nLab quantum harmonic oscillator




physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



“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.

Classical oscillator

First the harmonic oscillator in classical mechanics.

The force exerted by a spring is proportional to how far you stretch it:

F=kx.F = k x.

The potential energy stored in a stretched spring is the integral of that:

V 0=12kx 2+C,V_0 = \frac{1}{2}k x^2 + C,

and to make things work out nicely, we’re going to choose C=1/2.C = -1/2. The total energy H 0H_0 is the sum of the potential and the kinetic energy:

H 0=V 0+T=12kx 2+12mv 212.H_0 = V_0 + T = \frac{1}{2}k x^2 + \frac{1}{2}m v^2 - \frac{1}{2}.

By choosing units so that k=m=1,k = m = 1, we get

H 0=x 22+p 2212,H_0 = \frac{x^2}{2} + \frac{p^2}{2} - \frac{1}{2},

where pp is momentum.

Quantum harmonic oscillator

Now the harmonic oscillator in quantum mechanics.

We quantize, getting a quantum harmonic oscillator, or QHO. We set p=ix,p = -i \frac{\partial}{\partial x}, taking units where =1.\hbar = 1. Now

[x,p]x n = (xppx)x n = (xix+ixx)x n = i(nx n(n+1)x n) = ix n.\array{ [x, p]x^n & = & (x p - p x)x^n \\ & = &(- x i \frac{\partial}{\partial x} + i \frac{\partial}{\partial x} x)x^n \\ & = & -i(n x^n - (n+1)x^n) \\ & = & i x^n. }

If we define a new observable z=p+ix2,z = \frac{p + ix}{\sqrt{2}}, then

zz * = (p+ix)2(pix)2 = 12(p 2+i(xppx)+x 2) = 12(p 21+x 2) = H 0.\array{ z z^* & = & \frac{(p + i x)}{\sqrt{2}} \frac{(p - i x)}{\sqrt{2}} \\ & = & \frac{1}{2}(p^2 + i(x p - p x) + x^2) \\ & = & \frac{1}{2}(p^2 -1 + x^2) \\ & = & H_0. }

We can think of z *z^* as ddz\frac{d}{dz} and write the energy eigenvectors as polynomials in z:z:

H 0z n=zddzz n=nz n.H_0 z^n = z \frac{d}{dz} z^n = n z^n.

The creation operator zz adds a photon to the mix; there’s only one way to do that, so zz n=1z n+1.z\cdot z^n = 1 z^{n+1}. The annihilation operator ddz\frac{d}{dz} destroys one of the photons; in the state z nz^n, there are nn photons to choose from, so ddzz n=nz n1.\frac{d}{dz} z^n = n z^{n-1}.

Schrödinger's equation? says iddtψ=H 0ψ,i \frac{d}{dt} \psi = H_0 \psi, so

ψ(t)= n=0 e itna nz n.\psi(t) = \sum_{n=0}^{\infty} e^{-itn} a_n z^n.

This way of representing the state of a QHO is known as the Fock basis.



Geometric quantization

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

Last revised on July 14, 2022 at 01:22:13. See the history of this page for a list of all contributions to it.