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Quantum Operators in the Fock Space and Wave Function Languages

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Quantum Operators in the Fock Space and Wave Function Languages Quantum Operators in the Fock Space and Wave Function Languages A quantum operator acting on identical bosons can be described in terms of N{particle wave functions (the first-quantized formalism) or in terms of the creation and annihilation operators in the Fock space (the second-quantized formalism). In these notes I explain how to translate between the two formalisms. The Fock space formalism is explained in my notes on free fields, harmonic oscillators, and identical bosons. On pages 8{12 of those notes, I define the bosonic Fock space as the Hilbert space of states containing arbitrary numbers of identical bosons, 1 M F = H(N identical bosons); (1) N=0 then, starting with an arbitrary basis of one-particle states jαi I build the occupation-number basis jfnαgi for the whole Fock space, and eventually define the creation operators and the annihilation operators in terms of that occupation number basis, y def p 0 a^α fnβg = nα + 1 fnβ = nβ + δαβg ; ( p E (2) 0 def nα fnβ = nβ − δαβg for nα > 0, a^α fnβg = 0 for nα = 0. In the present notes, I shall start by translating these definition to the language of N{ y boson wave function. And then I shall use the wave-function definitions of thea ^α anda ^α operators to translate the more complicated one-body, two-body, etc., operators between the Fock-space and the wave-function languages. 1 Creation and Annihilation Operators in the Wave Function Language First, a quick note on multi-boson wave functions. A wave function of an N{particle state depends on all N particles' positions, (x1; x2;:::; xN ), where by abuse of notations th each xi includes all the independent degrees of freedom of the i particle, xi = (xi; yi; zi; spini; isospini;:::): Moreover, the wave function must be totally symmetric WRT any permutations of the iden- tical bosons, (any permutation of the x1;:::; xN ) = (x1;:::; xN ): (3) This Bose symmetry plays essential role in the wave-function-language action of the creation and annihilation operators. Definitions: Let the φα(x) be the wave function of the one-particle states α which we want y to be created by thea ^α operators and annihilated by thea ^α operators. Then, given an N{boson state jN; i with a totally symmetric wave function (x1;:::; x1), we construct the 0 00 totally symmetric wave functions (x1;:::; xN+1) and (x1;:::; xN−1) of the (N + 1){ 0 y 00 boson state jN + 1; i =a ^α jN; i and the (N − 1){boson state jN − 1; i =a ^α jN; i according to: N+1 0 1 X (x1;:::; xN+1) = p φα(xi) × (x1;:::; 6xi;:::; xN+1); (4) N + 1 i=1 p Z 00 3 ∗ (x1;:::; xN−1) = N d xN φα(xN ) × (x1;:::; xN−1; xN ): (5) y 0 In particular, for N = 0 the statea ^α j0i has (x1) = φα(x1), while for N = 1 the state 00 def a^α jβi has (no arguments) = hφαj i. Also, for N = 0 we simply definea ^α j0i = 0. y Let me use eqs. (4) and (5) as definitions of the creation operatorsa ^α and the annihilation operatorsa ^α. To verify that these definitions are completely equivalent to the definitions (2) in terms of the occupation-number basis, I am going to prove the following lemmas: 2 y Lemma 1: The creation operatorsa ^α defined according to eq. (4) are indeed the hermitian conjugates of the operatorsa ^α defined according to eq. (5). Lemma 2: The operators (4) and (5) obey the bosonic commutation relations y y y [^aα; a^β] = 0; [^aα; a^β] = 0; [^aα; a^β] = δαβ : (6) Lemma 3: Let φαβ···!(x1; x2 :::; xN ) be the N{boson wave function of the state 1 jα; β; ··· ;!i = p a^y ··· a^y a^y j0i (7) T ! β α y where the creation operatorsa ^α act according to eq. (4) while T is the number of trivial permutations between coincident entries of the list (α; β; : : : ; !) (for example, α $ β when Q α and β happen to be equal). In terms of the occupation numbers nγ, T = γ nγ!. Then distinct permutations of (α,β;:::;!) 1 X φαβ···!(x1; x2 :::; xN ) = p φα~(x1) × φ ~(x2) × · · · × φ!~ (xN ) D β (~α;β;:::;~ !~) (8) all N! permutations of (α,β;:::;!) 