Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 36E / c / Solution

For the first two sine modes in a width-$a$ infinite well with
$V(x)=9\hbar^2x/(ma^3)$, the Hamiltonian in units $\hbar^2/(ma^2)$ is
$$
\begin{pmatrix}
(\pi^2+9)/2&-16/\pi^2\\
-16/\pi^2&(4\pi^2+9)/2
\end{pmatrix}.
$$
Its lower eigenvalue gives the upper bound
$$
E_0\leq\frac{\hbar^2}{ma^2}
\left[
\frac{5\pi^2+18}{4}
-\sqrt{\frac{9\pi^4}{16}+\frac{256}{\pi^4}}
\right].
$$


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 Incoming links: Two-mode variational bound for a linearly tilted square well