Harmonic Oscillator Family

The harmonic oscillator is both an exactly solvable one-particle system and the local language of stable quantum fluctuations. Its quadratic Hamiltonian supports two complementary exact solutions, a complete number basis, classical-looking coherent packets, squeezed and displaced Gaussian states, and a normal-mode construction that reaches from molecules to free fields.

This chapter owns the single-mode oscillator and its immediate wave-mechanics extensions. General coherent-state phase-space dynamics belongs in Quantum Dynamics. Many-mode occupation language belongs in Composite Systems and Entanglement. Molecular-vibration and optical implementations route through Atomic, Molecular, and Optical Physics, while the oscillator-to-field dictionary is maintained in Harmonic Oscillator to Fields.

The canonical model

For a particle of mass $m$ in a quadratic potential,

$$\hat H = \frac{\hat p^2}{2m} + \frac12m\omega^2\hat x^2, \qquad [\hat x,\hat p]=i\hbar.$$

The natural length and energy scales are

$$\ell = \sqrt{\frac{\hbar}{m\omega}}, \qquad E_{\mathrm{osc}} = \hbar\omega.$$

The oscillator length is not the ground-state position uncertainty. Instead,

$$\Delta x_0 = \frac{\ell}{\sqrt2}, \qquad \Delta p_0 = \frac{\hbar}{\sqrt2\,\ell}.$$

The exact spectrum is

$$E_n = \hbar\omega \left( n+\frac12 \right), \qquad n=0,1,2,\ldots.$$

It is discrete, equally spaced, and bounded below by $E_0=\hbar\omega/2$. Quantum Harmonic Oscillator owns the physical setup, scales, spectrum, eigenfunctions, expectation values, classical comparison, and first statement of universality.

Two exact solution routes

The differential and algebraic solutions answer the same spectral problem but reveal different structures.

Route	Starting point	Where quantization enters	What it exposes best
Differential equation	coordinate-space Schrödinger equation	normalizability forces the Hermite series to terminate	asymptotics, parity, nodes, and explicit wavefunctions
Ladder operators	canonical commutator and factorization of $\hat H$	positivity requires a lowest state annihilated by $\hat a$	spectrum-generating algebra, matrix elements, and number states

With

$$\xi = \frac{x}{\ell}, \qquad \epsilon = \frac{2E}{\hbar\omega},$$

the stationary equation becomes

$$\left[ -\frac{d^2}{d\xi^2} +\xi^2 \right] \psi(\xi) = \epsilon\psi(\xi).$$

Normalizable large-$\lvert\xi\rvert$ behavior supplies a Gaussian envelope. Writing $\psi=h e^{-\xi^2/2}$ produces Hermite’s equation, and polynomial termination gives $\epsilon=2n+1$. Differential-Equation Solution owns that derivation and explains why termination is a physical normalizability condition rather than an algebraic trick.

The algebraic route defines

$$\hat a = \frac{1}{\sqrt2} \left( \frac{\hat x}{\ell} + \frac{i\ell}{\hbar}\hat p \right),$$

and

$$\hat a^\dagger = \frac{1}{\sqrt2} \left( \frac{\hat x}{\ell} - \frac{i\ell}{\hbar}\hat p \right).$$

Then

$$[\hat a,\hat a^\dagger]=1, \qquad \hat H = \hbar\omega \left( \hat a^\dagger\hat a+\frac12 \right).$$

Positivity of $\hat N=\hat a^\dagger\hat a$ requires a lowest state $\hat a\lvert0\rangle=0$. Repeated action of $\hat a^\dagger$ builds the entire spectrum. Ladder-Operator Solution: First Encounter owns the factorization, lower-bound argument, normalized ladder actions, and ground-state Gaussian.

Hermite functions and the number basis

The normalized coordinate-space eigenfunctions are

$$\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left( \frac{1}{\pi\ell^2} \right)^{1/4} H_n\left(\frac{x}{\ell}\right) e^{-x^2/(2\ell^2)}.$$

They obey

$$\psi_n(-x) = (-1)^n\psi_n(x),$$

and the $n$th state has $n$ nodes. Hermite Functions is the focused reference for normalization, scaling, orthogonality, recurrences, completeness, matrix elements, and the Fourier-transform eigenfunction property. The underlying polynomial identities remain canonical in Hermite Polynomials.

Abstractly, the same eigenstate is the number state $\lvert n\rangle$:

$$\hat N\lvert n\rangle = n\lvert n\rangle,$$

with

$$\hat a\lvert n\rangle = \sqrt n\,\lvert n-1\rangle, \qquad \hat a^\dagger\lvert n\rangle = \sqrt{n+1}\,\lvert n+1\rangle.$$

Number States owns basis completeness, energy probabilities, ladder normalization, neighboring-level matrix elements, and the distinction between an oscillator quantum and literal particle number.

