QM1 and IQM, 2023
Problems for student presentation session 1, on 20th of September 2022 at 08.15.
Instructions:
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You get 1/3 of a point for every solved problem, 1 point in total for all three.
You can team up with up to 3 other classmates (max 4 students per group).
Everyone has the same probability to be selected for presentation
If anyone from the team is selected, the whole team must participate in solving the problem.
A team member that is not present at the presentation will not receive any points.
The presenter/presenting team must hand in written solutions to all solved problems at
the end of the presentation.
Problem 1. A particle of mass 𝑚 is confined to a one-dimensional potential 𝑉(𝑥):
∞,
𝑥<0
𝑉(𝑥) = {−𝑉0 , 0 ≤ 𝑥 ≤ 𝑎
0,
𝑥>𝑎
where 𝑉0 is a real and positive constant.
a) Consider only bound states and derive the allowed energy levels.
b) Is there always a bound state? Find the minimum value of 𝑉0 which will produce a
bound state.
c) Find as many constants as possible and sketch the wave function in the three regions.
Problem 2. A particle with mass m is constrained to move on a loop with radius R. For such a
i 
system the momentum operator is defined as Pˆ =
where  is the angle measured
R 
counter-clockwise. The eigenfunctions of P̂ are  P () and the obey the periodic boundary
condition  P (0) =  P (2) . Note that the inner product is defined as
2
n m =
 
*
n
m
Rd  .
0
a) Prove that P̂ represents an observable (i.e. P̂ is Hermitian) in this Hilbert space.
b) Solve the eigenvalue equation to find the eigenfunctions  P () of P̂ as a function of
the eigenvalue p and  . If possible, find the value of the normalization constant.
c) Show that the spectrum of eigenvalues p is quantized and give a formula for this
spectrum in terms of an integer, say n. Is the spectrum degenerate or non-degenerate
i.e. are there more than one states with the same momentum? Explain why.
d) Show that c) is equivalent to the requirement that an integer number of de Broglie
wavelengths fit around the circumference of the loop.
Problem 3. Consider the three-dimensional harmonic oscillator for which the potential is
1
V (r ) = m2 r 2 .
2
a) Show that separation of variables in Cartesian coordinates turns this into three onedimensional harmonic oscillators and exploit your knowledge of the latter to determine
the allowed energies. Answer:
En = (n + 3 / 2)  .
b) Determine the degeneracy d(n) of En.