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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2023/paper-2/section-b/21/iii/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2023 / Paper 2 / Section B / 21 / iii / Solution by

Codex  0 2026-10-20

Inside the rectangle the effective-mass Schrödinger equation is

- \frac{ℏ ^{2}}{2 m ^{*}} (\partial_{x}^{2} + \partial_{y}^{2}) ψ = E ψ,

with

ψ = 0

at all four walls because the confining potential is infinite outside. Separate

ψ (x, y) = X (x) Y (y)

. Each factor obeys a one-dimensional particle in a box equation. For

x

,

X^{''} + k_{x}^{2} X = 0, X (0) = X (L_{x}) = 0,

so

X

is proportional to

sin (n_{x} π x / L_{x})

with

n_{x} = 1, 2, \dots

; the zero and negative integers give no new physical states. The same argument applies in

y

.

The normalized spatial states and their energies are

ψ_{n_{x}, n_{y}} (x, y) = \frac{2}{L _{x} L _{y}} sin (\frac{n _{x} π x}{L _{x}}) sin (\frac{n _{y} π y}{L _{y}}),

E_{n_{x}, n_{y}} = E_{x} + E_{y} = \frac{h ^{2}}{8 m ^{*}} (\frac{n _{x}^{2}}{L _{x}^{2}} + \frac{n _{y}^{2}}{L _{y}^{2}}) .

The product form and energy sum follow from separability. Each spatial orbital can accommodate two opposite-spin electrons under the stated spin degeneracy. These are single-particle confinement energies; the electrostatic charging contribution must still be added when interpreting transport addition energies.

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