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“Harmonic oscillator” is a fancy name for a rock on a spring:
in classical mechanics it is the physical system given by a point mass in a parabolic potential, feeling forces driving it back to a specified origin that are propertional to the distance of the mass from that origin.
in quantum mechanics and in particular quantum field theory the quantum harmonic oscillator governs not just the dynamics of idealized point masses but crucially appears in the dynamics of all free massive quantum fields.
To quote the field theorist Sidney Coleman,
The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.
First the harmonic oscillator in classical mechanics.
The force exerted by a spring is proportional to how far you stretch it:
The potential energy stored in a stretched spring is the integral of that:
and to make things work out nicely, we’re going to choose $C = -1/2.$ The total energy $H_0$ is the sum of the potential and the kinetic energy:
By choosing units so that $k = m = 1,$ we get
where $p$ is momentum.
Now the harmonic oscillator in quantum mechanics.
We quantize, getting a quantum harmonic oscillator, or QHO. We set $p = -i \frac{\partial}{\partial x},$ taking units where $\hbar = 1.$ Now
If we define a new observable $z = \frac{p + ix}{\sqrt{2}},$ then
We can think of $z^*$ as $\frac{d}{dz}$ and write the energy eigenvectors as polynomials in $z:$
The creation operator $z$ adds a photon to the mix; there’s only one way to do that, so $z\cdot z^n = 1 z^{n+1}.$ The annihilation operator $\frac{d}{dz}$ destroys one of the photons; in the state $z^n$, there are $n$ photons to choose from, so $\frac{d}{dz} z^n = n z^{n-1}.$
Schrödinger's equation? says $i \frac{d}{dt} \psi = H_0 \psi,$ so
This way of representing the state of a QHO is known as the Fock basis.
Discussion of geometric quantization of the harmonic oscillator is in
Adrian Lim, A non-standard geometric quantization of the harmonic oscillator (pdf)
G. Sherry, Geometric quantization of the 3-dimensional harmonic oscillator, Quaestiones Mathematicae, 8 (1986)
Sergey V. Zuev, Geometric quantization of generalized oscillator (arXiv:math-ph/9902024)
A program initiated by John Baez aims to identify a categorification of sorts of the quantum harmonic oscillator
The notes
relate wavefunctions expressed in the Fock basis to structure types.
This originates in
For more along these lines see