Further Quantum

Question 1

Define a wave function that doesn’t distinguish between particles

Answer 1

Question 2

Define Q_λψ(r₁,…,r_n)

Answer 2

Question 3

Prove that Q_λ satisfies ψ(r₁,…,r_n) = λ(π)ψ(r_π(1),…,r_π(n)) ∀π∈S_n, and is a projection

Answer 3

Question 4

What is a boson?

Answer 4

When λ(π)=1, spin is integral

Question 5

Choosing a basis of N separable wave functions, describing n bosons, how many states are there? Prove it

Answer 5

Question 6

What is a Fermion?

Answer 6

When λ(π) = ε(π), spin is not integral

Question 7

What is the Pauli Exclusion Principle?

Answer 7

That no 2 Fermions can simultaneously occupy the same state. Proof as otherwise Q_λψ = 0

Question 8

Choosing a basis of N separable wave functions describing n Fermions, how many states are there? Prove it

Answer 8

Question 9

What is the Schrödinger Picture?

Answer 9

Question 10

What is the Heisenberg Picture?

Answer 10

Question 11

What is the Interaction Picture?

Answer 11

Question 12

Prove that the expectation is the same in all quantum pictures

Answer 12

Question 13

By what equations is the time evolution of wave functions and observables governed in the Interaction Picture? Prove it

Answer 13

Question 14

Use an equation governing time evolution in the interaction picture to set up an expansion for ψ_t

Answer 14

Question 15

What is the Feynman-Dyson Expansion? Prove it

Answer 15

Question 16

What is Fermi’s Golden Rule? Prove it

Answer 16

Question 17

Set up Rayleigh-Schrödinger Perturbation Theory

Answer 17

Question 18

If φ₁,…,φ_D form an orthonormal basis with H₀φ_r = E₀φ_r and ψ₀ = Σc_rφ_r, then what? Prove it

Answer 18

Question 19

What is the k’th order correction to the energy, E^(k)? Prove it

Answer 19

Question 20

Prove: <ψ_u|H’ψ_v> = (E_u-E_v)/(u-v)

Answer 20

Question 21

Expand (E_u-E_v)/(u-v)

Answer 21

Question 22

If ψ_α, α=0,1,… is an orthonormal basis such that H₀ψ_α = E_αψ₀ and E_α ≠ E₀ for α≠0, then what is ψ’ and E’’? Prove it

Answer 22

Question 23

Define the Rayleigh Quotient

Answer 23

Question 24

For a subset K of the Hilbert Space of wave functions, what is the result behind Variational methods? Prove it

Answer 24

^d/_du(f_H(ψ+uφ))|_u=0 = 0 ∀φ∈K if and only if <φ|H-f_H(ψ)|ψ> = 0 ∀φ∈K ie: Critical value of f_H(φ) is an eigenvalue with the critical point being the eigenvector

Question 25

State and prove the Virial Theorem

Answer 25

Question 26

Using a Variational approach, if f_H(ψ) > -∞, then what is the ground state energy?

Answer 26

E_ground = inf_ψ∈K{0} f_H(ψ), obtained at ψ_ground

Question 27

If ψ₀ is the ground state of H₀ in {ψ_u}, prove that Variational methods will give a better approximation for the energy than regular perturbation theory

Answer 27

Question 28

What is the k’th lowest energy eigenstate from a Variational point of view?

Answer 28

If it is attained, then: inf_{K_k} {max{f_H(ψ) : ψ∈K_k} : dim(K_k)=k}

Question 29

If V_j is a basis of K, then what provides an upper bound for the k’th lowest energy (k’th lowest eigenvalue of H)? Prove it

Answer 29

If Ē_k is the k’th lowest root of det(i|HV_k> - Ei|V_k>) = 0, then Ē_k ≥ E_k