Quantum Harmonic Oscillator

Equivalent forms

E_n = (n + 1/2) h f

\hat{H} = \frac{\hat{p}^2}{2m} + \tfrac{1}{2}m\omega^2\hat{x}^2

Every quantum field — photons, phonons, gravitons — is built from harmonic oscillators, one at every point in space.

Symbol	Name	SI unit	Dimension	Range
$E_n$	Energy of level noutput Allowed energy of the nth excited state	J	ML^2T^-2	0 – 1e-18
$hbar$	Reduced Planck constant h/(2*pi)	J*s	ML^2T^-1	1.054571817e-34 – 1.054571817e-34
$omega$	Angular frequency Classical angular frequency sqrt(k/m)	rad/s	T^-1	10000000000 – 10000000000000000
$n$	Quantum number Non-negative integer labeling the level	1	1	0 – 20

Unit systems

Symbol	SI	CGS	Imperial
$E_n$	J	erg	ft*lbf
$hbar$	J*s	erg*s	ftlbfs
$omega$	rad/s	rad/s	rad/s
$n$	1	1	1

Where it holds

Exact for purely quadratic potentials. Real bonds (Morse potential) deviate at high n. Relativistic corrections appear when hbar*

\omega \sim m*c^2

.

Dimensional analysis

E_n \to [M*L^2*T^-2]

hbar*

\omega \to [M*L^2*T^-1]*[T^-1] = [M*L^2*T^-2]

Both sides are energy.

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Discovery

Werner Heisenberg / Paul Dirac · 1925

Heisenberg solved the oscillator with matrix mechanics in 1925; Dirac introduced ladder operators in 1930, exposing its algebraic beauty. The model became the universal language of quantum fields.

Try this

Why does a quantum oscillator never sit still — even at absolute zero?

A spring obeys Hooke's law classically, but its quantum version forbids zero motion. What is its energy spectrum, and why is it the most important model in physics?

Research status: stable

Real-world applications

Molecular vibrational spectroscopy (IR, Raman)
Phonons in condensed matter (specific heat, thermal transport)
Quantum field theory — every Fourier mode is an oscillator
Trapped-ion quantum computing (qubits ride oscillator modes)

Common misconceptions

Zero-point energy is real and measurable — Casimir effect proves it
Levels are equally spaced (unlike hydrogen) only because the potential is exactly quadratic
x and p both have zero mean in stationary states, but nonzero variance

Experimental verification

Infrared spectroscopy of diatomic molecules (HCl absorbs at

2990\,\mathrm{cm}^-1 \approx

hbar*omega). Optical phonons in solids. Trapped-ion mode spectroscopy resolves individual n with <1% error.

Derivation

Define ladder operators

a = \sqrt(m*\omega /(2

*hbar))*(x + i*p/(m*omega)) and a† its conjugate.

Then [a, a†

] = 1

and

H =

hbar*omega*(a†*a + 1/2).

Acting on the ground state a|0>

= 0

gives

E_0 =

hbar*omega/2.

Applying a† repeatedly raises energy by hbar*omega, giving

E_n =

hbar*omega*(n + 1/2).

Limiting cases

n = 0

⟶

E_0 =

hbar*omega/2Zero-point energy — nonzero even in the ground state, mandated by uncertainty.

n \to

infinity⟶ Quantum behavior approaches classical SHMBohr correspondence — high-n probability density mirrors the classical 1/sqrt(x_max^2 - x^2) distribution.

\omega \to 0

⟶ Spacing collapses, continuum returnsA free particle has no quantization.

What if…

What if zero-point energy were zero?

Atoms could collapse to point particles, helium would freeze even at 0 K, and the Casimir force would vanish.

What if the potential were quartic instead of quadratic?

Levels become unequally spaced — anharmonic oscillator, requires perturbation theory.

What if you couple two oscillators?

Normal modes appear with shifted frequencies — basis of phonon and photon band structures.

1

Ground state of HCl bond

Given ·

omega:: 563000000000000

Find · E_0

Steps

$E_0 = 0.5 * 1.054571817e-34 * 5.63e14$
$= 2.97e-20\,\mathrm{J}$
$= 0.186\,\mathrm{eV}$ — matches IR absorption $\sim 2990\,\mathrm{cm}^-1$

Result ·

E_0 = (1/2)

*hbar*

\omega \approx 2.97e-20\,\mathrm{J} \approx 0.186\,\mathrm{eV}

2

Level spacing for a quantum dot oscillator

Given ·

omega:: 10000000000000

Find ·

\Delta _E = E_1 - E_0

Steps

$\Delta _E = 1.054571817e-34 * 1e13$
$= 1.05e-21\,\mathrm{J}$
$= 6.58\,\mathrm{meV}$

Result ·

\Delta _E =

hbar*

\omega \approx 1.05e-21\,\mathrm{J} \approx 6.58\,\mathrm{meV}

Time-Independent Schrödinger Equation→Heisenberg Uncertainty Principle→Photon Energy (Planck Relation)→