Quantum Harmonic Oscillator

A diatomic molecule vibrates somewhat like two masses on a spring with a potential energy that depends upon the square of the displacement from equilibrium. But the energy levels are quantized at equally spaced values.

The energy levels of the quantum harmonic oscillator are

and for a diatomic molecule the natural frequency is of the form

where the reduced mass is given by

This form of the frequency is the same as that for the classical simple harmonic oscillator. The most surprising difference for the quantum case is the so-called zero-point vibration" of the n=0 ground state. This implies that molecules are not completely at rest, even at absolute zero temperature.


Schrodinger equation for harmonic oscillatorWavefunctions
Correspondence principleProbability distributions
Energy minimum from uncertainty principle

Schrodinger equation concepts

Beiser, Perspectives
Sec 8-7
Sec 7-9
HyperPhysics***** Quantum Physics R Nave
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