Superposition of wavefunctions

Question

Suppose you have 2 normalized wavefunctions $\psi_1=Ne^{iax}e^{if(x)}e^{i\omega t}$ and $\psi_2=Ne^{-iax}e^{if(x)}e^{i\omega t}$ defined on $x\in [-x_0,x_0]$ and vanishes for $|x|>x_0$. What then is the normalization factor for the superposed wavefunction? Perhaps an argument by the symmetry?

Answer 1

$e^{i\theta} + e^{-i\theta}$ is just $2\cos \theta$. The superposed wavefunction is

$$\Psi(x,t) = 2N\cos(ax) e^{i(f(x) + \omega t)}$$

Then

$$\Psi^*\Psi = 4N^2\cos^2(ax)$$

The average height is $2N^2$ if $x_0a = n\pi/2$, in which case $N = \frac{1}{2}\sqrt{1/x_0}$. Otherwise you can do this integral.