assignment Homework
Central Forces 2023 (3 years)
You know that the normalized spatial eigenfunctions for a particle in a 1D box of length
\(L\) are \(\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}\). If you want the eigenfunctions for a particle in a 2D
box, then you just
multiply together the eigenfunctions for a 1D box in each
direction. (This is what the separation of variables procedure tells you to do.)

Find the normalized eigenfunctions for a particle in a 2D box with sides of length \(L_x\)
in the \(x\)direction and length \(L_y\) in the \(y\)direction.
 Find the Hamiltonian for a 2D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
Any sufficiently smooth spatial wave function inside a 2D box can be expanded in a double sum of the product wave functions, i.e.
\begin{equation}
\psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y)
\end{equation}
Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing
energy.
You may find it easier to work in bra/ket notation:
\begin{align*}
\left{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left{n}\right\rangle \left{m}\right\rangle \\
&=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left{nm}\right\rangle
\end{align*}

Find a formula for the \(c_{nm}\)s in part (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.