Harmonic Oscillator Hamiltonian

The one-dimensional harmonic-oscillator Hamiltonian is

$$\hat H =\frac{\hat p^2}{2m} +\frac12m\omega^2\hat x^2, \qquad m>0, \quad \omega>0.$$

It defines an exactly soluble bound system on the real line and the leading local approximation near a smooth, nondegenerate stable equilibrium. The same operator algebra describes vibrational normal modes, trapped particles, electromagnetic modes, phonons, and free-field modes, although the physical meanings of $\hat x$, $\hat p$, $m$, and $\omega$ change from one realization to another.

The full physical treatment is Quantum Harmonic Oscillator. This card identifies the Hamiltonian, its conventions, and the limits within which its familiar formulas apply.

Quick Reference

Property	Standard one-dimensional realization
Hilbert space	$L^2(\mathbb R,dx)$
Hamiltonian	$\hat p^2/(2m)+m\omega^2\hat x^2/2$
Configuration space	Full real line
Parameters	$m>0$ and angular frequency $\omega>0$
Characteristic length	$\ell=\sqrt{\hbar/(m\omega)}$
Characteristic momentum	$p_\ell=\hbar/\ell=\sqrt{m\hbar\omega}$
Algebraic form	$\hat H=\hbar\omega(\hat N+I/2)$
Spectrum	$E_n=\hbar\omega(n+1/2)$
Quantum numbers	$n=0,1,2,\ldots$
Degeneracy	Nondegenerate in one dimension
Eigenstate parity	$(-1)^n$
Solvability	Exact in coordinate or ladder-operator form

System and Assumptions

The standard model uses

$$\hat x=x, \qquad \hat p=-i\hbar\frac{d}{dx}$$

on $L^2(\mathbb R)$. In the position representation,

$$\hat H =-\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +\frac12m\omega^2x^2.$$

The displayed form assumes:

• one nonrelativistic canonical degree of freedom;
• the full line as configuration space;
• an exactly quadratic, time-independent potential;
• a positive curvature $m\omega^2$;
• no driving force, damping, or environmental coupling;
• no anharmonic terms or coupling to other modes.

The adjective harmonic refers to the exactly quadratic potential and the resulting sinusoidal classical motion. It does not mean that every state is stationary or that every quantum expectation value follows one classical orbit.

Parameters and Scales

Symbol	Meaning	SI units	Convention
$m$	Mass or effective inertia	$\mathrm{kg}$	Must be positive
$\omega$	Angular frequency	$\mathrm{s^{-1}}$	$\omega=2\pi\nu$ for ordinary frequency $\nu$
$\ell$	Oscillator length	$\mathrm m$	$\ell=\sqrt{\hbar/(m\omega)}$
$p_\ell$	Momentum scale	$\mathrm{kg\,m\,s^{-1}}$	$p_\ell=\hbar/\ell$
$\hbar\omega$	Energy spacing	$\mathrm J$	Equals $h\nu$

With

$$\hat \xi=\frac{\hat x}{\ell}, \qquad \hat \pi=\frac{\ell\hat p}{\hbar},$$

the commutator and Hamiltonian become

$$[\hat\xi,\hat\pi]=i, \qquad \frac{\hat H}{\hbar\omega} =\frac12\left(\hat\pi^2+\hat\xi^2\right).$$

Thus every one-dimensional oscillator with positive $m$ and $\omega$ has the same dimensionless spectral problem.

The length convention is a frequent source of factors of $\sqrt2$. This card uses

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

for which the ground-state standard deviations are

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

Some sources call $\Delta x_0$ rather than $\ell$ the oscillator length.

Domain and Spectral Character

The operator is essentially self-adjoint on the Schwartz space $\mathcal S(\mathbb R)$. Its self-adjoint closure may be characterized by square-integrable wavefunctions for which the kinetic and quadratic-potential terms are also square integrable:

$$\mathcal D(\hat H) = \left\lbrace \psi\in L^2(\mathbb R): \psi''\in L^2(\mathbb R), \quad x^2\psi\in L^2(\mathbb R) \right\rbrace,$$

with derivatives understood weakly.

Because the potential tends to $+\infty$ as $\lvert x\rvert\to\infty$, the resolvent is compact and the spectrum is purely discrete. In one dimension the bound-state energies are nondegenerate.

The real-line domain matters. Placing the same differential expression on a finite interval adds boundary data and generally destroys the usual equally spaced spectrum.

