MATH3078: PDEs and Waves

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].
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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.
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• 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?