The Harmonic Oscillator

The harmonic oscillator model is so important in physics. Consider the Taylor series expansion of an arbitrary potential with a minimum at :We can ignore higher order terms for small amplitude oscillations. We can also subtract the constant from the potential without changing the force experienced by the particle, leaving This is why harmonic oscillator is so important!

The Analytical Method

Lemma

The Hamiltonian for a harmonic oscillator is given by

Proof Simply by and the potential operator of a harmonic oscillator is with .

Theorem

The energy eigenstates of a harmonic oscillator is given by with corresponding energy eigenvalues: where is the Hermite polynomial, and .

Proposition

The -th excited state of a harmonic oscillator has nodes.

Proposition

On average, the energy is split evenly between the potential and the kinetic energy.

The Algebraic Method

Creation Operator & Annihilation Operator

The creation and annihilation operators are defined as: Clearly .

Proposition

We have the following equations hold:

• .
• , .
• .
• .
• and .

Theorem

The creation and annihilation operators act on quantum states as follows:

Here is a summary of the roles of these operators:

Harmonic Oscillator Operator	Role
Number operator	Counts energy levels
Creation (raising) operator	Raises energy level
Annihilation (lowering) operator	Lowers energy level

From these operators, one forms quadratic combinations:These quadratic combinations satisfy the commutation relations of the Lie algebra . i.e.,

Non-Stationary States

Proposition

All energy eigenstates of the harmonic oscillator in 1D have no time dependent observables, thus are stationary states.

Coherent State

A coherent state is a specific quantum state of a quantum harmonic oscillator, often described as a state that has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. Mathematically, we have with the mean of the quantum numbers and .

Proposition

The probability distribution of a coherent state forms a Poisson distribution. Thus it has mean and standard deviation . The expectation value of the energy of a such state is and the uncertainty is If is large, then observe that

Remark

So the energy uncertainty becomes less important. The position and momentum uncertainties become “less important” as well, and therefore the motion of the particle described by the wavefunction approaches classical harmonic oscillation.

3D Harmonic Oscillator

Prop The 3D harmonic oscillator have energy eigenstates characterized by three quantum numbers , and : The solution in each spatial dimension can be solved independently and the spatial form of the wavefunction is just the product of the solution for each dimension:

Prop We can also solve 3D harmonic oscillator in terms of a radial quantum number and the angular momentum quantum numbers and :