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

Inside the rectangle the effective-mass Schrödinger equation iswith $\psi =0$ at all four walls because the confining potential is infinite outside. Separate $\psi (x,y)=X(x)Y(y)$. Each factor obeys a one-dimensional particle in a box equation. For $x$,so $X$ is proportional to $\sin (n_x\pi x/L_x)$ 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 areThe 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.