1 X = p φα~(x1) × φ ~(x2) × · · · × φ!~ (xN ); T × N! β (~α;β;:::;~ !~) where D = N!=T is the number of distinct permutations. In other words, the state (7) is precisely the symmetrized state of N bosons in individual states jαi ; jβi ;:::; j!i. Together, the lemmas 1{3 establish that the definitions (4) and (5) of the creation and annihilation operators completely agree with the definitions (2)of the same operators in terms of the occupation number basis. 3 y Proof of Lemma 1: To prove that the operatorsa ^α anda ^α are hermitian conjugates of each other, we need to compare their matrix elements and verify that for any two states jN; i and jN;e ei in the Fock space we have y ∗ hN;e ej a^α jN; i = hN; j a^α jN;e ei : (9) y Since thea ^α always lowers the number of particles by 1 while thea ^α always raises it by 1, it is enough to check this equation for Ne = N − 1 | otherwise, we get automatic zero on both sides of this equation. 00 00 Let (x1;:::; xN−1) be the wave function of the state jN − 1; i =a ^α jN; Ψi according to eq. (5). Then, on the LHS of eq. (9) we have 00 hN − 1; ej a^α jN; i = N − 1; ejN − 1; Z Z 3 3 ∗ 00 = d x1 ··· d xN−1 e (x1;:::; xN−1) × (x1;:::; xN−1) Z Z 3 3 ∗ = d x1 ··· d xN−1 e (x1;:::; xN−1) × p Z 3 ∗ × N d xN φα × (x1;:::; xN ) p Z Z 3 3 ∗ ∗ = N d x1 ··· d xN e (x1;:::; xN−1) × φα(xN ) × (x1;:::; xN ): (10) 0 0 y Likewise, let e (x1;:::; xN ) be the wave function of the state N; e =a ^α jN − 1; ei ac- cording to eq. (4). Then the matrix element on the RHS of eq. (9) becomes Z Z y 3 3 ∗ 0 hN; j a^α jN − 1; ei = d x1 ··· d xN (x1;:::; xN ) × e (x1;:::; xN ) Z Z 3 3 ∗ = d x1 ··· d xN (x1;:::; xN ) × N 1 X × p φα(xi) × e(x1;::: 6xi;:::; xN ) N i=1 N Z Z 1 X 3 3 ∗ = p d x1 ··· d xN (x1;:::; xN ) × N i=1 × φα(xi) × e(x1;::: 6xi;:::; xN ): (11) By bosonic symmetry of the wavefunctions and e, all terms in the sum on the RHS are 4 equal to each other. So, we may replace the summation with a single term | say, for i = N | and multiply by N, thus Z Z y N 3 3 ∗ hN; j a^ jN − 1; ei = p × d x1 ··· d xN (x1;:::; xN )×φα(xN )× e(x1;::: xN−1): α N (12) By inspection, the RHS of eqs. (10) and (12) are complex conjugates of each other, thus y ∗ hN − 1; ej a^α jN; i = hN; j a^α jN − 1; ei : (9) This completes the proof of Lemma 1. Proof of Lemma 2: Let's start by verifying that the creation operators defined according y y to eq. (4) commute with each other. Pick any two such creation operatorsa ^α anda ^β, and 000 pick any N-boson state jN; i. Consider the (N + 2)-boson wavefunction (x1;:::; xN+2) 000 y y of the state jN + 2; i =a ^αa^β jN; i. Applying eq. (4) twice, we immediately obtain φα(xi) × φ (xj) × 000 1 X β (x1;:::; xN+2) = p (N + 1)(N + 2) i;j=1;:::;N+2 × (x1;:::; xN+2 except xi; xj): i6=j (13) On the RHS of this formula, interchanging α $ β is equivalent to interchanging the summa- y y tion indices i $ j | which has no effect on the sum. Consequently, the statesa ^αa^β jN; i y y anda ^βa^α jN; i have the same wavefunction (13), thus y y y y a^αa^β jN; i =a ^βa^α jN; i : (14) Since this is true for any N and any totally-symmetric wave function , this means that the y y creation operatorsa ^α anda ^β commute with each other. Next, let's pick any two annihilation operatorsa ^α anda ^β defined according to eq. (5) and show that they commute with each other. Again, let jN; i be an arbitrary N-boson 5 state . For N < 2 we have y a^αa^β jN; i = 0 =a ^βa^α jN; i ; (15) so let's focus on the non-trivial case of N ≥ 2 and consider the (N − 2)-boson wavefunction 0000 0000 of the state jN − 2; i =a ^αa^β jN; i. Applying eq. (5) twice, we obtain ∗ ∗ Z Z φ (xN ) × φ (xN−1) × 0000 p 3 3 α β (x1;:::; xN−2) = N(N − 1) d xN d xN−1 × (x1;:::; xN−2; xN−1; xN ): (16) On the RHS of this formula, interchanging α $ β is equivalent to interchanging the integrated-over positions of the N th and the (N − 1)th boson in the original state jN; i. Thanks to bosonic symmetry of the wave-function , this interchange has no effect, thus a^αa^β jN; i =a ^βa^α jN; i : (17) Therefore, when the annihilation operators defined according to eq.
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