time evolution Schrödinger equation
The wave functions so far were pictures with time stopped. The engine that moves the picture is the time-dependent Schrodinger equation iℏ ∂Ψ/∂t = HΨ. H is the Hamiltonian that carries the total energy. A stationary state of definite energy keeps the shape |ψ|² and only the phase e−iEt/ℏ turns. Overlay two states of different energy and the phases fall out of step, and the probability begins to slosh from side to side. The first picture pulls time and follows the phase of a stationary state as it turns.
One stationary state follows time as Ψ(x,t) = ψ(x) e−iEt/ℏ. The real part Re(Ψ) sloshes and even flips sign as time runs, but the probability density |Ψ|² does not move, because the size of a phase is always 1 and squaring removes it. Change the time and the quantum number n, and for any stationary state only the phase turns while the shape stays still.
Mix two stationary states ψ₁ and ψ₂ of different energy with amplitudes c₁, c₂. Expanding |Ψ|² adds the squares of c₁ψ₁ and c₂ψ₂ to an interference term 2 Re[c₁c₂* ψ₁ψ₂* e−i(E₁−E₂)t/ℏ] where the two meet. That term changes sign with time and pushes the probability lump from side to side. Change the ratio of the two amplitudes and pull time, and the sloshing grows when the mix is even. ω=(E₂−E₁)/ℏ is the beat frequency made by the two phases falling out of step, not the oscillator ω and not the dispersion ω(k).
The time for one round trip of the sloshing is the beat period. The formula is T = h/(E₂−E₁), so a larger gap between the two energies makes a shorter period. Grow the gap (E₂−E₁) and read 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 the energies, the faster the slosh, and when the two energies coincide the sloshing stops.
Draw why the sloshing happens with two clock hands. The amplitudes c₁, c₂ are the lengths of the hands, 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 hands point the same way the probability leans one way, and when they point opposite it leans the other way. Turn time and each time the fast hand laps the slow one, one beat is finished.
Plot the average position ⟨x⟩(t) of the particle in the two-state mix against time and a sine curve comes out. The center of mass of the sloshing probability lump traveling back and forth is this oscillation. The frequency is the beat ω=(E₂−E₁)/ℏ, and the amplitude is proportional to the product c₁c₂ of the two amplitudes. With only one state the product is 0 and the average sits still, and when the two are mixed evenly the product is largest and the swing is largest. Change the mix and the amplitude grows and shrinks.