Problems

1. Suppose that a particle is represented by the wave function $\psi = \sin (kx - \omega t) + \sin (-kx - \omega t)$.
   1. Use trigonometry to simplify this wave function.
   2. Compute the probability of finding the particle by squaring the wave function.
   3. Explain what this result says about the time dependence of the probability of finding the particle. Does this make sense?
2. Repeat the above problem for a particle represented by the wave function $\psi = \exp [i(kx - \omega t)] + \exp [i(-kx - \omega t)]$.
3. Realizing that $\cos (kx - \omega t)$ can be written in terms of complex exponential functions, give a physical interpretation of the meaning of the above cosine wave function. In particular, what are the possible values of the associated particle's momentum and energy?
4. Suppose that a particle of mass $M$ and total energy $E$ moving in a region of variable potential energy has wavenumber varying with position according to $k(x) = a + b \sin (cx)$ where $a$, $b$, and $c$ are positive constants. We assume that $a \ll b$, so that equation (9.12) is valid.
   1. Find the wave function $\psi (x,t)$ for the corresponding matter wave.
   2. Compute the potential energy $U(x)$.
5. Make an energy level diagram for the case of a massless particle in a box.
6. Compare $\vert \Pi \vert$ for the ground state of a non-relativistic particle in a box of size $a$ with $\Delta \Pi$ obtained from the uncertainty principle in this situation. Hint: What should you take for $\Delta x$?
7. Imagine that a billiard table has an infinitely high rim around it. For this problem assume that $\hbar = 1 \mbox{ kg} \mbox{ m}^2 \mbox{ s}^{-1}$.
   1. If the table is $1.5 \mbox{ m}$ long and if the mass of a billiard ball is $M = 0.5 \mbox{ kg}$, what is the billiard ball's lowest or ground state energy? Hint: Even though the billiard table is two dimensional, treat this as a one-dimensional problem. Also, treat the problem nonrelativistically and ignore the contribution of the rest energy to the total energy.
   2. The energy required to lift the ball over a rim of height $H$ against gravity is $U = MgH$ where $g = 9.8 \mbox{ m} \mbox{ s}^{-2}$. What rim height makes the gravitational potential energy equal to the ground state energy of the billiard ball calculated above?
   3. If the rim is actually twice as high as calculated above but is only $0.1 \mbox{ m}$ thick, determine by what factor the wave function decreases going through the rim.

   Figure 9.7: Real part of the wave function $\psi$, corresponding to a fixed total energy $E$, occurring in a region of spatially variable potential energy $U(x)$. Notice how the wavelength $\lambda$ changes as the kinetic energy $K = E - U$ changes.

   

8. The real part of the wave function of a particle with positive energy $E$ passing through a region of negative potential energy is shown in figure 9.7.
   1. If the total energy is definitely $E$, what is the dependence of this wave function on time?
   2. Is the wave function invariant under displacement in space in this case? Why or why not?
   3. Does this wave function correspond to a definite value of momentum? Why or why not?
   4. Is the momentum compatible with the energy in this case? Why or why not?
9. Assuming again that $\hbar = 1 \mbox{ kg} \mbox{ m}^2 \mbox{ s}^{-1}$, what are the possible speeds of a toy train of mass $3 \mbox{ kg}$ running around a circular track of radius $0.8 \mbox{ m}$?
10. If a particle of zero mass sliding around a circular loop of radius $R$ can take on angular momenta $L_m = m \hbar$ where $m$ is an integer, what are the possible kinetic energies of the particle? Hint: Remember that $L = \Pi R$.