Harmonic oscillator

The reason why we want to study the harmonic potential is that all potentials can be approximated with a harmonic one near one of their local minima.

Given the potential

we can write the Hamiltonian as

Which, introducing the constants

can be rewritten as

We can now introduce the two ladder operators

that allow us to rewrite the Hamiltonian as

By evaluating the commutators

we can easily find how the ladder operators act on a state. Supposing to know an eigenstate of the Hamiltonian, we can look at the action of on :

This means that is still an eigenvector of with eigenvalue . The same can be done with obtaining that the new eigenvalue is .

This shows that the ladder operators allow us to build an infinite number of eigenvectors with evenly spaced eigenvalues starting from a single eigenvector.

Since the energy cannot be negative there must be a minimum eigenvalue and a corresponding eigenstate which cannot be lowered:

Since the solution is not normalizable, we can use the condition above to find the ground state

or

which has solution

with which can be found applying the normalization condition.