Probability to measure momentum of a certain range (eigenfunctions and such)

Question

At a certain point in time a particle of mass $m$ has the corresponding function (function of $x$)

$$\psi(x)=\begin{cases}Nx \exp[-bx]~~&\text{for}& x\geq 0 \\ 0 ~~&\text{for}& x<0\end{cases}~~~~~~~~b>0~~ N\text{ is a norm a coefficient}$$

What's the probability that during a measurement of the momentum for that point in time the momentum has a value between $-\hbar b$ and $+\hbar b$.

This is an interesting exercise I found in one of the textbooks in the library.

So, by intuition I want to compute $\langle p\rangle$, which is

$$\langle p \rangle =\int_0^\infty\psi^*p\psi dx=-\int_0^\infty\psi^*i\hbar\frac{\partial}{\partial x}\psi dx$$

Plugging in $\psi$ and after some fiddling around I still can't seem to solve this problem. Specifically, I don't know how to explicitly find the probability for the given range of $-\hbar b$ to $+\hbar b$.

Any ideas?

Answer 1

You have the wavefunction $\Psi(x)$ of the particle given in position representation. In this representation, $\int_a^b |\Psi(x)|^2 dx$ gives you the probability to find the particle somewhere in the interval $[a, b]$.

You can convert this function into the equivalent momentum representation (by taking the Fourier transform of $\Psi(x)$), in which the wavefunction is usually denoted $\Phi(p_x)$, where $p_x$ is the momentum of the particle in the $x$-direction. As you might guess, you can now calculate the probability to measure the momentum of the particle somewhere between to values $a$ and $b$ as $\int_a^b |\Phi(p_x)|^2 dp_x$

So you simply need to calculate $\int_{-\hslash b}^{\hslash b} |\Phi(p_x)|^2 dp_x$, where, $\Phi(p_x) = \dfrac{1}{\sqrt{2\pi\hslash}}\int_{-\infty}^{+\infty} e^{\frac{-ip_x x}{\hslash}}\Psi(x) dx$ (The Fourier transform of $\Psi(x)$).