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A SMALL EXPERIMENT / Quantum mechanics

Particle in a box.

Change an electron’s confinement and connect its energy, wavefunction, and probability density.

01 / THE CONFINED ELECTRONIdeal model

A wave with boundaries.

Electron wavefunction and probability densityPlots load when JavaScript is enabled.

Position is shown as a fraction of the box width. Both vertical scales are dimensionless. The shaded interval is where we calculate the probability.

Change the confinement.

A narrower box raises the electron’s energy.

The relationship

At fixed n, doubling the box width reduces the energy to one quarter. At fixed width, doubling n raises the energy fourfold.

Electron energy
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Interior nodes
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Probability in interval
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What changes

Change the box width or quantum number to explore the model.

The model, explained.

En=n2h28meL2E_n = \frac{n^2h^2}{8m_{\mathrm e}L^2}
ψn(x)=2Lsin⁡ ⁣(nπxL)\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right)

Here L is the box width, mₑ is the electron mass, h is Planck’s constant, and n is a positive integer. The probability in an interval is the area under |ψ|² within it. A negative wavefunction is allowed; probability density is always nonnegative.

This is one electron in a one-dimensional infinite well, with zero potential inside and impenetrable walls. It is a stationary energy state, so its probability density does not move. Real molecules have finite, more complicated potentials. Width changes compare separate wells, not a time-dependent compression.

Read the derivation in OpenStax University Physics