Harmonic Oscillator Spectrum

Formula

For the one-dimensional Hamiltonian

$$H = \frac{P^2}{2m} +\frac12 m\omega^2X^2, \qquad m>0, \quad \omega>0,$$

the exact energy spectrum is

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

Adjacent levels have constant spacing

$$E_{n+1}-E_n = \hbar\omega,$$

and the ground-state energy is

$$E_0 = \frac12\hbar\omega.$$

At a Glance

Quantity	Formula
Oscillator length	$\ell=\sqrt{\hbar/(m\omega)}$
Momentum scale	$p_\ell=\hbar/\ell=\sqrt{m\hbar\omega}$
Energy scale	$\hbar\omega$
Spectrum	$E_n=\hbar\omega(n+1/2)$
Level spacing	$\Delta E=\hbar\omega$
Ground-state width	$\Delta x_0=\ell/\sqrt2$
Ground-state momentum width	$\Delta p_0=\hbar/(\sqrt2\,\ell)$
Parity	$(-1)^n$
Number of real nodes	$n$
Classical turning points	$x_{\mathrm t}^{(n)}=\pm\ell\sqrt{2n+1}$

The length convention matters. Some authors call $\Delta x_0$ the oscillator length; this card uses

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

Meaning

The quadratic oscillator has a purely discrete, nondegenerate one-dimensional spectrum. Its levels are equally spaced and begin at a nonzero zero-point energy.

The formula is exact only for an exactly quadratic Hamiltonian on the real line. Near a smooth stable minimum of a more general potential, it gives the leading approximation, with anharmonic terms producing corrections.

The angular frequency is $\omega$. If ordinary frequency is $\nu$, then

$$\omega=2\pi\nu,$$

and the spacing can equivalently be written

$$\hbar\omega = h\nu.$$

Natural Scales

The parameters $m$, $\omega$, and $\hbar$ define

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

With

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

the canonical commutator is

$$[\xi,\pi]=i$$

and the Hamiltonian becomes

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

All one-dimensional oscillators therefore share the same dimensionless eigenvalue problem. Changing $m$ and $\omega$ rescales position, momentum, energy, and time.

Ladder-Operator Form

Define

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

Then

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

and

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

Number states satisfy

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

The normalized construction from the vacuum is

$$\lvert n\rangle = \frac{ (a^\dagger)^n }{ \sqrt{n!} } \lvert0\rangle, \qquad a\lvert0\rangle=0.$$

The operators $a$ and $a^\dagger$ are dimensionless under this convention.

Position-Space Eigenfunctions

Using the physicists’ Hermite polynomials $H_n$,

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

They obey

$$\int_{-\infty}^{\infty} \psi_m^*(x)\psi_n(x)\,dx = \delta_{mn},$$ $$\psi_n(-x) = (-1)^n\psi_n(x).$$

The state $\psi_n$ has exactly $n$ simple nodes on the real line. Its tails extend beyond the classical turning points; those turning points are not boundary conditions.

The full stationary solution is

$$\Psi_n(x,t) = \psi_n(x) e^{-iE_nt/\hbar}.$$

Its probability density is time independent even though the state vector carries a phase.

Ground State

The normalized ground-state wavefunction is

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

Its uncertainties are

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

so

$$\Delta x_0 \Delta p_0 = \frac{\hbar}{2}.$$

The ground state saturates the position-momentum uncertainty bound. Excited number states do not.

Moments and Virial Check

For the number state $\lvert n\rangle$,

$$\langle X\rangle_n = 0, \qquad \langle P\rangle_n = 0,$$ $$\langle X^2\rangle_n = \left( n+\frac12 \right) \ell^2,$$ $$\langle P^2\rangle_n = \left( n+\frac12 \right) m\hbar\omega.$$

Therefore

$$\Delta X_n \Delta P_n = \hbar \left( n+\frac12 \right).$$

The virial theorem gives

$$\left\langle \frac{P^2}{2m} \right\rangle_n = \left\langle \frac12m\omega^2X^2 \right\rangle_n = \frac{E_n}{2}.$$

These equations are strong checks on analytic and numerical eigenstates.

