Harmonic Oscillator

The quantum harmonic oscillator is one canonical degree of freedom with a positive quadratic Hamiltonian. As an exact model it has equally spaced number states; as an approximation it describes each independent normal mode near a stable equilibrium.

Its importance is therefore broader than its analytic solvability. The oscillator supplies the local language of vibrations, Gaussian states, bosonic modes, phonons, photons, and free-field excitations.

Model at a Glance

Field	Standard one-dimensional model
Degrees of freedom	One canonical pair $(\hat x,\hat p)$
Hilbert space	$L^2(\mathbb R,dx)$
Hamiltonian	$\hat p^2/(2m)+m\omega^2\hat x^2/2$
Parameters	$m>0$, $\omega>0$
Configuration space	Full real line
Spectrum	$E_n=\hbar\omega(n+1/2)$
Eigenstate label	$n=0,1,2,\ldots$
Degeneracy	None in one dimension
Solvability	Exact by differential equation or ladder algebra
Key symmetry	Parity
Characteristic length	$\ell=\sqrt{\hbar/(m\omega)}$
Canonical home	Quantum Harmonic Oscillator

The Harmonic Oscillator Hamiltonian gives the operator-domain and convention details. This card focuses on what physical model those details define.

Physical Setup

The exact particle realization is a nonrelativistic coordinate $x\in\mathbb R$ in the potential

$$V(x)=\frac12m\omega^2x^2.$$

The restoring force is linear,

$$F(x)=-m\omega^2x,$$

and the potential is exactly quadratic at every displacement.

In a normal-mode realization, $x$ is a generalized coordinate rather than necessarily the position of one particle. It may be:

• a molecular normal-mode amplitude;
• a collective displacement in a crystal;
• the quadrature of an electromagnetic cavity mode;
• a flux or charge coordinate in an ideal linear circuit;
• one normal coordinate obtained from coupled mechanical degrees of freedom;
• one Fourier amplitude of a free field.

The mathematical oscillator is the same after canonical normalization, but the parameter dictionary and measured observables differ.

Hilbert Space and Hamiltonian

The standard Hilbert space is

$$\mathcal H=L^2(\mathbb R,dx).$$

The Hamiltonian is

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

In the position representation,

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

It is essentially self-adjoint on a standard dense core such as the Schwartz space. The quadratic potential confines the system, so the spectrum is discrete and bounded below.

The full-line domain is part of the model. Adding hard walls, compactifying the coordinate, or changing the sign of the quadratic term defines a different spectral problem.

Parameters and Natural Scales

Symbol	Meaning	SI units
$m$	Mass or effective inertia	$\mathrm{kg}$ in a mechanical coordinate
$\omega$	Angular frequency	$\mathrm{s^{-1}}$
$\ell$	Oscillator length $\sqrt{\hbar/(m\omega)}$	Same units as $x$
$p_\ell$	Momentum scale $\hbar/\ell$	Same units as $p$
$\hbar\omega$	Level spacing	Energy

Using

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

the dimensionless Hamiltonian is

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

All positive one-dimensional quadratic oscillators therefore share one dimensionless eigenvalue problem. The parameters set only the conversion back to physical length, momentum, energy, and time.

This card uses $\ell=\sqrt{\hbar/(m\omega)}$. The ground-state position width is

$$\Delta x_0=\frac{\ell}{\sqrt2}.$$

Some sources call $\Delta x_0$ the oscillator length, creating a factor-of-$\sqrt2$ convention difference.

Solvability

The model is exactly solvable in two complementary representations:

1. The coordinate-space Schrödinger equation reduces to the Hermite equation after nondimensionalization.
2. The ladder-operator algebra factors the Hamiltonian into a number operator.

Define

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

Then

$$[\hat a,\hat a^\dagger]=I, \qquad \hat H=\hbar\omega \left( \hat N+\frac12I \right).$$

Exact solvability here includes the complete spectrum, normalized eigenbasis, propagator, and unitary evolution for arbitrary initial states. The derivations remain at Differential-Equation Solution and Ladder-Operator Solution: First Encounter.

Spectrum and Eigenstates

The number states satisfy

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

with energies

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

The ground state is Gaussian,

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

The $n$th state:

• has parity $(-1)^n$;
• has $n$ real nodes;
• is nondegenerate;
• has classical turning points $x_{\mathrm t}=\pm\ell\sqrt{2n+1}$;
• is stationary up to the phase $e^{-iE_nt/\hbar}$.

The compact formula set, including normalized Hermite functions, is Harmonic Oscillator Spectrum.

