Stationary states and superpositions

Put Ψₙ(x, t) = ψₙ(x)e^{(−iEₙt/ħ)} into Born's rule. The density is Ψₙ*Ψₙ = ψₙ*ψₙ e^(+iEₙt/ħ)e^{(−iEₙt/ħ)} = |ψₙ|², because a real energy makes the phase factor a unit complex number. The same cancellation runs through any sandwich integral ⟨A⟩ = ∫Ψ*ÂΨ dx whose operator carries no explicit t, so ⟨x⟩, ⟨p⟩, ⟨H⟩ and every variance are constants of the motion. The probability current is static too, and for a real bound eigenfunction it is zero everywhere. What "stationary" does not mean is at rest: in the infinite well ⟨p⟩ = 0 but ⟨p²⟩ = 2mEₙ, so an electron in the n = 2 state of a 0.50 nm well carries 6.02 eV of kinetic energy while its picture never twitches. The description is frozen; the energy is not zero.

To evolve an arbitrary start, first write it on the eigenbasis: Ψ(x,0) = Σ cₙψₙ(x) with cₙ = ∫ψₙ*Ψ(x,0)dx. That integral is an inner product, and it isolates one coefficient only because the ψₙ are orthonormal — multiply the sum by ψₘ*, integrate, and every term but m collapses. Take the parabolic start Ψ(x,0) = A x(L−x) in an infinite well. Normalising with ∫₀^L x²(L−x)²dx = L⁵/30 gives A = √(30/L⁵), and the standard integral ∫₀^L x(L−x)sin(nπx/L)dx = 2L³[1−(−1)ⁿ]/n³π³ then gives cₙ = 4√60/n³π³ for odd n and exactly zero for even n. So c₁ = 0.9993 and |c₁|² = 0.9986: the parabola is 99.86% ground state. The even levels vanish for a reason worth naming — Ψ(x,0) is symmetric about L/2 while ψ₂ is antisymmetric, so their overlap cancels term by term.

Because the moduli |cₙ| are fixed, the probability of each energy is fixed, and ⟨H⟩ = Σ|cₙ|²Eₙ never changes — that is energy conservation written in this language. For the parabola above, ⟨H⟩ = E₁Σ_odd 0.9986/n⁴ = 0.9986 × (π⁴/96) E₁ = 1.0132E₁, about 1.3% above the ground-state energy and constant forever. What does change is phase, and only relative phase matters. Factor e^{(−iE₁t/ħ)} out of the whole sum: it multiplies every term equally, and a global phase cancels in Ψ*Ψ and in every expectation value, so it is not physics. What survives is exp[−i(Eₙ−E₁)t/ħ] riding on each higher term. Two states differing by a global phase are the same state; two states differing by a relative phase are different states with different densities.

Square a two-term sum and three pieces appear. With real eigenfunctions and cₙ = |cₙ|e^(iφₙ), |Ψ|² = |c₁|²ψ₁² + |c₂|²ψ₂² + 2|c₁||c₂|ψ₁ψ₂cos(ω₂₁t − δ), where ω₂₁ = (E₂−E₁)/ħ and δ = φ₂ − φ₁. The first two terms are the frozen densities of the separate levels; every bit of the motion lives in the third. Three consequences follow. The beat carries the energy difference and nothing else — add 10 eV to both levels and the density is unchanged, because the shift is a global phase. Kill either coefficient and the cross term dies with it. A statistical mixture of the same two levels fails for a different reason: it keeps both weights but holds no definite relative phase, so averaging over that phase erases the third term and leaves a density that never moves. Preparing the pair coherently or statistically gives identical energy statistics and different densities, and the beat is the measurement that tells them apart. And the cross term integrates to zero over x by orthogonality, so the norm stays at 1 while the density sloshes: probability is moved around, never created.

Take L = 0.50 nm. E₁ = h²/8mL² = 2.41×10⁻¹⁹ J = 1.50 eV and Eₙ = n²E₁, so E₂ = 6.02 eV and ΔE = 3E₁ = 4.51 eV. Then ω₂₁ = ΔE/ħ = 6.86×10¹⁵ rad s⁻¹ and the beat period is T = h/ΔE = 0.917 fs, a sloshing frequency of 1.09 PHz — fast enough that only attosecond pulses can strobe it. With equal weights on ψ₁ and ψ₂, ⟨x⟩ = L/2 − (16L/9π²)cos(ω₂₁t) runs from 0.160 nm to 0.340 nm about the midpoint 0.250 nm, while ⟨H⟩ = 2.5E₁ = 3.76 eV sits still throughout. And because every Eₙ here is an integer multiple of E₁, all the phases realign together: the whole wavefunction, not merely the 1–2 pair, revives exactly after Tᵣₑᵥ = h/E₁ = 2.75 fs, three beats later. That exact revival is a peculiarity of the n² spectrum; a general potential gives only partial ones.

Three assumptions are doing work here. Completeness: that every admissible Ψ(x,0) really is Σcₙψₙ. Sturm–Liouville theory delivers it for a bound problem on a finite interval, but it is imported, not proved, and it fails as stated once the spectrum has a continuous part — free and scattering states need ∫dE φ(E)ψ_E, an integral, not a sum. A time-independent potential: if V depends on t there are no stationary states to expand on and the scheme is void from the first line. And a resolved basis inside degenerate subspaces: when two independent eigenfunctions share an energy, any combination of them is also an eigenfunction, so you must first choose an orthonormal pair — Gram–Schmidt will do — before the overlap integral means anything. One practical caveat too: the series is truncated in practice, and a start with a corner in it converges only as slowly as its coefficients decay.