Spring 2025 Homework Assignments
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No late homework accepted after Sunday, May 4
This week's homework will involve finding some constants to find the answers to the questions.
Provide references to the sources where you obtain the relevant numbers.
1. (a) Two particles with total spins 2 and 3/2 are in the angular momentum outer product state
|2 1⟩ |3/2 1/2⟩ . Find the expansion in terms of the total angular momentum eigenstates.
(b) Find the expansion of the total angular momentum state |1/2 1/2⟩ in terms of the s=2 and s=3/2
outer product states.
2. Find the energy eigenvalues Enℓ for a particle of mass m confined to a spherical box of radius a
with an infinite potential wall. Find 9 energy values corresponding to 1≤n≤3 and 0≤ℓ≤2.
Give your the results in multiples of the quantity ħ2π2/(8ma2), the ground state energy
of a cubic box of size 2a. (Yes, you will have to find the numerical values of the roots βnℓ.)
Consider the full Hydrogen wavefunction Ψnlm(r,θ,φ) = Rnl(r) Ylm(θ,φ). We know that if l≠l' or m≠m' the angular part of the wavefunctions Ylm(θ,φ) and Yl'm'(θ,φ) are orthonormal. So, for two wavefunctions with same l and m values, but different n values, we must have ∫ Rnl(r) Rn'l(r) r2 dr equal to zero. Choose your favorite n≠n' and l values for which you can find the radial functions from your textbook, and verify that they are orthogonal.
Consider three identical, non-interacting particles in a one-dimensional quantum box of size a.
(You can refer to the wavefunctions of one-particle states as ua(x), ub(x), etc.)
Write down all possible full (space + spin) 3-particle wavefunctions, for the ground state and the
first excited state, when the particles are
o s=1/2 Fermions,
o s=0 Bosons.
A quantum system has two energy states, one with energy ε and the other with 2ε. The system has three identical, non-interacting particles in it. The system is known to be at temperature T. (Although temperature is a quantity defined for macroscopic systems, you can think of the existence of many systems of this type, and we are looking at the statistics of those.) Calculate and plot the average energy of the system as a function of kBT/ε when o the particles are Fermions (spin 1/2) o the particles are Bosons (spin 0). Discuss the limits for T∼0 and T→∞.
Consider the three-Fermion problem given in Homework 4.
(a) List the possible three-particle states as |ψ1>, |ψ2>, etc.
(b) Write down the matrix representation of the Hamiltonian H in the basis of the states in (a).
(c) Consider an operator N with eigenfunctions as states in part (a), and eigenvalues as the number of
particles in the ground state. i.e. N |ψi> = ni |ψi> where ni is the number of particles in the
ground state for the three-particle state |ψi>.
Write down the matrix representation of N in the basis of the states in (a).
(d) The system is at thermal equilibrium at temperature T so that its density matrix is ρ = exp(-H/kBT)/Z
where Z is necessary for normalization: Z = Tr(exp[-H/kBT])
Write down the matrix representation of ρ in the basis of the states in (a).
(e) Find the ensemble average [N] = Tr(ρN) and plot it as a function of kBT/ε.
Consider the three dimensional quantum harmonic oscillator with the potential energy
V = k r2/2
where r2 = x2 + y2 + z2.
(a) Using separation of variables in the Cartesian coordinate system, find the energy levels
of the system.
(b) Find the expectation value <V2> for a general energy eigenstate of this system.
Consider a particle of mass m in a potential
{ ∞ for x < 0
V(x) = {
{ α |x| for x > 0
(a) Obtain an estimate for the ground state energy using a variational approach.
(b) Obtain another estimate using the WKB method.
(c) Obtain the exact value of the ground state energy. (You will need to find the
value of the zero of the Ai(x) function by some means - numerically, reading off a
graph or a table, etc.)
Each homework is to be placed as a single PDF document (which may contain photos of your work) and submitted through moodle.
Please name the file in the format:
No late homework accepted after Sunday, May 4
Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.