Key Observables

For an energy eigenstate,

$$\langle\hat x\rangle_n =\langle\hat p\rangle_n=0,$$

and

$$\langle\hat x^2\rangle_n =\left(n+\frac12\right)\ell^2,$$ $$\langle\hat p^2\rangle_n =\left(n+\frac12\right)m\hbar\omega.$$

The kinetic and potential energies are equal:

$$\left\langle \frac{\hat p^2}{2m} \right\rangle_n = \left\langle \frac12m\omega^2\hat x^2 \right\rangle_n =\frac{E_n}{2}.$$

Useful observables and diagnostics include:

• the occupation number $\hat N$;
• quadratures proportional to $\hat x$ and $\hat p$;
• parity $(-1)^{\hat N}$;
• position and momentum variances;
• two-time correlation functions;
• transition matrix elements of $\hat x$, $\hat p$, or nonlinear perturbations.

Because $\hat x$ is linear in $\hat a+\hat a^\dagger$, a linear perturbation connects only $n$ to $n\pm1$ within the exact oscillator basis.

Dynamics and State Families

An arbitrary initial state evolves as

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

All relative phases recur after $2\pi/\omega$, so isolated oscillator observables are periodic. The system supports several important state families:

State family	Defining feature	Canonical link
Number states	Exact energy and occupation eigenstates	Number States
Coherent states	Minimum-uncertainty Gaussian packet with classical phase-space center	Coherent States
Squeezed states	Reduced variance in one quadrature with increased conjugate variance	Squeezed States: First Encounter
Displaced number states	Translated phase-space versions of number states	Displaced Oscillator

Number states, coherent states, and squeezed states are not interchangeable. They answer different preparation and measurement questions.

What the Model Teaches

The oscillator is the minimal model for:

• quantization of a bound canonical degree of freedom;
• zero-point energy and uncertainty-limited localization;
• factorization and ladder-operator methods;
• exact correspondence of mean motion with classical linear dynamics;
• Gaussian states and phase-space rotations;
• number-state selection rules;
• normal-mode decomposition;
• bosonic occupation algebra;
• analytic benchmarks for approximation and numerical methods.

Its uniform level spacing also makes it atypical. Generic confining potentials are anharmonic and have level spacings that depend on excitation.

Exact Model versus Harmonic Approximation

Let a general smooth potential have a stable equilibrium at $x_0$:

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

With $q=x-x_0$,

$$V(x) =V(x_0) +\frac12V''(x_0)q^2 +\frac{1}{3!}V^{(3)}(x_0)q^3 +\frac{1}{4!}V^{(4)}(x_0)q^4 +\cdots.$$

The leading oscillator frequency is

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

The approximation is controlled only while the occupied wavefunctions remain in a region where the omitted terms are small. A rough displacement scale for level $n$ is

$$q_n\sim\ell\sqrt{2n+1}.$$

Comparing the cubic and quartic terms with the quadratic term at $q_n$ gives a practical warning that higher excitations usually amplify anharmonic corrections. Symmetry may remove the cubic term, but it does not remove generic quartic corrections.

Oscillator as a Universal Local Model owns the approximation analysis. The exact spectrum should not be attached to an arbitrary smooth well without stating the truncation.

Physical Realizations

Molecular vibrations

A diatomic molecule near its equilibrium bond length has one leading vibrational coordinate with an effective reduced mass and curvature-defined frequency. The oscillator predicts the first level spacing, while anharmonicity explains nonuniform vibrational lines and dissociation. See Vibrations of Diatomics.

Coupled mechanical and lattice modes

A positive quadratic form in several coordinates can be diagonalized into independent normal modes. Each mode is an oscillator with its own $\omega_j$. In a crystal, quantized normal modes become phonons. See Coupled Oscillators: First Encounter and Phonons.

Quantized electromagnetic modes

After a cavity mode is isolated and canonically normalized, its two quadratures obey oscillator dynamics. Number states count photons in that mode, and coherent states approximate classical single-mode fields in an operationally precise sense. See Quantized Electromagnetic Modes.

Traps and linear circuits

The low-energy motion of a trapped particle and the ideal mode of an $LC$ circuit are oscillator realizations after their coordinates and effective inertia are identified. Nonlinearities, drive, loss, and coupling to other modes define extensions rather than properties of the isolated model.

Variants

Variant	Change	Consequence
Shifted oscillator	Replace $\hat x$ by $\hat x-x_0$	Translated eigenstates; unchanged spacing
Driven oscillator	Add $-F(t)\hat x$	Time-dependent displacement and phase
Parametric oscillator	Let $\omega$ depend on time	Squeezing and nontrivial time ordering
Coupled oscillators	Add quadratic cross terms	Normal-mode transformation
Anharmonic oscillator	Add cubic, quartic, or higher terms	Nonuniform spacing; perturbative or numerical treatment
Inverted oscillator	Reverse the quadratic sign	Unstable, unbounded motion; no number-state spectrum
Damped oscillator	Couple to an environment	Open-system generator and noise data required

The Harmonic Oscillator Hamiltonian gives the operator forms of these nearby cases.

Benchmark Targets

The oscillator is a standard validation problem because several independent checks are exact.

