Homework due March 29.
The purpose of these exercises is to consider the free quantum evolution generated by the pure kinetic Hamiltonian:
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
Exercise 1 Show 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
(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 3 Show that
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 4 Now 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.
Exercise 5 Assume (7). Prove that , , , and .
Exercise 6 Assume (7) and show that
Exercise 7 Still assuming (7), argue that
Exercise 8 Show that
(Hint: complete the square and use (5).)
Exercise 9 Finally, assuming (7), prove that
for , where
with if and if .
Exercise 10 Assuming (7), prove that
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).