PAPER CODE
MATH 325
EXAMINER: Prof T Teubner
DEPARTMENT: Mathematical Sciences
NOVEMBER 2025 MARKED HOMEWORK
Quantum Mechanics
Time allowed: One week
INSTRUCTIONS TO CANDIDATES: This paper contains 4
questions. Candidates should attempt all questions. Full marks will
be awarded for complete solutions to all questions.
Online submission on Canvas by Monday, 17.11.2025, 5pm.
By submitting solutions to this assessment you affirm that you have
read and understood the Academic Integrity Policy detailed in Ap-
pendix L of the Code of Practice on Assessment,
https://www.liverpool.ac.uk/media/livacuk/tqsd/code-of-
practice-on-assessment/appendix L cop assess.pdf
and have successfully passed the Academic Integrity Tutorial and
Quiz. The marks achieved on this assessment remain provisional un-
til they are ratified by the Board of Examiners.
Paper Code MATH 325 Page 1 of 4 CONTINUED
1. A particle moves on the positive x-axis and is described, at some moment in
time, by the wave function
ψ(x) = Ae−x sinx ,
with A a real and positive constant.
(i) Determine A such that ψ is correctly normalised. [2 marks]
(ii) Calculate the average position and momentum of the particle. Deter-
mine the position of the global maximum of the probability density.
Briefly discuss this in comparison to the particle’s average position.
[6 marks]
(iii) State Heisenberg’s uncertainty principle and show that it is satisfied for
the particle with the wave function ψ(x). [6 marks]
Hint: You may use the integrals
∫∞
0
e−2x sin2 x dx =
∫∞
0
xe−2x sin2 x dx = 1
8
,∫∞
0
x2e−2x sin2 x dx = 5
32
,
∫∞
0
e−2x sinx cosx dx = 1
8
,
∫∞
0
e−2x cos2 x dx = 3
8
.
2. A particles of mass m and energy E moves on the x-axis under the influence
of the potential V given by
V (x) =

∞ for x ≤ 0 ,
0 for 0 < x < L ,
V1 for x ≥ L ,
where V1 and L are real and positive constants.
(i) By solving the (time independent) Schro¨dinger equation in the three
regions, find the general form of bound state solutions for this potential.
There is no need to determine the normalisation constants. [4 marks]
(ii) Now impose the required continuity conditions and show that the al-
lowed eigenenergies must satisfy the quantisation condition
tan
√
2mEL
h¯
= −
√
E
V1 − E .
[6 marks]
(iii) Deduce the condition which V1 must fulfill for there to be at least one
bound state. [4 marks]
(iv) By referring to general symmetry properties of energy eigenfunctions (as
discussed in the lectures) and the form of the potential in this problem,
explain why the solutions are neither symmetric nor antisymmetric w.r.t.
x = L/2. [3 marks]
Paper Code MATH 325 Page 2 of 4 CONTINUED
3. A beam of particles of mass m and energy E moves in the positive direction
along the x-axis and interacts with the potential step
V (x) =
{
V1 for x < 0 ,
0 for 0 ≤ x < ∞ ,
where V1 is a real and positive constant and E > V1.
(i) Solve the Schro¨dinger equation for this setup in the regions x < 0 and
x > 0. By imposing the required continuity conditions, determine the
normalisation constants in both regions in terms of the normalisation
of the incoming wave and hence give the wave function in both regions.
[6 marks]
(ii) Using your results from part (i), derive the reflection and transmission
coefficients, R and T , defined as the ratios between the probability fluxes
of the reflected and the incoming, and the transmitted and the incoming
particles, respectively. Briefly discuss your results in comparison with
the same setup in classical mechanics. [6 marks]
(iii) A scattering experiment with this setup but unknown V1 uses incoming
particles with known energy E > V1. If the measurements give the result
R = 1/4, derive the value of V1 in terms of E. [3 marks]
Paper Code MATH 325 Page 3 of 4 CONTINUED
4. The Hamiltonian Hˆ for the Simple Harmonic Oscillator is
Hˆ =
(
a†a + 12
)
h¯ω ,
where ω is a real, positive constant and a and a† are the annihilation and
creation operators, respectively.
(i) The orthonormal energy eigenstates |ψn⟩ (n ≥ 0) can be obtained from
the (normalised) ground state |ψ0⟩ by
|ψn⟩ = 1√
n!
(
a†
)n |ψ0⟩ .
Using this relation, the commutator of the simple harmonic oscillator,
[a, a†] = 1, and a|ψ0⟩ = 0, verify that ⟨ψ1|ψ1⟩ = 1 .
Show that ⟨ψ3|ψ1⟩ = 0 .
[4 marks]
(ii) Now consider a system in the normalised state |ψ⟩, where
|ψ⟩ = A
(
3 |ψ0⟩ − i
√
5 |ψ2⟩+
√
6 |ψ4⟩
)
and A > 0 is a real constant.
Determine A.
Using the number operator Nˆ = a†a, with Nˆ |ψn⟩ = n|ψn⟩, calculate the
energy expectation value for the state |ψ⟩.
[4 marks]
(iii) The operators a and a† can be defined in terms of the position and
momentum operators, xˆ and pˆ, by
a =
α√
2
(
1
mω
pˆ− ixˆ
)
and a† =
α√
2
(
1
mω
pˆ+ ixˆ
)
,
where α2 = mω/h¯.
First express xˆ2 in terms of a and a†, then use the relations derived in
the lectures,
aψn =
√
n ψn−1 and a†ψn =
√
n+ 1 ψn+1 ,
to calculate
⟨ψ5|xˆ2|ψ7⟩ and ⟨ψ6|xˆ2|ψ6⟩ .
[5 marks]
Paper Code MATH 325 Page 4 of 4 END

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