Harmonic Oscillator Family

The harmonic oscillator is an exact model of a quadratic potential and a useful approximation near a stable minimum. It connects differential equations to ladder-operator algebra and provides simple examples of number states, coherent packets, squeezing, and collective normal modes.

One spectrum, two solution methods

For $H=P^2/(2m)+m\omega^2X^2/2$ with $\omega>0$,

$$E_n=\hbar\omega(n+1/2),\qquad n=0,1,2,\ldots, \qquad \ell=\sqrt{\frac{\hbar}{m\omega}}.$$

The differential-equation method imposes normalizability on a Gaussian times a Hermite polynomial. The ladder method writes $H=\hbar\omega(a^\dagger a+1/2)$ and builds excited states from $a\lvert0\rangle=0$. Both solve the same problem and expose different useful features.

The ground-state position width is $\ell/\sqrt2$, not $\ell$. Its nonzero energy and momentum spread do not imply a hidden classical orbit. A number-state density is stationary; a coherent-state center oscillates while its Gaussian width stays fixed under the ideal harmonic Hamiltonian.

Displacement moves the center of a Gaussian state. Squeezing changes its covariance, narrowing one quadrature while widening a conjugate one. An ideal displaced squeezed vacuum remains a pure superposition of number states, not an incoherent mixture. For a rotated squeezing ellipse, the position–momentum covariance must be included in the strengthened uncertainty relation.

Reading sequence

Page	Central question
Quantum Harmonic Oscillator	What are the model, natural scales, spectrum, and eigenstates?
Differential-Equation Solution	How does normalizability produce Hermite functions and quantized energy?
Ladder-Operator Solution: First Encounter	How do commutators and a lowest state generate the spectrum?
Hermite Functions	Which normalized functions, recurrences, and transform identities represent number states?
Zero-Point Energy	Why can the oscillator not occupy the classical minimum with zero energy?
Number States	What does oscillator excitation number count, and how do ladders act?
Coherent States	Which Gaussian packet preserves shape while its center follows classical motion?
Squeezed States: First Encounter	How can one quadrature be narrowed without violating uncertainty?
Displaced Oscillator	What changes when a static linear force moves the equilibrium?
Coupled Oscillators: First Encounter	How do interacting quadratic coordinates become independent collective modes?
Oscillator as a Universal Local Model	Why and when is a stable system locally harmonic?

Extend the model carefully

Learn one complete solution method first, then compare it with the other. Continue from number states to coherent and squeezed states, or from a displaced oscillator to coupled oscillators and normal modes.

The local oscillator approximation requires positive curvature and states concentrated in the nearly quadratic region. Flat minima, unstable directions, high excitations, and separated wells need additional terms or methods. For many-mode occupation language, continue to many-body quantum mechanics; a single trapped particle’s ladder operator changes its excitation level, not its particle number.

Exercises

Hold the potential curvature $K=m\omega^2$ fixed and multiply $m$ by four. How do the frequency, level spacing, and ground-state position width change?

Solution

$\omega=\sqrt{K/m}$ and $\hbar\omega$ are halved. The length $\ell=(\hbar^2/(mK))^{1/4}$ and width $\ell/\sqrt2$ are divided by $\sqrt2$. This differs from changing mass at fixed frequency, where the width is halved because the curvature changes too.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. R. Klauder and B.-S. Skagerstam, Coherent States, World Scientific, 1985.
• L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed., Butterworth-Heinemann, 1976.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.