Question

The wavefunction of a particle in one dimension is given by \[ \psi(x) = \begin{cases} M, & \text{for } -a < x < a, \\ 0, & \text{otherwise}. \end{cases} \] Here $M$ and $a$ are positive constants. If $\varphi(p)$ is the corresponding momentum space wavefunction, which one of the following plots best represents $|\varphi(p)|^2$?

• A. A
• B. B
• C. C
• D. D

Hint:

The Fourier transform of a constant function in position space results in a sharply peaked function in momentum space. This corresponds to a sharp peak at \(p = 0\).

Answer 1

The Correct Option is C

- The particle's position space wavefunction is a constant \(M\) within the region \(-a<x<a\). This implies the corresponding momentum space wavefunction \(\varphi(p)\) is a Fourier transform of a constant function, which results in a sharply peaked function in momentum space. The graph that represents a sharply peaked distribution centered around zero momentum is (C).