Zero-point energy and motion

The ground state has

$$E_0 = \frac12\hbar\omega,$$

with equal kinetic and potential contributions,

$$\langle T\rangle_0 = \langle V\rangle_0 = \frac14\hbar\omega.$$

This is compatible with a stationary probability density: zero-point “motion” means nonzero spreads in position and momentum, not a hidden classical orbit. Localizing the state more tightly raises kinetic energy, while spreading it farther raises potential energy. Their balance fixes the Gaussian width and lower energy.

Zero-Point Energy owns the spectral and uncertainty arguments, energy-zero conventions, observable energy differences, physical examples, and the warning that one finite oscillator does not by itself settle QFT vacuum-energy questions.

Gaussian state families

The ground-state Gaussian can be displaced, squeezed, or both. These operations change different data.

State	Mean phase-space point	Covariance shape	Energy-basis structure
Ground state	origin	circular in dimensionless quadratures	$\lvert0\rangle$
Coherent state	displaced	ground-state widths	Poisson superposition of all $n$
Squeezed vacuum	origin	one quadrature narrowed, the conjugate widened	even-number superposition
Displaced squeezed state	displaced	squeezed ellipse	displaced squeezed-number mixture

Coherent states

A coherent state satisfies

$$\hat a\lvert\alpha\rangle = \alpha\lvert\alpha\rangle,$$

and has number-basis expansion

$$\lvert\alpha\rangle = e^{-\lvert\alpha\rvert^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} \lvert n\rangle.$$

Its center follows the classical oscillator orbit while its shape and ground-state uncertainties remain fixed. The mean occupation is $\lvert\alpha\rvert^2$, and under undriven harmonic evolution $\alpha(t)=\alpha(0)e^{-i\omega t}$. Coherent States owns the Poisson distribution, displacement operator, first moments, time evolution, nonorthogonality, and bounded classical interpretation.

Squeezed states

Define dimensionless quadratures

$$\hat Q = \frac{\hat a+\hat a^\dagger}{\sqrt2}, \qquad \hat P = \frac{\hat a-\hat a^\dagger}{i\sqrt2},$$

so $[\hat Q,\hat P]=i$. For one real squeezing convention,

$$(\Delta Q)^2 = \frac12e^{-2r}, \qquad (\Delta P)^2 = \frac12e^{2r}.$$

Ideal pure squeezing redistributes uncertainty while preserving $\Delta Q\,\Delta P=1/2$. It does not reduce all noise or violate the uncertainty relation. Squeezed States: First Encounter owns the squeeze operator, phase-space ellipse, even-number content, energy cost, mixed-state caveat, and applications preview. General covariance and multimode entanglement belong in the continuous-variable chapters.

Static displacement

For a constant force,

$$\hat H_F = \frac{\hat p^2}{2m} + \frac12m\omega^2\hat x^2 -F\hat x,$$

completing the square gives the shifted equilibrium

$$x_F = \frac{F}{m\omega^2},$$

and spectrum

$$E_n(F) = \hbar\omega \left( n+\frac12 \right) - \frac{F^2}{2m\omega^2}.$$

The level spacing and wavefunction width do not change. Relative to the old oscillator, the new ground state is coherent; relative to the shifted Hamiltonian, it is simply the ground state. Displaced Oscillator owns the coordinate, translation-operator, and ladder-operator descriptions and distinguishes a static shift from time-dependent driving.

Coupled oscillators and normal modes

Quadratic coupling does not destroy solvability. It changes which coordinates are independent. For two identical oscillators coupled through $\kappa(x_1-x_2)^2/2$, the symmetric and antisymmetric coordinates

$$Q_+ = \frac{x_1+x_2}{\sqrt2}, \qquad Q_- = \frac{x_1-x_2}{\sqrt2}$$

diagonalize the Hamiltonian, with frequencies

$$\omega_+ = \omega_0, \qquad \omega_- = \sqrt{ \omega_0^2+ \frac{2\kappa}{m} }.$$

Quantization gives independent mode occupations $\lvert n_+,n_-\rangle$. The quanta belong to collective normal modes, not to either original coordinate alone.

Coupled Oscillators: First Encounter owns this two-mode derivation, canonical coordinate transformation, mode interpretation, spectrum, and bridge to phonons and field modes.

Why the oscillator is universal

Near a stable minimum $x_0$ of a smooth potential,

$$V'(x_0)=0, \qquad V''(x_0)\gt0,$$

and

$$V(x_0+\eta) = V(x_0) + \frac12V''(x_0)\eta^2 + \frac16V'''(x_0)\eta^3 +\cdots.$$

The leading nonconstant term is harmonic, with

$$\omega = \sqrt{ \frac{V''(x_0)}{m} }.$$

The approximation is local. It is strongest for low-lying states whose support remains in the nearly quadratic region. Anharmonicity, tunneling between separated minima, zero modes, unstable directions, and interactions between normal-mode quanta require additional analysis.

