Chapter 7.2, Problem 7.9P

(a)

To determine

Show that the stationary states can be written in the form ${\psi }_n\left(x\right)=\frac{1}{\sqrt{L}}{e}^{2\pi inx/L},\left(-L/2<x<L/2\right)$ and the allowed energies are $E_n=\frac{2}{m}{\left(\frac{n\pi \hslash }{L}\right)}^2$.

(b)

To determine

The first-order correction to $E_n$.

(c)

To determine

The good linear combinations of ${\psi }_n$ and ${\psi }_{-n}$ and show that first-order correction can be determined with these states using Equation 7.9.

(d)

To determine

The hermitian operator $A$ that fits the requirements of the theorem and show that the simultaneous eigenstates $H^0$ and $A$ are precisely the same used in part (c).