Quantum Chemistry and Spectroscopy - 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 --- About TMP Chem --- All TMP Chem content is free for everyone, everywhere, and created independently by Trent Parker. Email: [email protected] --- Video Links --- Chapter Playlist: https://www.youtube.com/playlist?list=PLm8ZSArAXicL4kdvchwcgETYWXlt-HyzH Course Playlist: https://www.youtube.com/playlist?list=PLm8ZSArAXicL3jKr_0nHHs5TwfhdkMFhh Course Review: https://www.youtube.com/playlist?list=PLm8ZSArAXicLTRn3cJyyU1TiU7n_Pp4X1 Other Courses: https://www.youtube.com/playlist?list=PLm8ZSArAXicIXArfap9Tcb8izqRPvE0BK Channel Info: https://www.youtube.com/playlist?list=PLm8ZSArAXicLlGO4Rvpz-D6vX8MFbOn4V --- Social Links --- Facebook: https://www.facebook.com/tmpchem Twitter: https://www.twitter.com/tmpchem LinkedIn: https://www.linkedin.com/in/tmpchem Imgur: https://tmpchem.imgur.com GitHub: https://www.github.com/tmpchem --- Equipment --- Microphone: Blue Yeti USB Microphone Drawing Tablet: Wacom Intuos Pen and Touch Small Drawing Program: Autodesk Sketchbook Express Screen Capture: Corel Visual Studio Pro X8