Mathematics

Problem

Problem 2.37 A particle in the infinite square well (Equation 2.22) has the initial wave function

\Psi(x, 0) = A \sin^3(\pi x/a) \quad (0 \le x \le a).

Determine

A

, find

\Psi(x, t)

, and calculate

\langle x \rangle

, as a function of time. What is the expectation value of the energy? Hint:

\sin^n \theta

and

\cos^n \theta

can be reduced, by repeated application of the trigonometric sum formulas, to linear combinations of

\sin(m\theta)

and

\cos(m\theta)

, with

m = 0, 1, 2, \dots, n