Wavefunction & potential well

The wavefunction ψ is complex-valued and cannot be read directly; what an instrument records is |ψ|², the probability density of finding the particle. In an infinite well, level n always carries exactly n−1 nodes and its energy climbs as n², so the levels spread further apart the higher you go. A harmonic oscillator does the opposite, with evenly spaced rungs; a finite well lets the wavefunction leak past the walls; and superposing n=1 with n=2 sets the probability cloud sloshing at the beat frequency of the two energies.

▸ Building the infinite square well (hard walls at x=0, x=L)…

▸ Solving the Schrödinger equation — DISCRETE energy levels Eₙ ∝ n²

▸ Computing the wavefunction ψₙ(x) = sin(nπx/L) · n−1 nodes

▸ Squaring |ψ|² = probability density (the Born rule)

▸ Loading the harmonic oscillator & state superposition…

▸ Ready — Online. ✅

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Potential type—

Energy level n—

Energy Eₙ—

Nodes (n−1)—

Mean ⟨x⟩—

Quantum note

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Pick a level n on the right to climb the energy ladder · switch Scenarios to see |ψ|², the oscillator, superposition, or the finite well.

Wavefunction ψ(x) & probability density |ψ(x)|² ψ(x)|ψ|²