Homework due March 29.

A PDF version of these exercises is available here.

The purpose of these exercises is to consider the *free quantum evolution* generated by the pure kinetic Hamiltonian:

You may assume throughout that . Since , the Schroedinger equation for the evolution of the wave function is

Our goal is to solve (1) to find in terms of the initial value of the wave function .

Given a wave function in the position representation, denote its momentum representation by . Recall that

In the momentum representation the Schroedinger equation reads

Exercise 1Show that the unique solution to (2) with given initial value is

In the next few exercises, you will consider the evolution of certain specific initial wave functions. We start by defining these special functions — called *coherent states — *in the momentum representation

where and . To express these wave functions in the position representation you will need the following formula (which you need not prove).

where is the complex square root of computed as follows, if with then

Exercise 2Show that if and that the position space representation of is

(Hint: expand and complete the square.)

Note that (6) can be expressed as saying for suitable . This wave function may be thought of as describing a particle with momentum approximately and position approximately . More specifically,

Exercise 3Show thatwhile

So the mean position and momentum in are and respectively, while the uncertainties are and respectively. Note that the product of the uncertainties is

as required by the uncertainty principle.

Exercise 4Now consider the Schroedinger equation (1) with initial condition for some and with . Prove that the solution is given by

Note that the mean position and mean momentum undergo the classical evolution and . However the parameters and also flow with the evolution. The change of shows that the wave packet *disperses*, i.e., the uncertainty in diverges

By contrast, the uncertainty in is constant in time.

In fact, position space dispersion of the wave function happens for any initial wave function. You will demonstrate it for initial wave functions such that

Exercise 5Assume (7). Prove that, , , and .

Exercise 6Assume (7) and show that

Exercise 7Still assuming (7), argue thatwhere

Exercise 8Show that

(Hint: complete the square and use (5).)

Exercise 9Finally, assuming (7), prove that

with if and if .

Exercise 10Assuming (7), prove that

(Hint: for the identity (10) use eq. (8) and facts about the Fourier transform; for the inequality (11) use eq. (9).)

You have assumed (7) in deriving eqs. (10) and (11). However, these expressions allow us to define a solution to the Schroedinger equation (1) for any integrable or square integrable . Furthermore, (11) can be interpreted as dispersion of , since the amplitude goes to zero as .

Here are two challenge problems, not required but if you want something even harder to think about then have a try:

- Prove that the uncertainty as for any initial state .
- Prove the magic formula (5) (actually this isn’t too hard if you know the right complex analysis).

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