No. Assignment 2 (due Monday, February 9) 1. (a) Show that [x, pn ] = i h op d n p = i npn1 , h op dpop op
and hence show that if f (pop ) is any function of pop that can be written as a power series, then d [x, f (pop )] = i h f (pop ). dpop (b) Show that [g(x), pop ] = i h d g(x). dx
2. (a) Show that if the Fourier transform of a function (x) is (k), then the Fourier transform of (x a) is eika (k). (b) Show that (c) Show that
dx 1 (x)2 (x) = dk 1 (k) (k). 2 dx eikx 1 (x)2 (x) = dk 1 (k ) (k k). 2
3. Show that if (x, t) is the wavefunction of a free particle (not necessarily a pure plane wave state), then 1 d < x >= < p >, dt m where < x > and < p > are the expectation values of position and momentum, that is d dt dx (x, t)x(x, t) = 1 m dx (x, t)pop (x, t)
(Hint: Use the Schroedinger equation to evaluate the time derivatives of and of .)
h 4. Consider the so-called translation operator eiapop / , where a is a constant. Show that when this operator acts on a function f (x), the result is f (x + a), that is, h eiapop / f (x) = f (x + a).
5. Suppose (a) A is a linear operator. Show that if f is any function that can be expressed as a power series, then f (A) is also a linear operator. (b) Show that if | is an eigenvector of A with eigenvalue a, then it is also an eigenvector of f (A) with eigenvalue f (a), f (A)| = f (a)| , where again f is any function that can be expressed as a power series.
6. Prove the following identity for two operators A and B: eA BeA = B + [A, B] + 1 1 [A, [A, B]] + [A, [A, [A, B]]] + 2! 3!
(Hint: Consider the operator g() = eA BeA , where is a numerical parameter. Show that at = 0, dg/d = [A, B], d2 g/d2 = [A, [A, B]], . . . Then develop g() ...