Momentum Wavefunction Calculation

The momentum space wavefunction $\phi(p)$ is obtained by taking the Fourier transform of the position space wavefunction $\psi(x)$. Given $\psi(x) = M$ for $-a < x < a$ and $0$ otherwise, the Fourier transform integral is:

$ \phi(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-a}^{a} M e^{-ipx/\hbar} dx $

Evaluating this integral yields:

$ \phi(p) = \frac{M}{\sqrt{2\pi\hbar}} \left[ \frac{e^{-ipx/\hbar}}{-ip/\hbar} \right]_{-a}^{a} $

$ \phi(p) = \frac{M\hbar}{-ip\sqrt{2\pi\hbar}} \left( e^{-ipa/\hbar} - e^{ipa/\hbar} \right) $

Using the trigonometric identity $e^{i\theta} - e^{-i\theta} = 2i \sin(\theta)$, which implies $e^{-i\theta} - e^{i\theta} = -2i \sin(\theta)$, we simplify $\phi(p)$:

$ \phi(p) = \frac{M\hbar}{-ip\sqrt{2\pi\hbar}} \left( -2i \sin(pa/\hbar) \right) = \frac{2M\hbar}{p\sqrt{2\pi\hbar}} \sin(pa/\hbar) $

$ \phi(p) = \sqrt{\frac{2\hbar}{\pi}} M \frac{\sin(pa/\hbar)}{p} $

Analyzing the Shape of $|\phi(p)|^2$

The probability density in momentum space is $|\phi(p)|^2$. We can express $\phi(p)$ in terms of the sinc function. Let $y = pa/\hbar$. Then $p = y\hbar/a$. Substituting this:

$ \phi(p) = \sqrt{\frac{2\hbar}{\pi}} M \frac{\sin(y)}{y\hbar/a} = \sqrt{\frac{2\hbar}{\pi}} M \frac{a}{\hbar} \frac{\sin(y)}{y} $

$ \phi(p) = \left( \sqrt{\frac{2a^2}{\pi\hbar}} M \right) \frac{\sin(pa/\hbar)}{pa/\hbar} $

Thus, $|\phi(p)|^2$ is proportional to the square of the sinc function:

$ |\phi(p)|^2 \propto \left( \frac{\sin(pa/\hbar)}{pa/\hbar} \right)^2 $

This function, the squared sinc function, has specific characteristics:

• It has a maximum value at $p=0$.
• It is symmetric with respect to the p-axis ($p=0$).
• It passes through zero when $pa/\hbar = n\pi$ for non-zero integers $n$, i.e., at $p = \pm \frac{n\pi\hbar}{a}$.
• It decays towards zero as $|p|$ becomes large ($|p| \to \infty$).

Matching the Plot

We need to find the plot that matches these characteristics. Examining the options:

• Options 1 and 2 start at $|\phi(p)|^2 = 0$ at $p=0$. This is incorrect; the function peaks at $p=0$.
• Option 4 shows a peak that is not centered at $p=0$ and does not decay symmetrically to zero in the expected manner.
• Option 3 displays a function with a clear maximum at $p=0$, symmetrical decay, zeros at specific intervals, and decay to zero at large $|p|$. This aligns perfectly with the properties of $|\phi(p)|^2 \propto \text{sinc}^2(pa/\hbar)$.

Therefore, the plot in Option 3 is the correct representation.