Homework due April 10.

A PDF version of these exercises is available here.

In these exercises you will consider the eigenstates of the Hamiltonian:

where and is the step function,

As always .

**Exercise 1** Suppose is a real valued, twice differentiable function of and that and both vanish faster than any polynomial as . Prove that

**Exercise 2** Consider .

- Show that there is a unique solution to of the form
for suitable and Find , , and in terms of and .

- Show that .
- Show that any
*bounded *solution of is of the form for some .

**Exercise 3** Consider .

- Show that there is a unique solution to of the form
for suitable and . Find , , and in terms of and .

- Show that there is another solution to of the form
for suitable and . Find , , and in terms of and .

- Show, for that
for that

and that .

- Prove that
*any *solution to is of the form for some .

**Exercise 4** Given any wavefunction the function

is interpreted as the *momentum density.* Prove that

- For , for all .
- For , the momentum density is a positive constant and that

**CHALLENGE PROBLEM: **Show that there are continuous functions for and for so that for suitable , say and vanishing faster than any power as , we have

where

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