Oscillator as a Universal Local Model owns the Taylor and Hessian arguments, validity estimates, multidimensional normal modes, examples, failure modes, and the precise bridge from quadratic systems to free fields.

Reading route

1. Quantum Harmonic Oscillator
2. Differential-Equation Solution
3. Ladder-Operator Solution: First Encounter
4. Hermite Functions
5. Zero-Point Energy
6. Number States
7. Coherent States
8. Squeezed States: First Encounter
9. Displaced Oscillator
10. Coupled Oscillators: First Encounter
11. Oscillator as a Universal Local Model

Page map

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?

Common mistakes

Mistake	Correction
Writing $E_n=n\hbar\omega$	retain the zero-point term and start at $n=0$
Equating $\ell$ with $\Delta x_0$	use $\Delta x_0=\ell/\sqrt2$
Treating polynomial termination as optional	connect it to normalizability at large $\lvert x\rvert$
Calling $\hat a^\dagger$ a particle-creation operator in every context	identify which oscillator mode and quantum are being counted
Imagining a number state as a mass following a classical orbit	remember that each number-state density is stationary
Calling coherent states classical	state the quantum width, nonorthogonality, and number fluctuations
Saying squeezing lowers uncertainty in all directions	track the conjugate anti-squeezed quadrature
Assuming a linear force changes the oscillator spacing	complete the square and separate displacement from curvature
Assigning a normal-mode quantum to one original coordinate	express the collective eigenvector or normal coordinate
Treating every bound potential as globally harmonic	state the local region and excitation range where quadratic terms dominate
Carrying one oscillator’s zero-point energy directly into a vacuum-energy claim	specify the mode set, observable differences, regularization, and later field theory

Exercises

1. Scaling at fixed frequency

At fixed $\omega$, the mass is multiplied by four. How do $\ell$, $\Delta x_0$, the level spacing, and $\Delta p_0$ change?

Solution

The oscillator length is

$$\ell = \sqrt{\frac{\hbar}{m\omega}},$$

so $m\mapsto4m$ sends $\ell\mapsto\ell/2$. Therefore $\Delta x_0=\ell/\sqrt2$ is also halved.

The level spacing $\hbar\omega$ is unchanged because the question holds $\omega$ fixed. Finally,

$$\Delta p_0 = \sqrt{\frac{m\hbar\omega}{2}},$$

so the momentum uncertainty doubles. The product remains $\Delta x_0\Delta p_0=\hbar/2$.

2. The lower-bound argument

Use $\hat H=\hbar\omega(\hat a^\dagger\hat a+1/2)$ to show that no normalized oscillator state can have energy expectation below $\hbar\omega/2$.

Solution

For any normalized $\lvert\psi\rangle$,

$$\langle\hat a^\dagger\hat a\rangle_\psi = \langle\hat a\psi\vert\hat a\psi\rangle = \lVert\hat a\lvert\psi\rangle\rVert^2 \geq0.$$

Hence

$$\langle H\rangle_\psi = \hbar\omega \left( \langle\hat a^\dagger\hat a\rangle_\psi +\frac12 \right) \geq \frac12\hbar\omega.$$

Equality requires $\hat a\lvert\psi\rangle=0$, which identifies the ground state up to phase.

3. Displacement is not squeezing

Explain why translating the oscillator ground state changes $\langle x\rangle$ but leaves $\Delta x$ unchanged, whereas squeezing can change $\Delta x$.

Solution

A position translation is generated by

$$\hat T(x_0) = \exp\left( -\frac{i}{\hbar}x_0\hat p \right).$$

It sends the probability density to the same shape centered at $x_0$. Both $\langle x\rangle$ and $\langle x^2\rangle$ shift, but their combination

$$(\Delta x)^2 = \langle x^2\rangle-\langle x\rangle^2$$

is unchanged.

A squeeze transformation rescales conjugate quadratures in reciprocal directions. It changes the covariance matrix, so $\Delta x$ can decrease while $\Delta p$ increases. Translation moves the center; squeezing changes the shape.

4. Stable, flat, and unstable directions

For a local coordinate $\eta$, classify the leading physics when the quadratic coefficient in

$$V(x_0+\eta) = V(x_0)+\frac12K\eta^2+\cdots$$

has $K\gt0$, $K=0$, or $K\lt0$.

Solution

For $K\gt0$, the point is a stable quadratic minimum. The local frequency is $\omega=\sqrt{K/m}$, and low-lying fluctuations are oscillator-like.

For $K=0$, the quadratic term supplies no restoring force. A symmetry may produce a zero mode, or higher-order terms may control the motion. The ordinary oscillator approximation is insufficient.

For $K\lt0$, the point is unstable. The local model is an inverted oscillator rather than a discrete bound oscillator, so the standard number-state spectrum does not apply.

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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed., Butterworth-Heinemann, 1976.
• J. R. Klauder and B.-S. Skagerstam, Coherent States, World Scientific, 1985.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.