Periodic Systems: Spring-2026
HW 7: Due Day 18

1. Width of momentum-space and position-space wavepackets

   1. Consider a momentum-space Gaussian:

      \[ \phi(p) = \Bigg(\frac{1}{2\pi\beta^2}\Bigg)^{1/4} e^{-(p-p_0)^2/4\beta^2}\]

      Calculate the corresponding position-space wavefunction.

   2. Use your favorite computational plotting tool to plot the momentum-space distribution and the position-space wavefunction. How are the widths of these two distributions related to each other? Include some plots to demonstrate the variation.
2. Momentum-Space Probability Distribution

   A beam of particles is described by the wave function:

   \[\psi(x) = \begin{cases} Ae^{ip_0x/\hbar}(b-|x|) & |x|<b \\ 0 & |x| > b\end{cases}\]

   where \(b>0\).

   1. Normalize the wave function.
   2. Plot the real and imaginary parts of the wavefunction.
   3. Plot the position probability density.
   4. Calculate and plot the momentum probability distribution.

3. Estimate Ground State Energy of the Hydrogen Atom Use the uncertainty principle to estimate the ground state energy of the hydrogen atom.