Consider a harmonic oscillator coupled to a heat bath at temperature T. The quantum mechanical treatment of a single harmonic oscillator shows that it has equally spaced, non-degenerate energy levels en =nhw where w is the vibrational frequency and n is a quantum number n = 0,1,2... Here the zero energy has been chosen to be the ground state energy, but it is hw/2 above the potential energy minimum (the so-called zero point energy).

Find the high temperature limit of the energy of the oscillator and compare with the average energy associated with translation of a particle.

Question

Consider a harmonic oscillator coupled to a heat bath at temperature T. The quantum mechanical treatment of a single harmonic oscillator shows that it has equally spaced, non-degenerate energy levels en =nhw where w is the vibrational frequency and n is a quantum number n = 0,1,2... Here the zero energy has been chosen to be the ground state energy, but it is hw/2 above the potential energy minimum (the so-called zero point energy). 

 Find the high temperature limit of the energy of the oscillator and compare with the average energy associated with translation of a particle.