Matrix Elements

The inverse ladder relations are

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

Hence

$$\langle m\rvert X\lvert n\rangle = \frac{\ell}{\sqrt2} \left[ \sqrt{n+1}\, \delta_{m,n+1} + \sqrt n\, \delta_{m,n-1} \right],$$ $$\langle m\rvert P\lvert n\rangle = \frac{i\hbar}{\sqrt2\,\ell} \left[ \sqrt{n+1}\, \delta_{m,n+1} - \sqrt n\, \delta_{m,n-1} \right].$$

A perturbation proportional to $X$ therefore has the ideal-oscillator selection rule

$$\Delta n=\pm1.$$

Other couplings have different rules. For example, $X^2$ connects $\Delta n=0,\pm2$.

Classical Turning Points

Setting the potential energy equal to $E_n$ gives

$$\frac12 m\omega^2 \left( x_{\mathrm t}^{(n)} \right)^2 = \hbar\omega \left( n+\frac12 \right),$$

so

$$x_{\mathrm t}^{(n)} = \pm \ell \sqrt{2n+1}.$$

The interval between these points is classically allowed. Quantum eigenfunctions remain nonzero outside it and decay in the classically forbidden region.

Shifted Oscillator

For

$$H = \frac{P^2}{2m} +\frac12 m\omega^2 (X-x_0)^2 +C,$$

the spectrum is

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

The displacement $x_0$ shifts the eigenfunctions but does not change level spacings. The constant $C$ shifts every energy equally.

If a Hamiltonian is instead written with a linear term,

$$H = \frac{P^2}{2m} +\frac12m\omega^2X^2 -FX,$$

completing the square gives

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

and

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

Several Dimensions and Normal Modes

For independent modes,

$$H = \sum_{j=1}^{d} \left[ \frac{P_j^2}{2m_j} +\frac12 m_j\omega_j^2X_j^2 \right],$$

the spectrum is

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

with

$$n_j=0,1,2,\ldots$$

For the isotropic case $\omega_j=\omega$, define

$$N = \sum_{j=1}^{d}n_j.$$

Then

$$E_N = \hbar\omega \left( N+\frac d2 \right),$$

and, ignoring additional internal degrees of freedom, the degeneracy is

$$g_N = \binom{N+d-1}{d-1}.$$

In three dimensions,

$$g_N = \frac{(N+1)(N+2)}{2}.$$

Anisotropic frequencies generally lift this shell degeneracy, although rationally related frequencies can produce accidental degeneracies.

Local Harmonic Approximation

Let a smooth potential have a stable nondegenerate minimum at $x_0$:

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

Then

$$V(x) = V(x_0) +\frac12 V''(x_0) (x-x_0)^2 +\cdots$$

with

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

The leading energy estimate is

$$E_n \approx V(x_0) +\hbar\omega \left( n+\frac12 \right).$$

This approximation is controlled only when cubic and higher terms are small over the region occupied by the state. Higher states typically probe farther from the minimum and become less harmonic.

Zero-Point Convention

Within an isolated oscillator, adding a constant to $H$ changes all energies by the same amount and leaves eigenstates and transition frequencies unchanged. One may therefore choose a shifted zero and write excitation energies as

$$\mathcal E_n = n\hbar\omega.$$

This notation does not mean the original Hamiltonian lacks zero-point energy. It means $E_0$ has been subtracted. State which Hamiltonian or energy zero is being used, especially when comparing different systems, boundary conditions, or gravitational settings.

Symbols

Symbol	Meaning	Units or range
$m$	Oscillator mass	mass, $m>0$
$\omega$	Angular frequency	inverse time, $\omega>0$
$\nu$	Ordinary frequency	inverse time
$\ell$	Oscillator length	length
$p_\ell$	Natural momentum scale	momentum
$n$	One-dimensional excitation number	$0,1,2,\ldots$
$N$	Total excitation number in several dimensions	nonnegative integer
$a$, $a^\dagger$	Lowering and raising operators	dimensionless
$N=a^\dagger a$	Number operator	dimensionless
$H_n$	Physicists’ Hermite polynomial	dimensionless function
$E_n$	Energy eigenvalue	energy

The symbol $N$ is overloaded: it denotes the number operator in an operator equation and a numerical total excitation label in an eigenvalue formula. Context and hats may be used to separate them.

