Consider a particle of mass m that can exist only between x = 0 m and x = +L on the x axis. We could say that this particle is confined to a “box” of length L. In this situation, imagine the standing de Broglie waves that can fit into the box. For example, the drawing shows the first three possibilities. Note in this picture that there are either one, two, or three half-wavelengths that fit into the distance L. Use Equation 29.8 for the de Broglie wavelength of a particle and derive an expression for the allowed energies (only kinetic energy) that the particle can have. This expression involves m, L, Planck’s constant, and a quantum number n that can have only the values 1, 2, 3, . . . .

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