Unitary evolution and exact revivals

Û(t) = exp(−iĤt/ℏ) is defined by its power series, and for a general Ĥ that series is real work. In an eigenbasis it is not: Ĥ|n⟩ = Eₙ|n⟩ gives Ĥᵏ|n⟩ = Eₙᵏ|n⟩, the series resums term by term, and Û(t)|n⟩ = exp(−iEₙt/ℏ)|n⟩. So Û is diagonal with entries of modulus one, and a diagonal matrix in an orthonormal basis is unitary precisely when every entry has modulus one. The recipe is three lines and never changes. Expand once, cₙ = ⟨n|ψ(0)⟩ = √(2/L) ∫₀ᴸ sin(nπx/L) ψ(x, 0) dx. Phase each coefficient, cₙ(t) = cₙ e^{(−iEₙt/ℏ)}. Resum, ψ(x, t) = Σ cₙ(t) √(2/L) sin(nπx/L). Numerically that is a truncation, not a time-stepper: hold the cₙ in a NumPy array of length N, build phase = numpy.exp(−1j * n * n * E1 * t / hb), and take one matrix product against the mode matrix. There is no timestep error to control, because you never take a step — t = 10⁶ Tᵣₑᵥ costs exactly what t = 0 costs. The only error is the tail you truncated, and since the cₙ of a smooth packet fall away fast, N of order 40 is usually plenty.

Read the propagator twice. First on the amplitudes: |cₙ(t)| = |cₙ(0)| for every n, because the phase has modulus one. So the energy distribution of the state is a constant of the motion, and with it ⟨Ĥ⟩ = Σ|cₙ|²Eₙ, the spread ΔE, and the expectation of any operator that commutes with Ĥ. For an electron in a 1.00 nm box prepared in (|1⟩ + |2⟩)/√2, ⟨Ĥ⟩ = 2.5E₁ = 0.940 eV and ΔE = 1.5E₁ = 0.564 eV, now and for ever. Second on the phases. The density matrix in the energy basis has elements ρₘₙ(t) = cₘ(t)c*ₙ(t) = ρₘₙ(0)e^{(−i(Eₘ − Eₙ)t/ℏ)}: the diagonal is untouched and every off-diagonal element rotates. That is the whole of the dynamics of a closed system with a known spectrum. If an observable's matrix is diagonal in the energy basis, its expectation cannot move; if it carries off-diagonal elements, they are the only route by which it can.

Square the evolved state and the phases stop hiding: |ψ(x, t)|² = Σ over m, n of c*ₘcₙψₘ(x)ψₙ(x)e^{(−i(Eₙ − Eₘ)t/ℏ)}. The m = n terms are static; every other term oscillates at a Bohr frequency ωₘₙ = (Eₘ − Eₙ)/ℏ. Because Eₙ = n²E₁ those frequencies are (m² − n²)E₁/ℏ, and m² − n² is an integer for every pair. With E₁ = 0.376 eV at L = 1.00 nm the fundamental is E₁/ℏ = 5.71 × 10¹⁴ rad s⁻¹, and the low pairs sit at ω₂₁ = 3E₁/ℏ = 1.71 × 10¹⁵, ω₃₂ = 5E₁/ℏ and ω₃₁ = 8E₁/ℏ rad s⁻¹. The same numbers are spectroscopy read the other way round: 3E₁ = 1.128 eV is the 2 → 1 photon, wavelength 1.10 μm. This is what is special about the box. A generic potential has gaps in no rational ratio, so its beats never all re-phase at once and the motion is merely quasi-periodic; here every beat is a harmonic of one frequency, and a harmonic series has a period.

Ask when every phase returns to 1 at once. Requiring exp(−in²E₁T/ℏ) = 1 for all n means n²E₁T/ℏ must be a multiple of 2π, and since n² is already an integer the single condition E₁T/ℏ = 2π suffices: Tᵣₑᵥ = 2πℏ/E₁ = h/E₁ = 4mL²/(πℏ). For an electron in a 1.00 nm box that is 11.0 fs; widen the box to 1.00 μm and the L² scaling stretches it to 11.0 ns. Nothing was approximated and no particular state was assumed — every state in this box returns exactly after the same Tᵣₑᵥ, even one such as (|1⟩ + |2⟩)/√2 whose density also repeats sooner. Half a revival is the more interesting instant. At t = Tᵣₑᵥ/2 each phase is exp(−iπn²) = (−1)ⁿ, and since ψₙ(L − x) = (−1)ⁿ⁺¹ψₙ(x), that combination is precisely a reflection: ψ(x, Tᵣₑᵥ/2) = −ψ(L − x, 0). A packet released near the left wall arrives, exactly mirrored, near the right wall at 5.50 fs. Rational fractions Tᵣₑᵥ p/q in between give fractional revivals, where the state is a finite superposition of shifted copies of itself — the fine structure of the quantum carpet.

A moving density is not the same thing as a moving mean. ⟨x⟩(t) = Σ over m, n of c*ₘcₙ xₘₙ e^{(−i(Eₙ − Eₘ)t/ℏ)}, so a frequency reaches ⟨x⟩ only if its matrix element survives. In this box xₘₙ = −8Lmn/(π²(m² − n²)²) when m − n is odd and vanishes when m − n is even with m ≠ n, because ψₘψₙ is then even about L/2 while x − L/2 is odd. Two consequences follow. A packet prepared symmetrically about L/2 excites only odd n, every pair then has m − n even, and ⟨x⟩ sits at L/2 for all time while |ψ(x, t)|² breathes visibly. And in (|1⟩ + |2⟩)/√2 the one surviving element gives ⟨x⟩(t) = L/2 − 0.1801L cos(3E₁t/ℏ), a swing from 0.320 nm to 0.680 nm at L = 1.00 nm. Differentiate and Ehrenfest's theorem becomes checkable: mₑ d⟨x⟩/dt = (8ℏ/3L) sin(3E₁t/ℏ), which is exactly 2Re[c*₁c₂p₁₂e^{(−iω₂₁t)}] built from p₁₂ = 8iℏ/(3L). Peak value 2.81 × 10⁻²⁵ kg m s⁻¹, by either route.

The revival is a property of this spectrum, not of quantum mechanics. It needs every gap Eₘ − Eₙ to be an integer multiple of one energy, and few systems oblige: the infinite box through n², the oscillator through n, the rigid rotor through l(l + 1). A cubic box still qualifies, since Eₙ = (nₓ² + n_y² + n_z²)E₁ is still an integer multiple, and its revival time is the same h/E₁. Round the walls off, though — a finite well, a real quantum dot — and the upper levels are pulled below n²E₁, the ratios stop being rational, and the exact revival degrades into a partial one that fades with each pass. Isolation is the second cost. Let the environment learn which level the particle occupies and ρₘₙ decays for m ≠ n while the diagonal survives, so the density relaxes to the static incoherent mixture Σ|cₙ|²|ψₙ(x)|² with ⟨x⟩ pinned at L/2. No energy has been absorbed; the state has simply stopped interfering with itself, and that is the failure mode a revival experiment is really testing.