Sample Test Problems

1. The absolute square of a wave function for a free particle is given as:
   
   $$\vert\psi(x,t)\vert^2 = \sqrt{a\over 2\pi (a^2+b^2t^2)} e^{-a(x-v_g t)^2/2(a^2+b^2t^2)}$$
   
   
   Find the expected value of $x$ as a function of time. Find the expected value of $x^2$ as a function of time. Compute the RMS x-width of this wave packet as a function of time.
2. Find the commutator $[p,e^{ik_0 x}]$ where $k_0$ is a constant and the second operator can be expanded as $e^{ik_0 x} = \sum\limits_{n=0}^\infty {(ik_0 x)^n\over n!}$.
3. Which of the following are linear operators?
   • $O_1 \psi (x) = 1/\psi (x)$
   • $O_2 \psi (x) = {\partial\psi (x)\over \partial x}$
   • $O_3 \psi (x) = x^2 \psi (x)$
   • $O_4 \psi (x) = -\psi (x+a)$
4. For a free particle, the total energy operator H is given by $H = p^2/2m$. Compute the commutators [H,x] and [H,p]. If a particle is in a state of definite energy, what do these commutators tell you about how well we know the particle's position and momentum?
5. Find the commutator $[x,p^3]$.
6. Compute the commutator $[H,x^2]$ where $H$ is the Hamiltonian for a free particle.