Chemistry 3730 assignment 12

Due: Friday, Nov. 28, 10:00a.m.

One method for obtaining the harmonic oscillator wavefunctions involves the ladder operators:

These operators have the property that and , i.e.\ generates the harmonic oscillator with the next highest energy and generates the previous harmonic oscillator wavefunction, but neither operator maintains normalization.

1. Confirm that

   

   with is a normalized harmonic oscillator wavefunction corresponding to the energy .

   Reminder: .

   Maple hints: You will have to tell Maple that , k and are positive. The easiest way to demonstrate that is a solution of Schrödinger's equation for a particular energy is to simplify the expression .

2. Use the raising operator to obtain from . Normalize this wavefunction and show that it is a solution of Schrödinger's equation for the energy .
3. What do you get when you apply to ?
4. What do you get when you apply to ?