##

The 1D Harmonic Oscillator

The **harmonic oscillator is an extremely important physics problem**.
Many potentials look like a harmonic oscillator near their minimum.
This is the first non-constant potential for which we will solve the
Schrödinger Equation.

The harmonic oscillator Hamiltonian is given by

which makes the **Schrödinger Equation for energy eigenstates**

Note that this potential also has a Parity symmetry.
The potential is unphysical because it does not go to zero at infinity, however,
it is often a very good approximation, and this potential can be solved exactly.
It is standard to remove the spring constant
from the Hamiltonian,
replacing it with the **classical oscillator frequency**.

The **Harmonic Oscillator Hamiltonian** becomes.
The **differential equation to be solved is**

To
solve the Harmonic Oscillator equation,
we will first change to dimensionless variables,
then find the form of the solution for
, then
multiply that solution by a polynomial, derive a recursion relation
between the coefficients of the polynomial, show that the polynomial series
must terminate if the solutions are to be normalizable, derive the energy
eigenvalues, then finally derive the functions that are solutions.

The **energy eigenvalues** are

for
.
There are a countably infinite number of solutions with **equal energy spacing**.
We have been forced to have quantized energies by the requirement that the
wave functions be normalizable.
The **ground state wave function** is.

This is a Gaussian (minimum uncertainty) distribution.
Since the HO potential has a parity symmetry, the **solutions either have even or odd parity**.
The ground state is even parity.
The **first excited state** is an odd parity state, with a first order polynomial multiplying
the same Gaussian.

The **second excited state** is even parity, with a second order polynomial multiplying
the same Gaussian.

Note that
is equal to the number of zeros of the wavefunction.
This is a common trend.
With more zeros, a wavefunction has more curvature and hence more kinetic energy.

The general solution can be written as

with the coefficients determined by the recursion relation
and the dimensionless variable
given by.
The series terminates with the last nonzero term having
.

Jim Branson
2013-04-22