Quantum Harmonic Oscillator: Energy Levels and Wave Functions

Energy eigenvalues and zero-point energy

Solving the Schrödinger equation for the parabolic potential V(x) = ½mω²x² requires the wavefunction to vanish at infinity, which forces the energy to take only discrete values:

Eₙ = ħω(n + ½),   n = 0, 1, 2, 3, …
E₀ = ½ħω           ← zero-point energy, non-zero even at T = 0
ΔE = ħω             ← equal spacing between every adjacent level

Unlike a classical spring, which can sit motionless with zero energy, the quantum oscillator cannot: Heisenberg's Δx·Δp ≥ ħ/2 forbids simultaneous certainty in position and momentum, so confining a particle forces a non-zero momentum spread and hence non-zero kinetic energy. This zero-point energy is directly measurable — it drives the Casimir effect between uncharged plates and keeps liquid helium liquid at atmospheric pressure down to absolute zero.

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Hermite polynomials, ladder operators, tunnelling

The full wavefunctions ψₙ(x) = Nₙ·Hₙ(ξ)·e^(−ξ²/2) combine a Gaussian envelope with Hermite polynomials Hₙ(ξ), each with exactly n nodes and alternating even/odd parity. An elegant algebraic shortcut avoids differential equations entirely: raising and lowering operators a⁺ and a⁻ satisfy [a⁻,a⁺] = 1 and step the quantum number up or down by exactly one, factorising the Hamiltonian as H = ħω(a⁺a⁻ + ½) — precisely the creation/annihilation operators used for photons in quantum electrodynamics.

A striking quantum feature: probability density is non-zero beyond the classical turning points x = ±xₙ, where a classical particle would have negative kinetic energy and instantly reverse. For the ground state, roughly 15.7% of the probability lies outside this classically forbidden region. At large n, Bohr's correspondence principle takes over: the rapidly oscillating quantum density averages out to match the classical probability distribution, which peaks at the turning points where a classical oscillator moves slowest.

Frequently asked questions

What is the zero-point energy of a quantum harmonic oscillator?

The zero-point energy is E₀ = ½ħω, the ground-state energy when n=0. It is non-zero because the Heisenberg uncertainty principle forbids a particle from simultaneously having zero position and zero momentum uncertainty, so even at absolute zero the oscillator retains this irreducible kinetic energy.

Why are the energy levels equally spaced?

The equal spacing ΔE = ħω follows from the ladder-operator algebra: the raising operator a⁺ and lowering operator a⁻ each change n by exactly 1, always changing the energy by exactly ħω. This is a direct consequence of the quadratic potential V = ½mω²x², and it is why phonons in a crystal and photons in a field mode both carry equally spaced energy quanta.

What is quantum tunnelling in the harmonic oscillator?

The wavefunction extends into regions beyond the classical turning points x = ±xₙ, where a classical particle could never be found because its kinetic energy would be negative there. For the ground state, about 15.7% of the probability lies outside the classical turning points.

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