Harmonic Oscillator Solution using Operators

Operator methods are very useful both for solving the Harmonic Oscillator problem and for any type of computation for the HO potential. The operators we develop will also be useful in quantizing the electromagnetic field.

The Hamiltonian for the **1D Harmonic Oscillator**

looks like it could be written as the square of a operator. It can be rewritten in terms of the operator

We will use the commutators between , and to solve the HO problem.

From these commutators we can show that is a raising operator for Harmonic Oscillator states

The actual wavefunctions can be deduced by using the differential operators for and , but often it is more useful to define the eigenstate in terms of the ground state and raising operators.

Almost **any calculation** of interest can be done without actual functions
since we can express the operators for position and momentum.

- Introducing and
- Commutators of , and
- Use Commutators to Derive HO Energies

- Expectation Values of and
- The Wavefunction for the HO Ground State
- Examples
- The expectation value of in eigenstate
- The expectation value of in eigenstate
- The expectation value of in the state .
- The expectation value of in eigenstate
- The expectation value of in eigenstate
- Time Development Example

- Sample Test Problems