5.9: Appendix I- Second Quantization
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Basis States and Creation/Annihilation Operators
Second quantization is a convenient scheme to label basis states of a many particle quantum system. We are ultimately interested in solutions of the many-body Schrödinger equation, ˆHΨ(x∗1,…,x∗N)=EΨ(x∗1,…,x∗N) where the Hamiltonian is ˆH=−ℏ22mN∑i=1∇2i+N∑j<kV(x∗j−x∗k). To the coordinate labels {x∗1,…x∗N} we may also append labels for internal degrees of freedom, such as spin polarization, denoted {ζ∗1,…,ζ∗N}. Since [ˆH,σ]=0 for all permutations σ∈S∗N, the many-body wavefunctions may be chosen to transform according to irreducible representations of the symmetric group S∗N. Thus, for any σ∈S∗N, Ψ(x∗σ(1),…,x∗σ(N))={1sgn(σ)}Ψ(x∗1,…,x∗N), where the upper choice is for Bose-Einstein statistics and the lower sign for Fermi-Dirac statistics. Here x∗j may include not only the spatial coordinates of particle j, but its internal quantum number(s) as well, such as ζ∗j.
A convenient basis for the many body states is obtained from the single-particle eigenstates {|α⟩} of some single-particle Hamiltonian ˆH∗0 , with ⟨x|α⟩=φ∗α(x) and ˆH∗0|α⟩=ε∗α|α⟩. The basis may be taken as orthonormal, ⟨α|α′⟩=δ∗αα′. Now define Ψ∗α∗1,…,α∗N(x∗1,…,x∗N)=1√N!∏αn∗α!∑σ∈S∗N{1sgn(σ)} φ∗α∗σ(1)(x∗1)⋯φ∗α∗σ(N)(x∗N). Here n∗α is the number of times the index α appears among the set {α∗1,…,α∗N}. For BE statistics, n∗α∈{0,1,2,…} , whereas for FD statistics, n∗α∈{0,1} . Note that the above states are normalized22: ∫ddx∗1⋯∫ddx∗N|Ψ∗α∗1⋯α∗N(x∗1,…,x∗N)|2=1N!∏αn∗α!∑σ,μ∈S∗N{1sgn(σμ)} N∏j=1∫ddx∗j φ∗α∗σ(j)(x∗j)φ∗α∗μ(j)(x∗j)=1∏αn∗α!∑σ∈S∗NN∏j=1δ∗α∗j,α∗σ(j)=1. Note that ∑σ∈S∗Nφ∗α∗σ(1)(x∗1)⋯φ∗α∗σ(N)(x∗N)≡per{φ∗α∗i(x∗j)}∑σ∈S∗Nsgn(σ)φ∗α∗σ(1)(x∗1)⋯φ∗α∗σ(N)(x∗N)≡det{φ∗α∗i(x∗j)}, which stand for permanent and determinant, respectively. We may now write Ψ∗α∗1⋯α∗N(x∗1,…,x∗N)=⟨x∗1,⋯,x∗N|α∗1⋯α∗N⟩, where |α∗1⋯α∗N⟩=1√N!∏αn∗α!∑σ∈S∗N{1sgn(σ)} |α∗σ(1)⟩⊗|α∗σ(2)⟩⊗⋯⊗|α∗σ(N)⟩. Note that |α∗σ(1)⋯α∗σ(N)⟩=(±1)σ|α∗1⋯α∗N⟩ , where by (±1)σ we mean 1 in the case of BE statistics and sgn(σ) in the case of FD statistics.
We may express |α∗1⋯α∗N⟩ as a product of creation operators acting on a vacuum |0⟩ in Fock space. For bosons, |α∗1⋯α∗N⟩=∏α(b†α)n∗α√n∗α!|0⟩≡|{n∗α}⟩, with [b†α,b†β]=0,[b†α,b†β]=0,[b†α,b†β]=δ∗αβ, where [∙,∙] is the commutator. For fermions, |α∗1⋯α∗N⟩=c†α∗1c†α∗2⋯c†α∗N|0⟩≡|{n∗α}⟩, with {c†α,c†β}=0,{c†α,c†β}=0,{c†α,c†β}=δ∗αβ, where {∙,∙} is the anticommutator.
Second Quantized Operators
Now consider the action of permutation-symmetric first quantized operators such as ˆT=−ℏ22m∑Ni=1∇2i and ˆV=∑Ni<jˆv(x∗i−x∗j). For a one-body operator such as ˆT, we have ⟨α∗1⋯α∗N|ˆT|α′1⋯α′N⟩=∫ddx∗1⋯∫ddx∗N(∏αn∗α!)−1/2(∏αn′α!)−1/2×∑σ∈S∗N(±1)σφ∗α∗σ(1)(x∗1)⋯φ∗α∗σ(N)(x∗N)N∑k=1ˆT∗iφ∗α′σ(1)(x∗1)⋯φ∗α′σ(N)=∑σ∈S∗N(±1)σ(∏αn∗α!n′α!)−1/2N∑i=1∏j(j≠i)δ∗α∗j,α′σ(j)∫ddx∗1φ∗α∗i(x∗1)ˆT∗1φ∗α′σ(i)(x∗1). One may verify that any permutation-symmetric one-body operator such as ˆT is faithfully represented by the second quantized expression, ˆT=∑α,β⟨α|ˆT|β⟩ψ†αψ†β, where ψ†α is b†α or c†α as the application determines, and ⟨α|ˆT|β⟩=∫ddx∗1φ∗α(x∗1)ˆT∗1φ∗β(x∗1). Similarly, two-body operators such as ˆV are represented as ˆV=12∑α,β,γ,δ⟨αβ|ˆV|γδ⟩ψ†αψ†βψ†δψ†γ, where ⟨αβ|ˆV|γδ⟩=∫ddx∗1∫ddx∗2φ∗α(x∗1)φ∗β(x∗2)v(x∗1−x∗2)φ∗δ(x∗2)φ∗γ(x∗1). The general form for an n-body operator is then ˆR=1n!∑α∗1⋯α∗nβ∗1⋯β∗n⟨α∗1⋯α∗n|ˆR|β∗1⋯β∗n⟩ψ†α∗n⋯ψ†α∗nψ†β∗n⋯ψ∗β∗1.
Finally, if the Hamiltonian is noninteracting, consisting solely of one-body operators ˆH=∑Ni=1ˆh∗i, then ˆH=∑αε∗αψ†αψ†α, where {ε∗α} is the spectrum of each single particle Hamiltonian ˆh∗i.