==> Due on Thursday 3/13 in class <==
(1) Sakurai Problem 1.28. (Do parts (b) and (c) only.)
(2) Sakurai Problem 1.33.
Cont'd:
c. To further illustrate the answer in part (b), let <x|u> be a real-space wave function and <p|u> be the corresponding momentum-space wave function. Let |v> be the state that results from applying exp(i*x*Q/hbar) (the operator in part (b)) to |u>. The corresponding momentum-space wave function is <p|v>. Relate <p|v> to <p|u>. Now suppose <x|u> is a one-dimensional Gaussian wave packet. (See Sakurai p57, and also next problem in this HW, about Gaussian wave packets.) Sketch <p|u> and <p|v>. Note that, as in part (b), Q is a number of dimension momentum and you can assume Q>0 for the sketch.(3) Sakurai Problem 1.18.
Cont'd:
d. Consider a free particle in the Gaussian wave packet state <x|a> in part (c) (of Problem 1.18), with <x>=0, at t=0. Write down its wave function a(x,t) for any time t>0. (Hint: How does a free particle in a plane wave state (i.e., momentum eigenstate) <x|p'> evolve with time?) What is the probability density |a(x,t)|^2 ? Sketch it (as a function of x) for three different times 0 = t1 < t2 < t3. Evaluate <(Delta x)^2> (i.e., <(x - <x>)^2>) as a function of t. What about <(Delta p)^2>? At t>0, do we still have a minimum uncertainty wave packet?(4) Sakurai Problem 2.9
Note that this deals with the time evolution of our electron on a one-dimensional crystal problem of HW1, problem 3, with N=2. This problem can come again, disguised in various ways, e.g., in terms of a spin-1/2 system, or with no disguise at all --- see Problem 2.8 in Sakurai! It's important to thoroughly understand it.