Physic Labs

Quantum mechanics

The wavefunction and its probabilistic interpretation

Mix the n=1n=1 and n=2n=2 stationary states of a particle in a one-dimensional box, watch the density ∣ψ∣2|ψ|^2 oscillate in time with beat ω=ΔE/ℏ\omega = \Delta E/\hbar, and check normalization ∫∣ψ∣2dx=1\int|ψ|^2dx = 1.

Undergraduate

Equipment

  • One-dimensional box hosting two eigenstates with their own time phases
  • “n=2 state fraction” slider setting the superposition
  • “Energy gap” slider between the two levels
  • Rotatable 3D figures of ψ and the density ∣ψ∣2|ψ|^2

Procedure

  1. Change the superposition mix

    Sweep the “n=2 state fraction” from 0 to 100: at 0 the density is the single-humped n=1 profile; at 100 the central node of n=2 appears. At intermediate values ∣ψ∣2|ψ|^2 sloshes back and forth between the two halves of the box in time.

  2. Change the energy gap and measure the beat

    Increase “Energy gap” and watch ∣ψ∣2|ψ|^2 oscillate faster. The eigenstates carry phases e−iEnt/ℏe^{-iE_nt/\hbar}, so the superposed density contains a cos⁡(ωt)\cos(\omega t) term with ω=(E2−E1)/ℏ\omega = (E_2-E_1)/\hbar — compare the observed period with this formula.

  3. Check normalization and the probability reading

    Whatever the mix, the area under ∣ψ∣2|ψ|^2 stays equal to 1 — the simulation notes this on the figure. Pick a mix, pause at two different instants, and predict which half of the box has the larger detection probability; explain that ∣ψ∣2|ψ|^2 is a probability density, not a particle trajectory.

Simulation

Experiment history

In early 1926 Erwin Schrödinger published his wave equation for the “state function” ψ, building on Louis de Broglie's matter-wave hypothesis (1924). Schrödinger himself initially hoped ψ was a real matter density, but Max Born proposed that same year — while analyzing scattering — that ∣ψ∣2|ψ|^2 is the probability density of finding the particle; he received the 1954 Nobel prize for this interpretation. The probabilistic reading turned quantum mechanics into a theory predicting ratios of measurement outcomes: superposing two eigenstates, as in this simulation, produces a density beat at frequency ΔE/ℏ\Delta E/\hbar — the origin of measurable Rabi beats and transition rates in atoms.

Related physicists

Max BornErwin SchrödingerLouis de Broglie

Related library topics

The wavefunction and its probabilistic interpretation