Periodic Systems: NoTerm-2022
HW 1 : Due Day 3 11/2

1. Particle in a Box Solve Schrodinger S0 4499S Consider a quantum particle with mass \(m\) in a 1-D box where the potential is: \begin{align*} V = \begin{cases} \infty & x< 0\\ 0 & 0 \leq x \leq L \\ \infty & L<x \end{cases} \end{align*}
   1. What is the Hamiltonian for this system?
   2. Use a separation of variables procedure to solve the Schrödinger Equation for this system and find the general solution \(\Phi(x,t)\). (You saw the solution in Ph425 but now I want you to solve it yourself.)
2. Particle in a Box Review S0 4499S

   As a review of the infinite square well system, consider a quantum particle with mass \(m\) is a 1-D box from \(0\leq x \leq L\). The initial state of the particle is:

   \[\psi(x,0) = \sqrt{\frac{2}{3L}}\sin\left(\frac{2\pi x}{L}\right) + \frac{2i}{\sqrt{3L}}\sin\left(\frac{3\pi x}{L}\right)\]

   1. Write down the energy eigenvalues and eigenstates for this system. (You don't have to re-solve for them in this problem.)
   2. If you measured the energy of the system at \(t=0\), what is the probability you would measure the value \(E = 9\pi^2\hbar^2/2mL^2\)?
   3. What is the expectation value of the energy at \(t=0\)
   4. What is the uncertainty of the energy at \(t=0\)
   5. Plot the probability density of the system.
   6. If instead you measured the position of the system at \(t=0\), what is the probability you would find the particle between \(x=L/4\) and \(x=L/2\)?
   7. What will be the state of this particle at a later time \(t\)?