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
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.