⚛️ Quantum Harmonic Oscillator

The quantum harmonic oscillator — a mass on a spring in quantum mechanics — is one of the most important exactly-solvable models in physics. Its energy levels are E_n = ℏω(n + ½), quantized in steps of ℏω with a non-zero zero-point energy ½ℏω at n = 0. The wavefunctions involve Hermite polynomials multiplied by a Gaussian envelope. The probability of finding the particle tunnels beyond the classical turning points ±√(2n+1) (shown as dashed lines). 🇺🇦 Українська

Display

Energy Levels

Showing all levels — select n to zoom in
n0
Energy Eₙ / ℏω0.5
Turning pts ±1.00
Nodes0
Tunnel probability—

Key Quantum Features

At n = 0 (ground state) the wavefunction is a pure Gaussian — the minimum uncertainty state. For large n, the probability density peaks near the classical turning points (where classical motion is slowest), recovering the correspondence principle. The n-th eigenstate has exactly n nodes. Tunneling probability decreases rapidly with n as the wavefunction becomes more classical. Applications: molecular vibrations (IR spectroscopy), phonons in solids, vacuum fluctuations in quantum optics, Casimir effect, squeezed states in quantum information.

About this simulation

This simulation renders the exact eigenstates of the quantum harmonic oscillator, the model of a particle in a parabolic potential V(x) = ½x² (in natural units where ℏ = ω = m = 1). Each wavefunction ψ_n(x) is computed directly from its closed-form solution — a Hermite polynomial H_n(x) multiplied by a Gaussian envelope e^(−x²/2) and normalised by 1/√(2ⁿn!√π) — for quantum numbers n from 0 to 10. The energy of each level is E_n = (n + ½)ℏω, so the ground state carries a non-zero zero-point energy of ½ℏω rather than sitting at zero like a classical particle at rest.

🔬 What it shows

The parabolic potential V(x) = ½x² is drawn in blue, with the selected level's energy shown as a dashed horizontal line at height E_n = n + ½. Red dashed vertical lines mark the classical turning points ±√(2n+1) — the boundary a classical ball on this spring could never cross. The oscillator's wavefunction ψ_n(x) or its probability density |ψ_n(x)|² is plotted around that energy line, and a live tunnelling-probability readout integrates |ψ_n|² beyond the turning points.

🎮 How to use

Drag the Quantum number n slider (0–10) to move between eigenstates and watch the number of wavefunction nodes increase by exactly one each step. Use the ψ(x), |ψ|² and Both buttons to switch between the raw wavefunction, its probability density, or an overlay of both. Click "Show all n ≤ 10" to stack every eigenstate's curve on the potential well at once, with the currently selected level highlighted, so you can compare node counts and energy spacing across states directly.

💡 Did you know?

Because the ground state (n = 0) is a pure Gaussian with no nodes, it is the unique minimum-uncertainty state allowed by the Heisenberg uncertainty principle. As n grows, the probability density piles up near the classical turning points — exactly where a classical mass on a spring moves slowest — which is the quantum-to-classical correspondence principle in action, visible directly with the "Show all" view.

Frequently asked questions

What exactly does this simulation compute?

It evaluates the exact analytic eigenstates of the 1-D quantum harmonic oscillator for quantum numbers n = 0 to 10, using natural units where ℏ = ω = m = 1. Each wavefunction is ψ_n(x) = N_n · H_n(x) · e^(−x²/2), where H_n is the physicist's Hermite polynomial computed by the standard three-term recurrence and N_n = 1/√(2ⁿn!√π) is the normalisation constant, so no numerical differential-equation solver is involved — the curves are the textbook closed-form solutions.

Why is the ground-state energy not zero?

The energy levels follow E_n = (n + ½)ℏω, so even the lowest state n = 0 has energy ½ℏω rather than zero. This zero-point energy is a direct consequence of the Heisenberg uncertainty principle: a particle confined to the potential well cannot have both zero position spread and zero momentum spread, so it can never sit perfectly still at the bottom of the well.

What are the dashed lines in the plot?

The dashed horizontal line marks the energy E_n = n + ½ (in units of ℏω) of the currently selected state. The dashed vertical lines mark the classical turning points at x = ±√(2n+1) — the positions where a classical particle with the same total energy would momentarily stop and reverse direction, since all its energy has become potential energy at that point.

What does the tunnelling probability readout mean?

The simulation numerically integrates the probability density |ψ_n(x)|² outside the classical turning points and divides it by the total probability, giving the fraction of the quantum particle's probability that lies in the classically forbidden region. This probability is largest for the ground state and shrinks rapidly as n increases, since higher-energy states become more classical in character.

Why does the number of nodes matter?

The n-th eigenstate's wavefunction ψ_n(x) crosses zero exactly n times, so the node count directly identifies the quantum number: the ground state (n = 0) has no nodes and is a single Gaussian hump, while each successive state adds one more node and one more oscillation. This node-counting rule is a general feature of bound-state eigenfunctions in one dimension, not specific to the harmonic oscillator.