Homework 2
(due Wednesday, September, 5)
1. Consider the wave function ψ(x) = C cos(πx/a) on the interval −a/2 ≤ x ≤ a/2.
(a) What is C so ψ is normalized to unity?
(b) Compute the expectation values of x and x2 as well as σx .
(c) Compute the expectation values of p and p2 as well as σp .
(d) Check that the uncertainty principle is satisfied.
2. Consider the wave function ψ(x) = Cx2 (a2 − x2 ) on the interval 0 ≤ x ≤ a.
(a) What is C so ψ is normalized to unity?
(b) Compute the expectation values of x and x2 as well as σx .
(c) Compute the expectation values of p and p2 as well as σp .
(d) Check that the uncertainty principle is satisfied.
3. Consider a Gaussian wave function of the form ψ(x) = Ce−ax
2 +bx
.
(a) Determine the normalization constant by completing the square in the exponential
using the the techniques shown in class (see Probability and Normalization).
(b) Check your normalization constant by evaluating the integral of |ψ(x)|2 numerically. You can see how to do this in Matlab or Octave by examining the code
used under Uncertainty Principle. (Choose particular values for a and b.)
(c) Compute the expectation values of x and x2 as well as σx analytically.
(d) Compute the expectation values of p and p2 as well as σp analytically.
(e) Check that the uncertainty principle is satisfied.