Question

Write down the wave function ψ(x) for an electron in the n = 3 state of a 10 nm wide infinite well with the center of the well being at x = 0. What numerical values does the function have at x = 0, 2, 4, 8, and 10 nm?

Answer 1

The well centred on the origin, hence for $x \in (-L/2, L/2) \colon$

$$\psi_n(x)=\begin{cases}{\sqrt {\frac {2}{L}}}\sin(k_{n}x)\quad {}{\text{for }}n{\text{ even}}\\{\sqrt {\frac {2}{L}}}\cos(k_{n}x)\quad {}{\text{for }}n{\text{ odd}}\end{cases}$$

where $k_n=\frac{n\pi}{L}, \,\, L=10\,nm$.

For $n=3\colon \psi_3(x)=\sqrt{0.2} \cos (0.3\pi x)$

$\psi_3(0)=\sqrt{0.2}\approx 0.447\\ \psi_3(2)=\sqrt{0.2}\cos(0.6\pi)\approx -0.138\\ \psi_3(4)=\sqrt{0.2}\cos(1.2\pi)\approx -0.362\\ \psi_3(8)=0\\ \psi_3(10)=0$