Answers for Assigned Problems

1.5 (a) A = Öl
      (b) <x> = 0, <x²> = 1/(2 l²)
      (c) s = 1/[lÖ2], P = e ^{- Ö2}

1.7 Answer given in text.

1.8 There is no effect on expectation values.

1.9 (a) A = [2am/(p hbar)]^1/4
        (b) V(x) = 2 m a² x²
        (c) <x> = 0, <x²> = hbar/(4am), <p> = 0, <p²> = a m hbar
        (d) s_x = Ö[hbar/(4am)], s_p = Ö[a m hbar] s_x s_p = hbar/2. The wave function in this problem is called a "minimum-uncertainty wave packet."

1.14 (b) J(x,t) = 0 . The wave function in problem 1.8 is of the form (real function of x) e^-iwt, and it is easy to see that when Y is in this form, J must vanish identically.

1.15 P(t) = P(0) e ^{- 2 G t / hbar}

Last revised 2002/08/09