Probability of finding a particle with a particular value of momentum Given a particle's wave function, what is the general method of finding the probability distribution of momentum (i.e., the probability of finding that particle with a particular value of momentum)?
For example, given
$$
\psi(x) = a e ^ { ikx } + b e ^ { -ikx },
$$
what is the probability of measuring $ p_x = \hbar k $ ?
 A: First of all, I assume that you know that the probability (density) distribution is given by the squared amplitude of the wavefunction.  In your example, you are given the wavefunction in the position basis, so it gives you the position probability density.  If you want the momentum probability density, you have to change basis to get $\psi(p)$.
The standard way to change basis is to use a "resolution of the identity".  The manipulation is easiest in bra-ket notation.  You start out with
\begin{equation}
  \psi(x) = \langle x | \psi \rangle = a\, e^{i k x} + b\, e^{-i k x}.
\end{equation}
But you want $\psi(p) = \langle p | \psi \rangle$.  You can find it with this manipulation:
\begin{align}
  \langle p | \psi \rangle
  &= \int_{-\infty}^{\infty} \langle p | x \rangle \langle x| \psi \rangle\, d x \\
  &= \int_{-\infty}^{\infty} \frac{1} {\sqrt{2\pi \hbar}} e^{-i p x / \hbar} \langle x| \psi \rangle\, d x \\
  &= \int_{-\infty}^{\infty} \frac{1} {\sqrt{2\pi \hbar}} e^{-i p x / \hbar} \left( a\, e^{i k x} + b\, e^{-i k x} \right)\, d x.
\end{align}
Here, the resolution of the identity I used was $\int_{-\infty}^\infty |x\rangle \langle x |\, dx$.  For a one-dimensional system, that's just the identity operator, so you should be able to just plonk it into any old expression you want.
I'll leave it as a homework exercise to do the integral and take the squared magnitude of the result.  But here are a couple hints.  First, I happen to know that your solution $\psi(x)$ is a sum of two momentum eigenstates, so you should almost always get zero probability of measuring any given momentum — except for those two particular momentum eigenvalues.  That is, your answer will be zero everywhere except for two values related to $k$ and $-k$.  Second, you might want to read up on the Dirac $\delta$ function — specifically how it can be expressed as an integral.
