Quantum Harmonic Oscillator: Energy Levels and Wave Functions

1. The Classical Harmonic Oscillator

A mass m attached to a spring with constant k obeys Newton's second law:

The potential energy is V(x) = (1/2)kx² = (1/2)mω²x², a parabola centred at the equilibrium. The total energy E = (1/2)mω²A² is continuous — a classical oscillator can have any energy whatsoever. The particle spends more time near its turning points (where it moves slowly) and less time near the centre (where it moves fastest), so its classical probability distribution peaks at x = ±A.

Quantum mechanics changes both of these facts dramatically: the allowed energies become discrete, and the spatial probability distribution shows interference structure absent from any classical picture.

2. Solving the Schrodinger Equation

The time-independent Schrodinger equation for the harmonic oscillator is:

Introducing the dimensionless coordinate ξ = x / x₀ where x₀ = sqrt(ℏ/mω) is the characteristic length of the ground state, the equation becomes:

For large |ξ|, the dominant behaviour is e²¹⁂². To get a normalisable solution we require the wave function to vanish at infinity, which forces ε to take only specific discrete values: ε = 2n + 1 for non-negative integers n = 0, 1, 2, …

3. Energy Eigenvalues and Zero-Point Energy

Substituting ε = 2n + 1 back into the energy relation gives the celebrated result:

The energy levels are equally spaced with gap ΔE = ℏω, the energy of one quantum of vibration (a phonon in a lattice, or a photon in a field mode). This equal spacing is a unique feature of the parabolic potential and does not hold for anharmonic potentials such as the Morse oscillator.

Zero-Point Energy

The ground state (n = 0) has energy:

This is the zero-point energy — the irreducible minimum energy that the oscillator retains even at absolute zero temperature. It has no classical analogue: a classical spring can be motionless at its equilibrium with zero energy. The quantum version cannot, because the Heisenberg uncertainty principle Δx Δp ≥ ℏ/2 forbids simultaneous certainty about both position and momentum. Confining a particle to a small region forces a large momentum spread, and hence a non-zero kinetic energy.

4. Hermite Polynomials and Wave Functions

The complete normalised wave functions in terms of dimensionless coordinate ξ = x / x₀ are:

The functions Hₙ(ξ) are the Hermite polynomials, an orthogonal polynomial family satisfying the differential equation:

They obey the three-term recurrence:

The first several Hermite polynomials are:

Notice that Hₙ is an even polynomial when n is even, and odd when n is odd. This means that even-n wave functions are symmetric about x = 0 (even parity) and odd-n wave functions are antisymmetric (odd parity). Each ψₙ has exactly n nodes — points where it crosses zero — reflecting the n quanta of excitation.

5. Probability Density and Quantum Tunnelling

The probability density of finding the particle at position x in state n is:

A striking feature is quantum tunnelling: the probability density is non-zero outside the classical turning points x = ±xₙ where xₙ = x₀ sqrt(2n + 1). At the turning points a classical particle has zero kinetic energy and immediately reverses; a quantum particle has a non-zero amplitude extending exponentially into the forbidden region. The probability of finding the ground-state particle beyond its classical turning points is about 15.7%.

The ground-state probability density is a pure Gaussian:

This is the minimum-uncertainty state: the product Δx Δp attains its lower bound ℏ/2 exactly for the harmonic oscillator ground state. No other quantum state is simultaneously as localised in both position and momentum.

6. Ladder Operators

An elegant algebraic approach avoids solving differential equations entirely by introducing the raising (creation) and lowering (annihilation) operators:

These operators satisfy the canonical commutation relation [a⁻, a⁺] = 1, and act on energy eigenstates as:

The Hamiltonian factorises as H = ℏω(a⁺a⁻ + 1/2), and the number operator N = a⁺a⁻ has eigenvalues n. The entire spectrum is generated algebraically: starting from a⁻|0⟩ = 0 and normalising, all higher eigenstates follow as |n⟩ = (a⁺)ⁿ |0⟩ / sqrt(n!).

7. The Correspondence Principle

Niels Bohr's correspondence principle demands that quantum mechanics reduces to classical mechanics in the limit of large quantum numbers. For the harmonic oscillator this is beautifully verifiable. At large n, the classical probability distribution of a particle oscillating between its turning points is:

This diverges at the turning points x = ±xₙ, reflecting the fact that the particle moves slowest there and spends the most time near its extremes. As n increases, the quantum probability density |ψₙ(x)|² develops an increasing number of rapid oscillations that average out to the classical distribution when smoothed over a few oscillation cycles — a phenomenon called dephasing or the Ehrenfest theorem in action.

Quantitatively, when n » 1 the energy spacing ℏω becomes negligible compared to the total energy nℏω, so the spectrum appears effectively continuous from the classical standpoint. The quantum-classical transition is not sharp but occurs smoothly as the de Broglie wavelength becomes much smaller than the oscillation amplitude.

8. Physical Applications

Molecular Vibrations

Diatomic molecules such as N₂ and HCl vibrate near their equilibrium bond length in a potential well that is approximately harmonic for small displacements. The vibrational energy levels Eₙ = ℏω(n + 1/2) are directly observed in infrared spectroscopy as absorption lines spaced ℏω apart. At room temperature, only the lowest few levels are significantly populated, so the harmonic approximation is excellent. For large displacements the true potential deviates (Morse potential), allowing molecular dissociation.

Crystal Lattice Phonons

In a solid crystal, each atom vibrates about its equilibrium lattice site. Treating the lattice as a collection of coupled harmonic oscillators and diagonalising the system yields normal modes called phonons, each with energy quanta ℏω. The heat capacity of a solid at low temperatures is governed by these phonon modes: the Einstein model (all modes at one frequency) and the Debye model (linear dispersion) both derive directly from the quantum harmonic oscillator energy spectrum.

Quantum Computing: Superconducting Qubits

Superconducting qubits such as the transmon are anharmonic quantum oscillators. The Josephson junction introduces a nonlinear inductance that tilts the equally spaced harmonic ladder into an unequal-spacing, allowing selective microwave addressing of the two-level |0⟩, |1⟩ subspace. The transmon is deliberately operated in the regime where only the first two energy levels are resonantly driven, making it a high-quality artificial qubit for quantum computing.