Units and Dimensions

The energy scale has dimensions

$$[\hbar\omega] = \text{energy}.$$

The oscillator length and momentum scale satisfy

$$[\ell]=L, \qquad [p_\ell]=MLT^{-1}, \qquad \ell p_\ell=\hbar.$$

In one-dimensional coordinate normalization,

$$[\psi_n(x)] = L^{-1/2}.$$

The Hermite-polynomial argument $x/\ell$ and ladder operators are dimensionless.

Assumptions

• The one-dimensional formula uses the full line $L^2(\mathbb R)$.
• The potential is exactly quadratic and confining.
• The mass and angular frequency are positive.
• Position and momentum obey $[X,P]=i\hbar I$.
• The Hamiltonian has the standard nonrelativistic kinetic term.
• Eigenfunctions are square-integrable and decay at both infinities.
• The displayed Hermite normalization uses the physicists’ polynomials and the stated $\ell$ convention.
• Multidimensional degeneracies assume independent isotropic modes and no extra internal degrees of freedom.

Validity and Limitations

The one-dimensional spectrum is exact for the displayed quadratic Hamiltonian. It is not exact for:

• anharmonic potentials;
• a harmonic well cut off by hard boundaries;
• an inverted oscillator with $\omega^2<0$;
• a time-dependent frequency without a separate dynamical treatment;
• coupled coordinates before normal-mode diagonalization;
• nonlinear or driven systems unless transformed to an equivalent quadratic form.

A local quadratic expansion is an approximation whose error depends on the higher derivatives of the potential and the spatial support of the state.

Calculation Checks

• The quantum number begins at $n=0$.
• Every energy has units of $\hbar\omega$.
• Adjacent one-dimensional levels differ by exactly $\hbar\omega$.
• The one-dimensional spectrum is nondegenerate.
• The $n$th eigenfunction has parity $(-1)^n$ and $n$ nodes.
• The ground-state width is $\ell/\sqrt2$, not $\ell$.
• Kinetic and potential expectations each equal $E_n/2$.
• Numerical eigenstates should be orthonormal and negligible at artificial box boundaries.
• Grid or basis results should converge toward the exact energies as resolution increases.
• A displaced quadratic potential changes the center but not the spacing.

Minimal Scaling Example

If the mass changes to $m'=4m$ while $\omega$ is fixed, then

$$\ell' = \sqrt{ \frac{\hbar}{4m\omega} } = \frac{\ell}{2}.$$

The wavefunctions become narrower in position, while

$$p_\ell' = \sqrt{ 4m\hbar\omega } = 2p_\ell.$$

The energies do not change:

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

This separates the spatial scale, which depends on $m\omega$, from the level spacing, which depends only on $\omega$.

Derivation and Canonical Home

Quantum Harmonic Oscillator owns the physical overview, scales, spectrum, eigenstates, moments, and approximation role.

Differential-Equation Solution derives the spectrum from square-integrability and Hermite-series termination. Ladder-Operator Solution derives it from factorization, positivity, and the lowest-weight state.

Worked Examples

• Hermite Functions
• Number States
• Displaced Oscillator
• Zero-Point Energy
• Oscillator as a Universal Local Model
• Degeneracy in Separable Systems

Common Mistakes

• Writing $E_n=n\hbar\omega$ without stating that the ground energy was subtracted.
• Starting the quantum number at $n=1$.
• Confusing angular frequency $\omega$ with ordinary frequency $\nu$.
• Confusing $\ell$ with the ground-state standard deviation.
• Treating classical turning points as quantum walls.
• Assuming equally spaced levels for a generic confining potential.
• Calling $a^\dagger$ a particle-creation operator in every context; here it raises oscillator excitation.
• Forgetting the additive constant when completing the square.
• Applying one-dimensional nondegeneracy to an isotropic multidimensional oscillator.
• Using the probabilists’ Hermite-polynomial convention with the physicists’ normalization.
• Applying the local harmonic approximation without estimating anharmonic corrections.

Related Formulas

• Harmonic Oscillator Ladder Operators
• Canonical Commutation Relations
• Particle-in-a-Box Spectrum
• Hydrogen Spectrum
• Variational Bound
• First-Order Perturbation Theory

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018, ch. 2.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, ch. 7.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, sec. 2.3.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, vol. I, Wiley, 1977, complement V A.