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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(x1,,xN)=EΨ(x1,,xN) where the Hamiltonian is ˆH=22mNi=12i+Nj<kV(xjxk). To the coordinate labels {x1,xN} we may also append labels for internal degrees of freedom, such as spin polarization, denoted {ζ1,,ζN}. Since [ˆH,σ]=0 for all permutations σSN, the many-body wavefunctions may be chosen to transform according to irreducible representations of the symmetric group SN. Thus, for any σSN, Ψ(xσ(1),,xσ(N))={1sgn(σ)}Ψ(x1,,xN), where the upper choice is for Bose-Einstein statistics and the lower sign for Fermi-Dirac statistics. Here xj may include not only the spatial coordinates of particle j, but its internal quantum number(s) as well, such as ζj.

A convenient basis for the many body states is obtained from the single-particle eigenstates {|α} of some single-particle Hamiltonian ˆH0 , with x|α=φα(x) and ˆH0|α=εα|α. The basis may be taken as orthonormal, α|α=δαα. Now define Ψα1,,αN(x1,,xN)=1N!αnα!σSN{1sgn(σ)} φασ(1)(x1)φασ(N)(xN). Here nα is the number of times the index α appears among the set {α1,,αN}. For BE statistics, nα{0,1,2,} , whereas for FD statistics, nα{0,1} . Note that the above states are normalized22: ddx1ddxN|Ψα1αN(x1,,xN)|2=1N!αnα!σ,μSN{1sgn(σμ)} Nj=1ddxj φασ(j)(xj)φαμ(j)(xj)=1αnα!σSNNj=1δαj,ασ(j)=1. Note that σSNφασ(1)(x1)φασ(N)(xN)per{φαi(xj)}σSNsgn(σ)φασ(1)(x1)φασ(N)(xN)det{φαi(xj)}, which stand for permanent and determinant, respectively. We may now write Ψα1αN(x1,,xN)=x1,,xN|α1αN, where |α1αN=1N!αnα!σSN{1sgn(σ)} |ασ(1)|ασ(2)|ασ(N). Note that |ασ(1)ασ(N)=(±1)σ|α1αN , 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 |α1αN as a product of creation operators acting on a vacuum |0 in Fock space. For bosons, |α1αN=α(bα)nαnα!|0|{nα}, with [bα,bβ]=0,[bα,bβ]=0,[bα,bβ]=δαβ, where [,] is the commutator. For fermions, |α1αN=cα1cα2cαN|0|{nα}, with {cα,cβ}=0,{cα,cβ}=0,{cα,cβ}=δαβ, where {,} is the anticommutator.

Second Quantized Operators

Now consider the action of permutation-symmetric first quantized operators such as ˆT=22mNi=12i and ˆV=Ni<jˆv(xixj). For a one-body operator such as ˆT, we have α1αN|ˆT|α1αN=ddx1ddxN(αnα!)1/2(αnα!)1/2×σSN(±1)σφασ(1)(x1)φασ(N)(xN)Nk=1ˆTiφασ(1)(x1)φασ(N)=σSN(±1)σ(αnα!nα!)1/2Ni=1j(ji)δαj,ασ(j)ddx1φαi(x1)ˆT1φασ(i)(x1). One may verify that any permutation-symmetric one-body operator such as ˆT is faithfully represented by the second quantized expression, ˆT=α,βα|ˆT|βψαψβ, where ψα is bα or cα as the application determines, and α|ˆT|β=ddx1φα(x1)ˆT1φβ(x1). Similarly, two-body operators such as ˆV are represented as ˆV=12α,β,γ,δαβ|ˆV|γδψαψβψδψγ, where αβ|ˆV|γδ=ddx1ddx2φα(x1)φβ(x2)v(x1x2)φδ(x2)φγ(x1). The general form for an n-body operator is then ˆR=1n!α1αnβ1βnα1αn|ˆR|β1βnψαnψαnψβnψβ1.

Finally, if the Hamiltonian is noninteracting, consisting solely of one-body operators ˆH=Ni=1ˆhi, then ˆH=αεαψαψα, where {εα} is the spectrum of each single particle Hamiltonian ˆhi.


This page titled 5.9: Appendix I- Second Quantization is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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