Ladder-Operator Form

Define

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

Then

$$[\hat a,\hat a^\dagger]=I, \qquad \hat N=\hat a^\dagger\hat a,$$

and

$$\hat H =\hbar\omega \left( \hat N+\frac12I \right).$$

Conversely,

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

The algebraic derivation and its domain assumptions live at Ladder-Operator Solution: First Encounter. Compact action formulas are collected at Harmonic Oscillator Ladder Operators.

Spectrum and Eigenstates

Number states satisfy

$$\hat N\lvert n\rangle=n\lvert n\rangle, \qquad n=0,1,2,\ldots,$$

and therefore

$$E_n =\hbar\omega \left( n+\frac12 \right).$$

The level spacing and ground energy are

$$E_{n+1}-E_n=\hbar\omega, \qquad E_0=\frac12\hbar\omega.$$

The normalized ground-state wavefunction is

$$\psi_0(x) =\frac{1}{\pi^{1/4}\sqrt{\ell}} \exp\left(-\frac{x^2}{2\ell^2}\right).$$

Higher eigenfunctions are Hermite polynomials times the same Gaussian envelope. Their exact form and normalization are tabulated at Harmonic Oscillator Spectrum and derived at Differential-Equation Solution.

The parity operator commutes with $\hat H$, and

$$\mathsf P\lvert n\rangle=(-1)^n\lvert n\rangle.$$

The $n$th wavefunction has $n$ nodes on the real line. The parity alternation and nondegeneracy are specific to the one-dimensional oscillator.

Matrix Elements and Selection Structure

The position and momentum matrix elements follow directly from their ladder-operator forms:

$$\begin{aligned} \langle m\vert\hat x\vert n\rangle &=\frac{\ell}{\sqrt2} \left( \sqrt n\,\delta_{m,n-1} +\sqrt{n+1}\,\delta_{m,n+1} \right),\\ \langle m\vert\hat p\vert n\rangle &=\frac{i\hbar}{\sqrt2\,\ell} \left( \sqrt{n+1}\,\delta_{m,n+1} -\sqrt n\,\delta_{m,n-1} \right). \end{aligned}$$

A perturbation linear in $\hat x$ or $\hat p$ therefore connects only states with

$$\Delta n=\pm1.$$

This is an operator selection rule, not a complete statement about an experimental transition. The interaction Hamiltonian, polarization, degeneracy, and other quantum numbers must also be specified.

Time Evolution

For a state

$$\lvert\psi(0)\rangle =\sum_{n=0}^{\infty}c_n\lvert n\rangle,$$

the exact evolution is

$$\lvert\psi(t)\rangle =\sum_{n=0}^{\infty} c_n e^{-i\omega(n+1/2)t} \lvert n\rangle.$$

In the Heisenberg picture,

$$\hat a_H(t) =e^{-i\omega(t-t_0)}\hat a_H(t_0).$$

Consequently, $\hat x_H$ and $\hat p_H$ rotate in phase space at angular frequency $\omega$. Their expectation values obey the classical oscillator equation for every state with the required moments:

$$\frac{d^2}{dt^2}\langle\hat x\rangle +\omega^2\langle\hat x\rangle=0.$$

The shape of a general state need not remain fixed. Coherent States are the distinguished Gaussian states whose phase-space center follows the classical orbit without changing their variances.

Exact Model or Local Approximation

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

$$V'(x_0)=0, \qquad V''(x_0)>0.$$

Writing $q=x-x_0$ gives the Taylor expansion

$$V(x) =V(x_0) +\frac12V''(x_0)q^2 +\frac{1}{3!}V^{(3)}(x_0)q^3 +\cdots.$$

The quadratic approximation has

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

It is accurate only when the wavefunction is concentrated where the omitted terms are small compared with the quadratic term. Highly excited states probe larger displacements and usually reveal anharmonic level shifts. Oscillator as a Universal Local Model develops this approximation and its validity checks.

Variants and Extensions

Shifted oscillator

For

$$\hat H =\frac{\hat p^2}{2m} +\frac12m\omega^2(\hat x-x_0)^2 +C,$$

the eigenfunctions are translated, while the energies are

$$E_n=C+\hbar\omega\left(n+\frac12\right).$$

The translation is implemented by a unitary displacement. See Displaced Oscillator.

Driven oscillator

Adding a time-dependent force,

$$\hat H(t) =\frac{\hat p^2}{2m} +\frac12m\omega^2\hat x^2 -F(t)\hat x,$$

preserves quadratic solvability but removes the stationary-spectrum description as the complete account of the dynamics. Time ordering and a time-dependent displacement are needed.

