Harmonic Oscillator

The quantum harmonic oscillator is the model with quadratic confinement. It is exactly solvable, but its importance is broader: any smooth system near a stable equilibrium is locally quadratic to leading order.

Formula Hook

In one dimension,

$$H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2.$$

The spectrum is

$$E_n = \hbar\omega \left( n+\frac{1}{2} \right), \qquad n=0,1,2,\ldots .$$

The ladder-operator form is

$$H = \hbar\omega \left( a^\dagger a+\frac{1}{2} \right).$$

Canonical Home

See Quantum Harmonic Oscillator for the teaching page, Harmonic Oscillator Spectrum for the formula card, and Harmonic Oscillator for the model card.

Common Confusions

• The oscillator is not important only because it is solvable; it is the universal local model near stable equilibria.
• Zero-point energy is part of the oscillator spectrum, but absolute energy offsets can be convention-dependent outside gravity.
• Number states are not coherent states.
• A field mode behaving like an oscillator is not the same object as a particle moving in a one-dimensional parabolic potential.

Related Entries

• Hamiltonian
• Eigenstate
• Jaynes–Cummings Model

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.