Chapter 1 Second Quantization 1.1 Creation and Annihilation Operators in Quan- tum Mechanics We will begin with a quick review of creation and annihilation operators in the non-relativistic linear harmonic oscillator. Let a and a† be two operators acting on an abstract Hilbert space of states, and satisfying the commutation relation a,a† = 1 (1.1) where by “1” we mean the identity operator of this Hilbert space. The operators a and a† are not self-adjoint but are the adjoint of each other. Let α be a state which we will take to be an eigenvector of the Hermitian operators| ia†a with eigenvalue α which is a real number, a†a α = α α (1.2) | i | i Hence, α = α a†a α = a α 2 0 (1.3) h | | i k | ik ≥ where we used the fundamental axiom of Quantum Mechanics that the norm of all states in the physical Hilbert space is positive. As a result, the eigenvalues α of the eigenstates of a†a must be non-negative real numbers. Furthermore, since for all operators A, B and C [AB, C]= A [B, C] + [A, C] B (1.4) we get a†a,a = a (1.5) − † † † a a,a = a (1.6) 1 2 CHAPTER 1. SECOND QUANTIZATION i.e., a and a† are “eigen-operators” of a†a. Hence, a†a a = a a†a 1 (1.7) − † † † † a a a = a a a +1 (1.8) Consequently we find a†a a α = a a†a 1 α = (α 1) a α (1.9) | i − | i − | i Hence the state aα is an eigenstate of a†a with eigenvalue α 1, provided a α = 0. Similarly,| ai† α is an eigenstate of a†a with eigenvalue α−+1, provided a†| αi 6 = 0. This also implies| i that | i 6 1 α 1 = a α (1.10) | − i √α | i 1 α +1 = a† α (1.11) | i √α +1 | i Let us assume that an α =0, n Z+ (1.12) | i 6 ∀ ∈ Hence, an α is an eigenstate of a†a with eigenvalue α n. However, α n< 0 if α<n, which| i contradicts our earlier result that all these− eigenvalues must− be non-negative real numbers. Hence, for a given α there must exist an integer n such that an α = 0 but an+1 α = 0, where n Z+. Let | i 6 | i ∈ 1 α n = an α a†a α n = (α n) α n (1.13) | − i an α | i ⇒ | − i − | − i k | ik where an α n = √α n (1.14) k | − ik − But 1 a α n = an+1 α =0 α = n (1.15) | − i − an α | i ⇒ k | ik In other words the allowed eigenvalues of a†a are the non-negative integers. Let us now define the ground state 0 , as the state annihilated by a, | i a 0 = 0 (1.16) | i Then, an arbitrary state n is | i 1 n = (a†)n 0 (1.17) | i √n! | i which has the inner product n m = δ n! (1.18) h | i n,m 1.1. CREATION AND ANNIHILATION OPERATORS IN QUANTUM MECHANICS3 In summary, we found that creation and annihilation operators obey a† n = √n +1 n +1 (1.19) | i | i a n = √n n 1 (1.20) | i | − i a†a n = n n (1.21) | i | i (1.22) and thus their matrix elements are m a† n = √n +1 δ m a n = √nδ (1.23) h | | i m,n+1 h | | i m,n−1 1.1.1 The Linear harmonic Oscillator The Hamiltonian of the Linear Harmonic Oscillator is P 2 1 H = + mω2X2 (1.24) 2m 2 where X and P , the coordinate and momentum Hermitian operators satisfy canonical commutation relations, [X, P ]= i~ (1.25) We now define the creation and annihilation operators a† and a as 1 mω P a = X + i (1.26) √2 ~ √ ~ r mω 1 mω P a† = X i (1.27) √ ~ − √ ~ 2 r mω which satisfy a,a† = 1 (1.28) Since ~ X = a + a† (1.29) r2mω mω~ a a† P = − (1.30) r 2 i the Hamiltonian takes the simple form 1 H = ~ω a†a + (1.31) 2 The eigenstates of the Hamiltonian are constructed easily using our results since all eigenstates of a†a are eigenstates of H. Thus, the eigenstates of H are the eigenstates of a†a, 1 H n = ~ω n + n (1.32) | i 2 | i 4 CHAPTER 1. SECOND QUANTIZATION with eigenvalues 1 E = ~ω n + (1.33) n 2 where n =0, 1,.... The ground state 0 is the state annihilated by a, | i 1 mω P a 0 X + i 0 = 0 (1.34) | i≡ 2 ~ √ ~ | i r mω Since d x P φ = i~ x φ (1.35) h | | i − dxh | i we find that ψ (x)= x 0 satisfies 0 h | i ~ d x + ψ (x) = 0 (1.36) mω dx 0 whose (normalized) solution is mω mω 1/4 x2 ψ (x)= e− 2~ (1.37) 0 π~ The wave functions ψ (x) of the excited states n are n | i 1 † n ψn(x) = x n = x (a ) 0 h | i √n! h | | i 1 mω n/2 ~ d n = x ψ0(x) (1.38) √n! 2~ − mω dx Creation and annihilation operators are very useful. Let us consider for instance the anharmonic oscillator whose Hamiltonian is P 2 1 H = + mω2X2 + λX4 (1.39) 2m 2 Let us compute the eigenvalues En to lowest order in perturbation theory in powers of λ. The first order shift ∆En is ∆E = λ n X4 n + O(λ2) n h | | i ~ 2 = λ n (a + a†)4 n + . 2mω h | | i ~ 2 = λ n a†a†aa n + other terms with two a′s and two a†′s + . 2mω h | | i ~ 2 = λ 6n2 +6n +3 + . (1.40) 2mω 1.1. CREATION AND ANNIHILATION OPERATORS IN QUANTUM MECHANICS5 1.1.2 Many Harmonic Oscillators It is trivial to extend these ideas to the case of many harmonic oscillators, which is a crude model of an elastic solid. Consider a system of N identical linear harmonic oscillators of mass M and frequency ω, with coordinates Q and { i} momenta Pi , where i = 1,...,N. These operators satisfy the commutation relations { } [Qj,Qk] = [Pj , Pk]=0, [Qj , Pk]= i~δjk (1.41) where j, k =1,...,N. The Hamiltonian is N N P 2 1 H = i + V Q Q (1.42) 2M 2 ij i j i=1 i i,j=1 X X where Vij is a symmetric positive definite matrix, Vij = Vji. We will find the spectrum (and eigenstates) of this system by changing vari- ables to normal modes and using creation and annihilation operators for the normal modes. To this end we will first rescale coordinates and momenta so as to absorb the particle mass M: xi = MiQi (1.43) Pi pi = p (1.44) √Mi 1 Uij = Vij (1.45) MiMj which also satisfy p [xj ,pj]= i~δjk (1.46) and N N p2 1 H = i + U x x (1.47) 2 2 ij i j i=1 i,j=1 X X We now got to normal mode variables by means of an orthogonal transformation −1 Cjk, i.e. [C ]jk = Ckj , N N xk = Ckj xj , pk = Ckj pj j=1 j=1 X X Ne eN CkiCji = δkj , CikCij = δkj (1.48) i=1 i=1 X X Sine the matrix Uij is real symmetric and positive definite, its eigenvalues, which 2 2 we will denote by ωk (with k = 1,...,N), are all non-negative, ωk 0. The eigenvalue equation is ≥ N 2 CkiCℓj Uij = ωkδkℓ, (no sum over k) (1.49) i,j=1 X 6 CHAPTER 1. SECOND QUANTIZATION Since the transformation is orthogonal, it preserves the commutation relations [xj , xk] = [pj , pk]=0, [xj , pk]= i~δjk (1.50) and the Hamiltoniane ise now diagonale e e e 1 N H = p2 + ω2 x2 (1.51) 2 j j j i=1 X We now define creation and annihilatione operatorse for the normal modes 1 i aj = √ωj xj + pj √ ~ ω 2 √ j † 1 i a = √ωj xej pej j √ ~ − ω 2 √ j ~ e † e xj = aj + aj s2ωj e ~ωj p = i a a† (1.52) j − 2 j − j r where, once again, e † † † [aj,ak] aj ,ak =0, aj ,ak = δjk (1.53) h i h i and the normal mode Hamiltonian takes the standard form N 1 H = ~ω a†a + (1.54) j j j 2 j=1 X † The eigenstates of the Hamiltonian are labelled by the eigenvalues of aj aj for each normal mode j, n ,...,n n . Hence, | 1 N i≡|{ j }i N 1 H n ,...,n = ~ω n + n ,...,n (1.55) | 1 N i j j 2 | 1 N i j=1 X where † N (a )nj n ,...,n = j 0,..., 0 (1.56) | 1 N i | i j=1 nj ! Y The ground state of the system, whichp we will denote by 0 , is the state in which all normal modes are in their ground state, | i 0 0,..., 0 (1.57) | i ≡ | i 1.1. CREATION AND ANNIHILATION OPERATORS IN QUANTUM MECHANICS7 Thus, the ground state 0 is annihilated by the annihilation operators of all normal modes, | i a 0 =0, j (1.58) j | i ∀ and the ground state energy of the system Egnd is N 1 E = ~ω (1.59) gnd 2 j j=1 X The energy of the excited states is N E(n1,...,nN )= ~ωjnj + Egnd (1.60) j=1 X We can now regard the state 0 as the vacuum state and the excited states | i n1,...,nN as a state with nj excitations (or particles) of type j.