Quantum Chemistry 5.10 – Harmonic Oscillator Even and Odd Functions

Short lecture on the use of even and odd functions with the harmonic oscillator.

Harmonic oscillator wavefunctions are composed of normalization constants, Hermite polynomials, and Gaussian functions. The normalization constants and Gaussian functions are always even functions. The Hermite polynomials are always either fully even or fully odd functions. Thus any harmonic oscillator wavefunction is always either completely even or completely odd. The position and momentum operators are both odd operators. The expectation value integral contains two wavefunctions and an operator. The product of two wavefunctions is always a product of even and even or odd and odd, and thus is always even. This gives an expectation value for position and momentum which is always odd, and thus are both zero for all harmonic oscillator wavefunctions.

Notes Slide: http://i.imgur.com/I63Hwio.png

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