(a) Using the integral in 40.42, determine the wave

Chapter 40, Problem 40.44

(choose chapter or problem)

(a) Using the integral in 40.42, determine the wave function c1x2 for a function B1k2 given by B1k2 = 0 k 6 0 1>k0, 0 k k0 0, k 7 k0 This represents an equal combination of all wave numbers between 0 and k0. Thus c1x2 represents a particle with average wave number k0>2, with a total spread or uncertainty in wave number of k0. We will call this spread the width wk of B1k2, so wk = k0. (b) Graph B1k2 versus k and c1x2 versus x for the case k0 = 2p>L, where L is a length. Locate the point where c1x2 has its maximum value and label this point on your graph. Locate the two points closest to this maximum (one on each side of it) where c1x2 = 0, and define the distance along the x-axis between these two points as wx, the width of c1x2. Indicate the distance wx on your graph. What is the value of wx if k0 = 2p>L? (c) Repeat part (b) for the case k0 = p>L. (d) The momentum p is equal to hk>2p, so the width of B in momentum is wp = hwk>2p. Calculate the product wpwx for each of the cases k0 = 2p>L and k0 = p>L. Discuss your results in light of the Heisenberg uncertainty principle