A particle on a ring feels no forces and meets no walls, yet its states form a set labelled by a number n. This sim shows each state twice: as a rung on an energy ladder, and as a wave drawn around the ring. It is the simplest model for ring-shaped molecules and tiny metal rings.

What's on screen

• Energy Levels (left panel): the rungs from n = 0 up to your |n|. Your rung is pink and drawn at the top, so the vertical scale changes with n.
• Real part of ψ (blue), plotted against x from 0 to 2π (R = 1), or wrapped on the ring.
• Imaginary part of ψ (green), drawn the same way.
• Yellow band: the probability density |ψ|². On the ring view a dashed purple circle marks the ring itself.
• Current State (sidebar): R, the energy E_n and the wavelength λ.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: R = 1, ħ = 1 and 2m = 1, so E_n is the number n² in units of ħ²/2mR². The sim offers only the circulating states e^inφ.

A bead on a frictionless wire can circle with any energy from zero upwards. A quantum particle on the ring is different because its wave must be single-valued: x and x + 2πR are the same point of the ring, so ψ has to return to its own value after one turn. With V = 0 the solutions are travelling waves:

Closing the wave onto itself keeps only whole numbers n, and the sign of n is allowed to be negative:

|n| whole wavelengths fit around the circumference 2πR. De Broglie's p = h/λ then gives the momentum, and on a ring momentum is angular momentum:

With V = 0 all the energy is kinetic, so E = p²/2m:

Doubling |n| halves the wavelength, doubles the momentum and quadruples the energy. In the sim's units (ħ = 1, 2m = 1, R = 1) E is simply n².

Periodicity allows n = 0. Then ψ₀ = 1/√(2πR) is a non-zero constant: it is normalised, it closes onto itself trivially, and it has E = 0, L_z = 0 and no wavelength (the sim prints "∞ (uniform)").

• No zero-point energy. A particle in a box must have ψ = 0 at both walls, so a constant is not allowed and its lowest state costs π²ħ²/2mL². On the ring a constant is allowed, so the lowest energy is zero.
• Normalisation. The density |ψ₀|² = 1/2πR, spread over the circumference 2πR, adds up to exactly 1.
• One state. This is the only rung that is not shared. The badge under the slider reads "Non-Degenerate (n=0)".

E = n²ħ²/2mR² does not see the sign of n, so E_−n = E_n. The two states are still different: p = nħ/R and L_z = nħ have opposite signs, so the sign of n is the sense of circulation. Several states with one energy are called degenerate. Every rung except n = 0 holds two, and the sim's badge reads "Doubly Degenerate (±n)".

• Frozen at t = 0 (Animate off): Re ψ = cos(nφ)/√2π is the same for ±n, and Im ψ = sin(nφ)/√2π changes sign.
• Running: the whole pattern slides towards +φ (rightwards on the plot) for n > 0, and the other way for n < 0.

A state with energy E_n evolves by one rotating phase factor. The curves in the sim are the real and imaginary parts of Ψ_n, so Animate Time Evolution makes them slide round the ring. The probability density is Ψ times its complex conjugate, so the time-dependent phase cancels:

• Stationary means |Ψ|² does not change in time. The curves move, but nothing you could measure does.
• |Ψ|² = 1/2πR is a probability density: the chance of finding the particle in a short arc is this value times the arc's length. It is the same for every arc, and for every n.
• Time Speed scales the screen clock only. The energy, the ladder and the band do not depend on it.

• Benzene. Treating its π electrons as free on a ring gives a first jump of about 5.9 eV, a photon near 210 nm. That lies in the ultraviolet region where benzene absorbs strongly, a fair estimate from a model with no chemistry in it.
• Bohr's orbits. De Broglie's argument for Bohr's allowed orbits is this closing condition: 2πr = nλ gives L = nħ. Real atoms are three-dimensional, so the ring is only the starting idea.
• Tiny metal rings. For a free electron on a ring of radius 50 nm, the first spacing is about 15 µeV (about 0.2 K), so experiments on such rings are done at millikelvin temperatures. Persistent currents set by a magnetic flux through the ring have been measured there.

Key points

• ψ(x + 2πR) = ψ(x) restricts n to whole numbers, including 0 and negative values.
• λ = 2πR/|n|, L_z = nħ and E ∝ n².
• n = 0 is a constant wave with E = 0; ±n share a rung and circulate in opposite directions.
• |ψ|² = 1/2πR for every circulating state.

Common mix-ups

• "n = −3 sits lower than n = +3." E depends on n², so both are on the same rung. The sign sets the direction of circulation.
• "At higher n the yellow band grows peaks." For the circulating states, |ψ|² = 1/2πR at every n. The curves wiggle faster, but their squares always add to a constant.
• "The wave is moving, so the state can't be stationary." Stationary refers to |Ψ|², which does not change. The motion is a phase.

Things to try on your own

• Step n from 0 up to 6 and write down E each time. Check that each gap is 2 larger than the one before.
• Set n = 6 and count the blue curve's crests across the plot. Compare with the wavelength readout.
• Turn Animate off and compare the Re and Im curves for n = 2 and n = −2.