Probability of measuring momentum [closed]

Question

Suppose we have this wavefunction:

$$ \psi = A \left( cos(kx) + cos (2kx) \right) $$

I have to find the possible results of measurement of momentum and their probabilities.

Attempt

For a momentum operator, $\hat p |\psi\rangle = p |\psi \rangle $ so $-i\hbar \frac{\partial \psi}{\partial x} = p \psi $.

This implies that in momentum space, $\psi_{(p)} \propto e^{i\frac{p}{\hbar}} $ and $p = \hbar k$

The wavefunction given is:

$$ \psi = \frac{A}{2} \left( e^{-i\frac{p}{\hbar}x} + e^{i\frac{p}{\hbar}x} \right) + \frac{A}{2} \left( e^{-i\frac{2p}{\hbar}x} + e^{i\frac{2p}{\hbar}x} \right) $$

Thus possible momentum measurements are $\pm \hbar k$ and $\pm 2\hbar k$. Probabilities are $\frac{A^2}{4}$ each?

Answer 1

Your answer is correct, but your terminology is a little off. The statement about the momentum space wave function being proportional to $e^{i \frac{p}{\hbar}}$ isn't correct and doesn't make any sense. In momentum space the wave function would have delta functions located at $\pm \hbar k$ and $\pm 2 \hbar k$.