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Mathematics for Physics

137 Quantum Harmonic Oscillator

A quantum system with a quadratic potential that is used as the standard model of small oscillations and of photonlike modes.

The quadratic Hamiltonian. The Hamiltonian is kinetic energy plus a quadratic potential. This principle is used to write the oscillator as the unique quadratic bound-state problem.

The oscillator Hamiltonian is

\[ H = \dfrac{P^{2}}{2m} + \dfrac{1}{2}m\omega^{2}X^{2} \]

where

  • \(m\) is the mass.
  • \(\omega\) is the angular frequency.
  • \(X\) and \(P\) are the position and momentum operators.

The equidistant spectrum. The allowed energies are equally spaced and start at \(\dfrac{1}{2}\hbar\omega\). This principle is used to write the spectrum of a mode or of a vibrational level.

The oscillator energies are

\[ E_{n} = \hbar\omega\bigl(n+\dfrac{1}{2}\bigr),\quad n = 0, 1, 2, \ldots \]

where

  • \(n\) is the quantum number.
  • \(\hbar\) is the reduced Planck constant.
  • \(\omega\) is the oscillator frequency.

The raising-lowering operator solution. The same Hamiltonian is \(\hbar\omega(a^{\dagger}a+\dfrac{1}{2})\) in terms of raising and lowering operators. This principle is used to climb the ladder of states without solving a differential equation.

The number-operator form is

\[ H = \hbar\omega\bigl(a^{\dagger}a+\dfrac{1}{2}\bigr) \]

where

  • \(a^{\dagger}\) is the raising operator.
  • \(a\) is the lowering operator.
  • \(a^{\dagger}a\) is the number operator.

Note: Also called the harmonic oscillator.

137.1 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — \(H=\dfrac{P^{2}}{2m}+\dfrac{1}{2}m\omega^{2}X^{2}\).
  2. Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — \(E_{n}=\hbar\omega(n+\dfrac{1}{2})\).
  3. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — \(H=\hbar\omega(a^{\dagger}a+\dfrac{1}{2})\).