In Time the Phase Turns and a Superposition Sloshes
Every wavefunction so far was a frozen snapshot. The engine that sets it in motion is the time-dependent Schrodinger equation iℏ ∂Ψ/∂t = HΨ, where H is the Hamiltonian, the total-energy operator. Solve it and a stationary state of definite energy keeps its shape |ψ|² fixed while only its phase e−iEt/ℏ spins around. But overlay two states of different energy and the two phases drift apart, and the probability begins to slosh from side to side. In the first figure, pull the time slider and follow how a stationary state turns its phase.
A single stationary state follows time as Ψ(x,t) = ψ(x) e−iEt/ℏ. The real part Re(Ψ) in the top panel sloshes and even flips sign as time runs, but the probability density |Ψ|² in the bottom panel stays dead still, because a phase has magnitude 1 and squares away to nothing. Move time with the slider and step the quantum number n, and you confirm that for any stationary state only the phase turns while the shape holds still.
Now mix two stationary states ψ₁ and ψ₂ of different energy with amplitudes c₁, c₂. Expanding |Ψ|² gives the squares of c₁ψ₁ and c₂ψ₂ plus an interference term 2 Re[c₁c₂* ψ₁ψ₂* e−i(E₁−E₂)t/ℏ] where the two meet. That term changes sign over time and pushes the probability lump from side to side. Change the ratio of the two amplitudes with the slider and pull time, and the sloshing grows largest when the mix is even. Here ω=(E₂−E₁)/ℏ is the beat frequency made by the two phases falling out of step, not the oscillator ω nor the dispersion ω(k).
The time for one full round trip of the sloshing is the beat period. Its formula is T = h/(E₂−E₁), so a larger gap between the two energies gives a shorter period. Grow the gap (E₂−E₁) with the slider and read off, in two numbers, how T shrinks while ω=(E₂−E₁)/ℏ grows. Here T is the period of the sloshing, not the transmission probability T of tunneling. The farther apart the energies, the faster the slosh, and when the two energies coincide the sloshing stops, an intuition you check by hand.
Why the sloshing happens is drawn with two clock hands. The amplitudes c₁, c₂ are the lengths of each hand, and the two hands turn at speeds proportional to their energies E₁, E₂. The interference term depends only on the relative angle of the two hands, namely (E₂−E₁)t/ℏ. When the two hands point the same way the probability leans one way; when they point opposite it leans the other. Turn the time slider, and each time the fast hand laps the slow one a full turn, one beat is complete.
Plot the average position ⟨x⟩(t) of the particle in the two-state mix against time and out comes a clean sine curve. The center of mass of the sloshing probability lump traveling back and forth is exactly this oscillation. Its frequency is again the beat ω=(E₂−E₁)/ℏ, and its amplitude is proportional to the product c₁c₂ of the two amplitudes. So with only one state present (product zero) the average sits still, and when the two are evenly mixed (product largest) it swings the most. Change the mix with the slider and watch the amplitude grow and shrink.