Separation of variables and stationary states

Let Ĥ be time-independent and try the product Ψ(x, t) = ψ(x)T(t) in iħ∂Ψ/∂t = ĤΨ. The time derivative hits only T, Ĥ hits only ψ, and dividing by ψT leaves iħT′(t)/T(t) = (Ĥψ)(x)/ψ(x). The left side is a function of t alone, the right of x alone, and two functions of independent variables can agree everywhere only if both equal one constant. Name it E: the single PDE has split into iħT′ = ET and Ĥψ = Eψ. Watch where the assumption earned its keep — if V depended on t, the right side would still contain t after the division and no separation constant would exist. Separation is not a trick that always works; it is a statement about closed, undriven systems.

iħT′ = ET is first-order and linear: T(t) = e^(−iEt/ħ), a pure phase of modulus one, so the norm ⟨Ψ|Ψ⟩ never drifts — and hermiticity has already forced E real. The clock's rate is enormous: the ground state of an electron in a 1.0 nm box carries E₁ = 0.376 eV, so its phase turns at E₁/ħ = 5.7 × 10¹⁴ rad s⁻¹, one full cycle every 11.0 fs. And the rate is not even physically well defined: add a constant V₀ to the potential and every Eₙ shifts by V₀, respinning every clock together while changing nothing measurable. Absolute phase rates are gauge; only differences between them will ever appear in an experiment.

Put the system in a single Ψₙ = ψₙe^{(−iEₙt/ħ)}. The density is Ψₙ*Ψₙ, and the phase meets its conjugate: |Ψₙ|² = |ψₙ|², frozen. The same cancellation runs through every expectation value — for any time-independent operator A, ⟨A⟩ = ⟨ψₙ|A|ψₙ⟩ with no t left anywhere, and ΔH = 0 exactly. This is why "stationary", and it does not mean at rest: the box ground state has ⟨p²⟩ = (πħ/L)² > 0, plenty of kinetic energy with no motion of any probability. It is also why, in this semiclassical picture, an atom parked in an energy eigenstate does not radiate: its charge density is a static distribution, and static distributions launch no waves; the spontaneous decay of real excited levels comes from coupling to the quantised field, which this Ĥ leaves out. Nothing observable in the world of one eigenstate ever happens.

Now superpose: Ψ = aψ₁e^{(−iE₁t/ħ)} + bψ₂e^{(−iE₂t/ħ)} with a, b real. Squaring gives |Ψ|² = a²ψ₁² + b²ψ₂² + 2abψ₁ψ₂ cos[(E₂ − E₁)t/ħ]. The diagonal terms are the frozen stationary densities; all motion lives in the cross term, oscillating at the single frequency ω₂₁ = (E₂ − E₁)/ħ — a difference, immune to the V₀ gauge. For the 1.0 nm box the gap is E₂ − E₁ = 3E₁ = 1.13 eV, so the density sloshes at ν = ΔE/h = 273 THz, and with a = b = 1/√2 the mean position swings as ⟨x⟩ = L/2 − 0.180L cos ω₂₁t. That sloshing charge is exactly what radiates: the frequencies a system emits are its level gaps over h, which is why spectroscopy measures differences and never absolute energies.

Self-adjointness delivers more than real eigenvalues: for a purely discrete spectrum, as in the box, the |n⟩ form a complete orthonormal basis (a continuous part adds an integral over generalised eigenstates, which are not vectors in the Hilbert space), so any initial state expands as |Ψ(0)⟩ = Σcₙ|n⟩ with cₙ = ⟨n|Ψ(0)⟩, and linearity evolves each term separately: |Ψ(t)⟩ = Σcₙe^{(−iEₙt/ħ)}|n⟩. Each |cₙ|² — the probability that an energy measurement returns Eₙ — is constant forever: energy statistics are conserved even while the density churns. Numerically this is the spectral method: discretise Ĥ on a grid, call E, U = scipy.linalg.eigh(H), project c = U.conj().T @ ψ0 (U†, the conjugate transpose), and the state at any t is U @ (np.exp(−1j*E*t/ℏ)*c). One diagonalisation buys every time at once, with no step-size error accumulating — t = 10 ps costs the same as 10 fs.

Two boundaries mark the method's edge. Drive the system — V = V(x, t), a laser field, an oscillating gate — and the very first step fails: no separation constant exists, there are no stationary states, and evolution needs the time-ordered propagator of the previous topic rather than one phase per level. Degeneracy is a softer edge: if E₂ = E₃, any combination of ψ₂ and ψ₃ is again an eigenstate, its cross term's Bohr frequency is zero, and that whole superposition is itself stationary — "the" stationary states are no longer unique. What survives everywhere is the moral: a superposition across distinct energies is never stationary, and its motion is a sum of cosines at the gaps (Eₘ − Eₙ)/ħ and nothing else.