Spring 2026 Homework Assignments
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No late homework accepted after Monday, May 4
(Assistants may set earlier deadlines for late homework!)
(a) An electron is inside a bubble which it cannot escape. The coefficient of surface
tension of the bubble is σ. What is the radius R of the bubble?
(b) A particle has angular momentum ℓ = 1. Its projection on some axis is m = 0.
Another axis is at the angle θ to the first one. Find the probability that the projection
of the momentum on the second axis is ħ, 0, or -ħ. Repeat for the cases m = 1, -1.
Two non-interacting electrons are placed in the n=1, l=0, m=0 and n=2, l=1, m=0 states of the Hydrogen atom potential.
(This then results in an H- ion. Ignore whether such a system will be stable or not.)
(a) What is the total energy of the system?
(b) Calculate the quantity (r1 - r2)2 for the particles when they are in the
singlet and triplet spin states.
[Hint: When calculating (r1 - r2)2 = r12 + r22 - 2 r1 ⋅ r2 you can show easily (please do) that the
angle γ between the vectors r1 and r2 obeys the relation
cos γ = cos θ1 cos θ2 + sin θ1 sin θ2 cos (φ1 - φ2)
where θi and φi are the spherical coordinate angles associated with the vector ri. ]
(a) N non-interacting electrons are placed in a three-dimensional harmonic oscillator potential.
Using the separation of variables in rectangular coordinates, identify the energy states associated
with the single-particle states.
(b) Find the Fermi energy (the maximum energy of the particles in the ground state).
(You can assume that ℏω is very small compared to the system energy so that summations
may be approximated by integrals.)
(c) What is the total energy of the system with N particles?
(a) A quantum system has three energy levels ε, 2ε, and 3ε, whwre ε is some positive energy.
Find the ground state (lowest energy state) when the system contains
(i) three identical non-interacting spin-½ particles
(ii) three identical non-interacting bosons.
(iii) For both of the above cases, plot the internal energy of the system as a function of temperature.
Identify the T=0 and T=∞ limits.
(b) A measurement of the x-component of the total angular momentum Jx=Lx+Sx of a number of Hydrogen atoms
is to be carried out. It is known that the atoms are a random admixture with following states:
1/2 are |1 0 0⟩ |↑z⟩
1/4 are ( |2 1 1⟩ |↑x⟩ + |1 0 0⟩ |↓x⟩ ) / √2
1/4 are |2 1 0⟩ |↑y⟩
where |n l m⟩ is the space part of the wavefunction.
Find the density matrix, and using the density matrix, find the ensemble average [Jx].
An eigenvalue equation is defined by the Hamiltonian
( 0 1 0 0 )
H(0) = E ( 1 0 0 0 )
( 0 0 0 2 )
( 0 0 2 0 )
(a) What are the eigenvalues and eigenvectors associated with this operator?
[You should be able to answer this question by inspection. No algebra is necessary.]
(b) A perturbation of the form
( 1 0 0 0 )
H(1) = ε ( 0 0 1 0 )
( 0 1 0 0 )
( 0 0 0 1 )
is now added to the system. Find the first order correction to the eigenvalues.
(a) Estimate the ground state of a particle in a one-dimensional potential of the form
V(x) = q x4
variationally, using your favorite trial function. q is a constant.
(b) Your result in part (a) should look like
Eest = A c
where A contains the physical constants such as ħ, m etc, and c is a numerical factor.
The exact result would look the same, obviously with the exact number in place of c.
Use the Feynman-Hellmann theorem to find the expectation value of the kinetic energy and the
potential energy of this system, within your approximation.
Is this result consistent with the expectation values you have evaluated?
A two-level system is exposed to a time-dependent perturbation of the form
{ 0 for t<0
H' = {
{ V(x) exp(-t/τ) sin(ωt) / τ for t>0
where τ and ω constants with appropriate units.
Discuss the dependence of transition probability (after a longtime) between the two
states, on τ and ω.
Find the time constant for spontaneous transition of the Hydrogen |3 2 1> state into the |2 1 0> state. Find the numerical result and express your result in seconds.
* These problems were taken from "The Theoretical Minimum" (relevant link).
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 Monday, 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.