ N, a k ] = a k a k a k a k a k a k = [ a k,a k ] a k = a k [ N, a k ] = a k and [ N, a k ] = + a k Therefore a k is an annihilation operator and a k is a creation operator. N op a N> = ( a + a N op ) N> = ( N 1 ) a N> Now consider fermionic operators { c k, c k' } = δ(k,k') { c k, c k' } = 0 and { c k, c k' } = 0 N op = c k c k Therefore [ N, c k ] = c k c k c k c k c k c k = c k c k c k = ( 1 c k c k ) c k = c k [ N, c k ] = c k and [ N, c k ] = + c k so again c k is annihilation and c k is creation. (Jordan and Wigner, 1928) 13

14 Homework due Friday, Feb. 3 Problems 11, 12, 13, 14 Problem 15: In the second quantized theory of 2 identical fermions, calculate 0 c α c β ψ (r 1 ) ψ (r 2 ) 0 where c α is the annihilation operator for a particle with wave function u α (r). 14