A particle trapped in a 1D infinite square well (ħ = m = 1) has exact stationary states ψₙ(x) = √2·sin(nπx) with energies Eₙ = n²π²/(2L²). The field here is built as a real superposition of the first four:
Ψ(x,t) = Σₙ cₙ·ψₙ(x)·exp(−iEₙt)
|Ψ(x,t)|² = Σₙ cₙ²ψₙ(x)² + 2Σₙ<ₘ cₙcₘψₙψₘ·cos((Eₙ−Eₘ)t)
The cross terms make the probability density genuinely oscillate ("quantum beating") even though each individual eigenstate is stationary — with c₁ and c₂ both nonzero, ⟨x⟩ visibly sloshes back and forth at the beat frequency E₂−E₁, a textbook signature of superposition that a single eigenstate alone cannot produce.
- Measure — draws a definite energy eigenstate n with the Born-rule probability pₙ = cₙ²/Σcₙ², then freezes Ψ into that pure, stationary ψₙ (density stops beating — a real collapse, not decoration).
- ⟨E⟩ — expectation energy, Σ pₙ·Eₙ, constant in time (energy is conserved even while the density beats).
- ⟨x⟩/L — expectation position, oscillates when more than one eigenstate is populated.