A boundary makes a quantum

Why are only certain energies allowed?

Why are only certain energies allowed?

1. Increase n from 1 to 3 and count internal nodes.
2. Double well width and inspect energy.
3. Pause and compare signed amplitude with probability.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

An electron confined by infinitely high walls must have zero wavefunction at both boundaries. Only standing sine waves with an integer number of half-wavelengths fit. Their allowed kinetic energies scale as n²/L². The signed ribbon shows the real part of ψ; the translucent envelope shows |ψ|². A stationary energy state has time-independent probability density even though its complex phase evolves. The display exaggerates transverse size for readability; the confinement is one-dimensional.

Build the argument

1. State the potential and boundary conditions, then solve or approximate the Schrödinger equation.
2. Normalise the wavefunction so the integral of |ψ|² is one.
3. Interpret measurable distributions through |ψ|²; colour indicates signed amplitude or phase only where labelled.

Worked example 1

An electron in a 1 nm well has E₁≈0.376 eV.

Worked example 2

For n=2, E₂=4E₁; doubling width divides all energies by four.

Applications

Confinement, tunnelling and orbital structure underpin semiconductor devices, scanning tunnelling microscopes and atomic spectroscopy. The scene shows probability or wave amplitude, never a hidden classical orbit.

Common misconception

A probability cloud is not a swarm of electrons. Wavefunction sign or phase is different from probability density. These are ideal single-particle models.

A further investigation

The nth state has how many internal nodes?

Check your understanding

The nth state has how many internal nodes?

• n−1
• n+1
• Always zero

Model limits

Single nonrelativistic electron in a one-dimensional infinite well; transverse thickness is illustrative.

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