(a) Solve the classical harmonic-oscillator equation (4.22) to find t as a function of x. Then differentiate this equation to find dt/dx as a function of x. (You will probably have to look up the formula for the derivative.) Then write your result in the form dt = f(x) dx, where f(x) is a certain function of x. This formula gives the infinitesimal amount of time that the oscillator spends in the infinitesimal region of width dx and located at x. The time the oscillator takes to go from x = -A to x = A is one-half the period, and if we divide dt by one-half the period we will get the probability that the oscillator is found in the region from x to x + dx. Show that this probability is

(b) What is the value of the classical probability density Ï€-1(A2 - x2)-1/2 at the turning points x = ± A?
(c) Plot A times the classical probability density versus x/A. Use the second or third online reference in the last paragraph of Section 4.2 to view graphs of |ψ|2 versus x for high quantum numbers such as v = 14, and compare your plot of the classical probability density with these plots.

Transcribed Image Text:

1. TA[1 - (*/A)°]2 dx