1. Consider the simple harmonic oscillator

Ĥ = \frac{p^{2}}{2m} + \frac{mω^{2}x^{2}}{2}

Write it in “second quantized” form , by expressing x and p in terms of creation and annihilation operators.

Ĥ = h̄ω(a^{†}a + \frac{1}{2}).

(i) Derive the equation of motion for a and show that

a(t) = a(0)e^{−iωt}.

Repeat this for a^{†}. (ii) Using the relationship

a^{†} = \frac{1}{\sqrt{2h̄}} [\frac{p̂}{\sqrt{mω}} + ix̂\sqrt{mω}] .

use the results of (i) to show that

p̂(t) = p̂(0) cos ωt − mωx̂(o)sin ωt
x̂(t) = x̂(0) cos ωt + \frac{p̂(0)}{mω}sin ωt.

(iii) The above operator expressions appear identical to the classical equations of motion.

Why?

2. Show that the Heisenberg spin operator S can be written in second quantized form as

S_{x} = \frac{1}{2}(c_{↑}^{†}c_{↓} + c_{↓}^{†}c_{↑}),
S_{y} = −\frac{i}{2}(c_{↑}^{†}c_{↓} − c_{↓}^{†}c_{↑}),
S_{z} = \frac{1}{2}(c_{↑}^{†}c_{↑} − c_{↓}^{†}c_{↓}).

Check the commutation relations!

3. Consider a system of fermions created by the field ψ ^{†} (r) interacting under the Yukawa potential

V (r) = \frac{Ae^{−λr}}{4πr} .

(i) Write the Hamiltonian in second quantized form, using the position basis.

(ii) Write the Hamiltonian in second quantized notation in the momentum basis, where

ck⃗ ^{†} = ∫ d^{3}rψ^{†}(r⃗)e^{ik⃗.r⃗}.

You will find it helpful to derive the Fourier representation

V (r) = ∫ \frac{d^{3}q}{(2π)^{3}} e^{iq⃗.r⃗} \frac{A}{(q^{2} + λ^{2})}.

4. For the spin 1/2 fermions considered in question 3, prove the “Hubbard Stratonovich” identity

e^{−a(n_{↑}−1/2)(n_{↓}−1/2)} = \frac{1}{2}e^{−a/4} ∑_{σ=±1} e^{aσ(n_{↑}−n_{↓})}

where cosha = e^{\frac{a}{4}} (Hint: consider the matrix representation of the operator on the left hand side in the four dimensional Fock space of the up and down electrons.) We shall later see how, considering a ∼ δt to be an infinitesimal time interval, we can use this relation to treat interacting spins as spins moving in a fluctuating Ising magnetic field.

6. Let a_{α}^{†} , a_{α} be Fermion creation and annihilation operators, and assume

H_{o} = ∑_{α} ϵ_{α}a_{α}^{†}a_{α}.

(i) Compute T r[e^{βH_{o}}a_{α}^{†}a_{α}]

(ii) Compute Tr [e^{−βH_{o}}a_{α}a_{α}^{†}].

(iii) Compute T r[e^{−βH_{o}}a_{α}a_{α}^{†}]. Using the identity e^{−βH_{o}}a_{α}^{†}a_{α} = −T \frac{∂}{∂ϵ_{α}} e^{−βH_{o}} check your answer to (i).

(iv) Confirm that the expectation

⟨n_{a}⟩ = Z^{−1}T r[e^{−βH_{o}}a_{α}^{†}a_{α}] = f(ϵ_{α})

where f(x) is the Fermi function.

(v) Repeat the above procedure assuming that a_{a}^{†} and a_{a} are boson operators. What are the restrictions on the values of ϵ_{a} and what happens when one of these energies ϵ_{a} becomes zero?

(vi) Calculate

( \begin{matrix}a_{α}^{†}(τ) \\ a_{α}(τ) \end{matrix} ) = e^{τH_{o}} ( \begin{matrix}a_{α}^{†} \\ a_{α} \end{matrix} ) e^{−τH_{o}}

4. Let γ be an oriented path from a to b in the complex plane. If A(z) is an operator valued function of z, then define the line integral

∫_{γ} A(z)dz ≡ ∑_{i=1}^{N} A(z_{i})(z_{i+1} − z_{i})

The time ordered product along the path γ orders the operators along γ. In the example

T_{γ}(A(z_{3})A(z_{2})A(z_{N})) = A(z_{N})A(z_{3})A(z_{1})

Prove that

(T_{γ} exp \frac{1}{i} ∫_{γ} H(τ)dτ )^{†} = (T_{γ} exp \frac{1}{i} ∫γ̃ H(τ)dτ )

where γ̃ is the path γ traversed in the reverse direction (i.e. if γ goes from a to b, then γ̃ goes from b to a.)

6. Define an antitime ordering operator T̃

T̃[A(t_{1})B(t_{2})] = \{ \begin{matrix}A(t_{1})B(t_{2}) & (t_{1} < t_{2}) \\ B(t_{2})A(t_{1}) & (t_{1} > t_{2}) \end{matrix}  (1)

Define

Ũ(t, t_{o}) = T̃ exp \frac{1}{i} ∫_{t_{o}}^{t} H(τ)dτ

What is the differential equation satisfied by Ũ(t,0)?

10. Consider two different representations of a Fermion Fock space defined by the creation operators

ψ^{†}a = c_{β}^{†}⟨β|α⟩

(i) Show that ψ_{α}^{†}, c_{β}^{†} = ⟨β|α⟩.

(ii) Show that ⟨0|ψ_{α}^{†}c_{β}^{†} = ⟨β|α⟩.

(III) Prove that if

|ϕ⟩ = c_{β_{N}}^{†} c_{β_{N−1}}^{†} … c_{β_{1}}^{†}|0⟩

is a state of N fermions that ϕ(α_{1}, α_{2} … α_{N}) = ⟨0|ϕ_{α_{1}}ϕ_{α_{2}} … ϕ_{α_{N}} |ϕ⟩ is a Slater determinant

ϕ(α_{1}, α_{2} … α_{N}) = det⟨α_{i}|β_{j}⟩

(iv) If a ≡ r⃗ and β ≡ k⃗ so that Ψ^{†}(r⃗) creates an electron in the position eigenstate and c† ; k⃗ creates an electron in a momentum eigenstate, what is the overlap matrix element ⟨r⃗|k⃗⟩, for states normalized in a box of volume V? Explicitly write out the Slater determinant in the position basis for a three particle state ϕ = |k⃗_{3}, k⃗_{2}, k⃗_{1}⟩.

11. (i) If M = M^{†} is an Hermitian N dimensional matrix, c_{i}^{†} and c_{i} (i = 1 … N) are creation and annihilation operators for Fermions (i.e.c_{i}, c_{j}^{†} = δ_{ij}, show that

c_{i}^{†}(t) = [e^{−M̂t}c^{†}e^{M̂t}]_{j} = c_{j}^{†} [e^{−M} ]_{ji} (M = c_{j}^{†}M_{jk}c_{k})

(Hint consider the differential equation satisfied by c^{†}(t)).

7. let U_{I}(t,t_{0}) be the evolution operator in the interaction picture and U_{S}(t,t_{0}) the corresponding operator in the Schrd̈inger picture. Prove that

U_{I}(t, 0) = e^{iH_{o}t}U_{S}(t, 0)