Coupled and multidimensional oscillators

A positive quadratic Hamiltonian can be transformed to independent normal modes under appropriate canonical coordinates:

$$\hat H =\sum_{j=1}^{f} \hbar\omega_j \left( \hat n_j+\frac12 \right).$$

Its energies are

$$E_{\mathbf n} =\sum_{j=1}^{f} \hbar\omega_j \left( n_j+\frac12 \right).$$

Degeneracies depend on frequency relations and spatial symmetry. The isotropic oscillator has more degeneracy than a generic anisotropic one. See Coupled Oscillators: First Encounter.

Inverted oscillator

Replacing $\omega^2$ by a negative curvature gives

$$\hat H =\frac{\hat p^2}{2m} -\frac12m\Omega^2\hat x^2.$$

This Hamiltonian is unbounded below and has no oscillator number-state spectrum. It models unstable motion and requires a separate scattering or resonance analysis.

Field-mode bridge

After normal-mode decomposition, every mode of a free bosonic field behaves algebraically like an oscillator. The bridge is summarized at Harmonic Oscillator to Fields. The continuum of modes and vacuum-energy regularization introduce structures absent from one isolated oscillator.

Common Mistakes

• Omitting the zero-point term $\hbar\omega/2$ from the Hamiltonian spectrum.
• Using ordinary frequency $\nu$ where the formula requires angular frequency $\omega=2\pi\nu$.
• Confusing $\ell=\sqrt{\hbar/(m\omega)}$ with the ground-state width $\ell/\sqrt2$.
• Treating $\hat a$ as dimensionful under the convention used on this page.
• Assuming equal spacing for a potential that is only approximately quadratic.
• Importing one-dimensional nondegeneracy into an isotropic multidimensional oscillator.
• Using the number-state spectrum for an inverted oscillator.
• Calling a damped oscillator a closed Hamiltonian system without including an environment or an effective nonunitary model.
• Dropping all zero-point energies in a setting where energy differences between Hamiltonians matter.
• Treating oscillator creation operators and relativistic particle-creation operators as physically identical merely because they obey related algebras.

Exercises

1. Static force and completed square

Consider

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

where $F$ is constant. Rewrite it as a shifted oscillator and find its spectrum.

Solution

Define

$$x_0=\frac{F}{m\omega^2}.$$

Completing the square gives

$$\hat H =\frac{\hat p^2}{2m} +\frac12m\omega^2(\hat x-x_0)^2 -\frac{F^2}{2m\omega^2}.$$

Translation does not change the oscillator spacing. Therefore

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

The eigenfunctions are the usual oscillator eigenfunctions centered at $x_0$.

2. Position selection rule

Use the ladder-operator action to evaluate $\langle m\vert\hat x\vert n\rangle$ and identify the allowed changes in $n$ for a perturbation proportional to $\hat x$.

Solution

Since

$$\hat x=\frac{\ell}{\sqrt2} \left(\hat a+\hat a^\dagger\right),$$

and

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

one obtains

$$\langle m\vert\hat x\vert n\rangle =\frac{\ell}{\sqrt2} \left( \sqrt n\,\delta_{m,n-1} +\sqrt{n+1}\,\delta_{m,n+1} \right).$$

The matrix element vanishes unless $m=n\pm1$. Thus a purely linear position perturbation has $\Delta n=\pm1$ within this one-dimensional basis.

3. Ground-state width convention

Starting from the normalized ground-state wavefunction, show that $\langle x\rangle_0=0$ and $\langle x^2\rangle_0=\ell^2/2$.

Solution

The density

$$\lvert\psi_0(x)\rvert^2 =\frac{1}{\sqrt\pi\,\ell} \exp\left(-\frac{x^2}{\ell^2}\right)$$

is even, so $\langle x\rangle_0=0$. Using the Gaussian moment

$$\int_{-\infty}^{\infty} x^2e^{-x^2/\ell^2}\,dx =\frac{\sqrt\pi}{2}\ell^3,$$

gives

$$\langle x^2\rangle_0 =\frac{\ell^2}{2}.$$

Therefore

$$\Delta x_0 =\sqrt{\langle x^2\rangle_0-\langle x\rangle_0^2} =\frac{\ell}{\sqrt2}.$$

Canonical Links

• Quantum Harmonic Oscillator is the canonical physical overview.
• Differential-Equation Solution derives the Hermite-function spectrum.
• Ladder-Operator Solution: First Encounter gives the algebraic derivation.
• Number States develops the Fock basis and occupation interpretation.
• Zero-Point Energy explains the lower bound and its interpretation.
• Creation and Annihilation Operators is the operator card.
• Harmonic Oscillator Spectrum is the compact spectrum formula card.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, 2 vols., Wiley, 1977.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.