16. •• A particle in box is constrained to move in one dimension, like a bead on a wire, as illustrated in

▼ Figure 28.14. Assume that no forces act on the particle in the interval 0 < x < L and that it hits a perfectly rigid wall. The particle will exist only in states of certain kinetic energies that can be determined by analogy to a standing wave on a string (Section 13.5). This means that an integral number n of half-wavelengths “fit” into the box’s length L, or n(λn/2) = L, where n = 1, 2, 3, ….

Using this relationship, show that the “allowed” kinetic energies of the particle are given by Kn = n2[h2/(8mL2)], where n = 1, 2, 3, … and m is the particle’s mass. [Hint:

Recall that kinetic energy is related to momentum by K = p2/(2m) and the de Broglie wavelength of the particle is related to its momentum.]