Physics 309 Homework

A Particle in a Box

Consider a particle of mass mthat is freely allowed to move along the x axis in the region $0\le x \le L$, but is strictly forbidden to be outside this region. The particle bounces between the impenetrable walls located at x=0 and x=L. The wave functions for the ground and first excited states are

$$\Psi_0(x,t) = \sqrt{2\over L} \sin \left( {\pi x \over L} \right) e^{-iE_0t/\hbar}$$

and

$$\Psi_1(x,t) = \sqrt{2\over L} \sin \left( {2 \pi x \over L}\right) e^{-iE_1t/\hbar}$$

for the region within the walls and are zero outside.

1.

Draw a schematic picture of the potential energy.

2.

Determine an expression for the energy in each case in terms of the other constants in the problem.

3.

Plot the spatial dependence of the wave functions for each state using Mathematica. Purely on the basis of these plots how could one identify which state is the high energy one?

4.

Calculate the expectation value of the position xfor each state of the system. Is your result consistent with the plots in part 3. Why?