Solution
= Solution
The electric potential is $\varepsilon(x-a/2)$, so inside the box
$$
H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}
+q\varepsilon(x-a/2),\quad
H^{(0)}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2},\quad
H^{(1)}=x-a/2,\quad \lambda=q\varepsilon.
$$
The infinite walls impose zero <wavefunction> at $0,a$. The normalized <particle in a box> states are $\psi_n(x)=\sqrt{2/a}\sin(n\pi x/a)$ and $E_n^{(0)}=n^2h^2/(8ma^2)$ for positive integers $n$.