Harmonic Oscillators

(7.136) |

where the first term on the right-hand side is the kinetic energy, involving the momentum, , and the mass, , and the second term is the potential energy, involving the displacement, , and the force constant, . Each of these terms is quadratic in the respective variable. So, in the classical approximation, the equipartition theorem yields:

(7.137) | ||

(7.138) |

That is, the mean kinetic energy of the oscillator is equal to the mean potential energy, which equals . It follows that the mean total energy is

According to quantum mechanics, the energy levels of a harmonic oscillator are equally spaced, and satisfy

where is a non-negative integer, and

(See Section C.11.) The partition function for such an oscillator is given by

(7.142) |

Now,

(7.143) |

is simply the sum of an infinite geometric series, and can be evaluated immediately to give

(7.144) |

(See Exercise 1.) Thus, the partition function takes the form

(7.145) |

and

(7.146) |

The mean energy of the oscillator is given by [see Equation (7.35)]

(7.147) |

or

Consider the limit

(7.149) |

in which the thermal energy, , is large compared to the separation, , between successive energy levels. In this limit,

(7.150) |

so

(7.151) |

giving

(7.152) |

Thus, the classical result, (7.139), holds whenever the thermal energy greatly exceeds the typical spacing between quantum energy levels.

Consider the limit

(7.153) |

in which the thermal energy is small compared to the separation between the energy levels. In this limit,

(7.154) |

and so

(7.155) |

Thus, if the thermal energy is much less than the spacing between quantum states then the mean energy approaches that of the ground-state (the so-called