The University of Sydney
School of Mathematics and Statistics
Solutions to Video Assignment 2
MATH3078: PDEs and Waves Semester 2, 2020
Web Page: https://canvas.sydney.edu.au/courses/27747
Lecturer: Daniel Hauer, Tutor: Timothy A. Collier
PDE and Waves
Directions
This video assignment is only for student enrolled into MATH3078!
Your recording to this assignment needs to be submitted by
15 November 2020, 11:59 pm.
Please record a video of 3 minutes or less with your mobile phone, webcam, or tablet explaining how
to solve the given problem below. Please upload this video on the Canvas page of this unit of study.
While it may be possible to identify students from the video, please do not put your name on your
assignment so that we can apply anonymous marking as far as possible.
The Schro¨dinger equation. The hydorgen atom consists of one nucleus and one electron. One can
split the montion of this atom into two parts:
(i) one related to the motion of the centre of mass, and
(ii) another one related to the relative motion between the electron and the nucleus.
If one focus only on the energy levels λn resulting from the electric attraction between the nucleus and
the electron, then the Schro¨dinger (eigenvalue) equation
− ~
2m
∆φ− e
r
φ = λn φ in R3, (1)
models the motion of an electron in an electric field generated by a nucleus with exactly one proton.
In (1), ~ denotes the Planck constant divided by 2pi, m is the reduced mass of the atom, r = r(x) the
distance between the elctron and the centre of the nucleus, e = e(x) the electric charge, and u = u(x) a
wave function/eigenfunction corresponding to the eigevalue λn.
The Schro¨dinger (eigenvalue) equation (1) was developed in 1925 by Erwin Schro¨dinger.
The problem
1. For simplicity, we assume that ~2m = e = 1. Explain why the set σ(−∆− 1r ) of eigenvalues λn to
the Schro¨dinger operator −∆− 1r is given by
σ(−∆− 1
r
) =
{
λn = − 1
4n2
∣∣∣n ∈ N, n ≥ 1}
and for each n ∈ N, n ≥ 1, the corresponding eigenfuctions are given by
φn,l,m(r, ϕ, θ) = r
l qn,l(r) e
−r/2nY ml (ϕ, θ)
for every l ∈ {0, . . . , n− 1}, m ∈ {−l, . . . , l}, qn,l(r) is a polynomial of degree n− l− 1, and Y lm is
a spherical harmonic.
To answer this question, follow the explanation in Exercise 5.7 of Borthwick [1].
Copyright© 2020 The University of Sydney 1
Marking criteria of the video assignment
Content (6 marks in total).
• 2 marks. Accurate presentation of mathematical ideas at the correct level of this unit.
• 2 marks. Flow of ideas – is the presentation in a logical order; is the content of the presentation
synthesized into a coherent whole.
• 2 marks. Evidence of understanding of the mathematics.
Presentation-style (4 marks in total).
• 2 marks. Verbal presentation is the delivery clear, confident and aurally interesting.
• 2 marks. Visual presentation (life written, prepared parts) are the visuals clear, well-designed
and confidently handled; does the visual presentation enhance the content of the presentation?
References
[1] D. Borthwick, Introduction to partial differential equations, Universitext, Springer, Cham, 2016.
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