Numerical or analytic test	Exact target
Ground-state energy	$E_0=\hbar\omega/2$
Adjacent level spacing	$\hbar\omega$
Ground-state variance	$\langle x^2\rangle_0=\ell^2/2$
Virial balance	$\langle T\rangle_n=\langle V\rangle_n=E_n/2$
Parity	$(-1)^n$
Position selection rule	$\Delta n=\pm1$
Time recurrence	Observables recur after $2\pi/\omega$

A coordinate-grid or finite-basis computation should report convergence with domain size, grid spacing, basis cutoff, and the number of low-lying levels used in the comparison. Matrix Diagonalization provides the general numerical workflow, while Harmonic-Oscillator Variational Estimate supplies an analytic benchmark.

QFT and Many-Body Bridge

A free bosonic field or harmonic lattice decomposes into independent oscillator modes:

$$\hat H =\sum_j \hbar\omega_j \left( \hat n_j+\frac12 \right),$$

with a sum replaced by an integral in an infinite-volume continuum.

In the single-particle oscillator, $\hat a^\dagger$ raises the excitation of the trapped degree of freedom; it does not create another copy of the trapped particle. In a quantized field mode, the corresponding creation operator creates one mode quantum. The algebra is shared, while the physical interpretation changes.

Harmonic Oscillator to Fields gives the full dictionary and the cautions about continuum normalization, vacuum energy, and interactions.

Common Mistakes

• Treating the oscillator as important only because it is solvable.
• Applying equal level spacing to a merely smooth confining potential at arbitrary excitation.
• Confusing angular frequency $\omega$ with ordinary frequency $\nu$.
• Confusing $\ell$ with the ground-state standard deviation $\ell/\sqrt2$.
• Omitting the zero-point energy in the isolated spectrum.
• Treating number states as localized classical orbits.
• Calling every minimum-uncertainty state coherent.
• Importing one-dimensional nondegeneracy into an isotropic multidimensional oscillator.
• Using the number-state spectrum for an inverted or damped oscillator.
• Interpreting $\hat a^\dagger$ as creating another trapped particle in first-quantized wave mechanics.
• Treating an infinite set of field modes as one oscillator without addressing normalization and vacuum-energy regularization.

Exercises

1. Dimensionless reduction

Starting from the oscillator Hamiltonian, use $\ell=\sqrt{\hbar/(m\omega)}$ to show that $\hat H/(\hbar\omega)$ contains no model parameters.

Solution

Write

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

Then

$$\frac{\hat p^2}{2m} =\frac{\hbar^2}{2m\ell^2}\hat\pi^2 =\frac{\hbar\omega}{2}\hat\pi^2,$$

and

$$\frac12m\omega^2\hat x^2 =\frac12m\omega^2\ell^2\hat\xi^2 =\frac{\hbar\omega}{2}\hat\xi^2.$$

Therefore

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

The mass and frequency determine the units, not the dimensionless spectral shape.

2. Quartic correction scale

For

$$\hat H =\hat H_0+\lambda\hat x^4,$$

construct a dimensionless coupling from $\lambda$, $\ell$, $\hbar$, and $\omega$. Use $\langle0\vert\hat x^4\vert0\rangle=3\ell^4/4$ to compare the first-order ground-state shift with $E_0$.

Solution

The dimensionless coupling is

$$g=\frac{\lambda\ell^4}{\hbar\omega}.$$

First-order perturbation theory gives

$$\Delta E_0^{(1)} =\lambda \langle0\vert\hat x^4\vert0\rangle =\frac34\lambda\ell^4.$$

Since $E_0=\hbar\omega/2$,

$$\frac{\Delta E_0^{(1)}}{E_0} =\frac32g.$$

Thus $g\ll1$ is a natural low-state criterion for weak quartic anharmonicity. Higher states have larger moments and can violate the harmonic approximation earlier.

3. What does the creation operator create?

Explain the meaning of $\hat a^\dagger$ for a particle in a one-dimensional quadratic trap and for one mode of a quantized electromagnetic field.

Solution

For one particle in a quadratic trap, $\hat a^\dagger$ maps the same particle’s oscillator state from $\lvert n\rangle$ to a multiple of $\lvert n+1\rangle$. It raises the motional excitation; the Hilbert space still describes one trapped particle.

For a quantized electromagnetic mode, the oscillator occupation number is interpreted as photon number in that mode. The corresponding $\hat a^\dagger$ creates one additional mode quantum. The commutation algebra is the same because both are oscillators, but the physical ontology follows from the model’s Hilbert-space construction.

Canonical Links

• Quantum Harmonic Oscillator is the canonical overview.
• Harmonic Oscillator Hamiltonian is the compact operator card.
• Harmonic Oscillator Spectrum is the compact spectrum card.
• Harmonic Oscillator Ladder Operators is the compact algebra card.
• Oscillator as a Universal Local Model owns the approximation analysis.
• Harmonic Oscillator to Fields owns the field-mode dictionary.

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.
• C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press, 2005.