Exercise 9.1

Equation (9.5) in the textbook is \[ - \frac{\hbar^2}{2m}\frac{\partial^2\psi(x)}{\partial x^2} = E \psi(x) \]

and the wave function is given by \[ \psi(x)= \mathrm{e}^{\frac{\mathrm{i}px}{\hbar}} \]

The partial dervivatives are \[ \frac{\partial \psi(x)}{\partial x} = \mathrm{e}^{\frac{\mathrm{i}px}{\hbar}} \frac{\mathrm{i}p}{\hbar} \] \[ \frac{\partial^2 \psi(x)}{\partial x^2} = \mathrm{e}^{\frac{\mathrm{i}px}{\hbar}} \frac{\mathrm{i}p}{\hbar} \frac{\mathrm{i}p}{\hbar} = - \frac{p^2}{\hbar^2} \mathrm{e}^{\frac{\mathrm{i}px}{\hbar}} \]

By plugging in we get:

\[ - \frac{\hbar^2}{2m} (- \frac{p^2}{\hbar^2} \mathrm{e}^{\frac{\mathrm{i}px}{\hbar}}) = E \mathrm{e}^{\frac{\mathrm{i}px}{\hbar}} \]

\[ \Rightarrow \frac{p^2}{2m} = E \]