The potential energy for a one-dimensional harmonic oscillator is m 2 x 2 (where is

The potential energy for a one-dimensional harmonic oscillator is ½ mω²x² (where ω is the angular frequency of its oscillation), so a classical harmonic oscillator with total energy E will oscillate back and forth between the classical turning points x = –x_T and x = +x_T, with xT = (2E/(mω²))^1/2.

a.) Suppose the wave associated with the oscillator is a standing wave, with n half waves between the two turning points. In terms of n, what is its deBroglie wavelength, what is its (average) momentum, and what is its (average) kinetic energy?

b.) Classically, the average kinetic energy of a harmonic oscillator is half its total energy. From this and your result from part (a), find the total energy in terms of n.