Separation of variables and the time-independent equation

The time-dependent Schrödinger equation, iħ ∂Ψ/∂t = −(ħ²/2m) ∂²Ψ/∂x² + V(x, t)Ψ, is a partial differential equation: one unknown function of two variables, differentiated once in t and twice in x. For an arbitrary V there is no general closed form, and even where one exists you would not reach it by direct attack. Separation of variables sidesteps the problem rather than solving it. Ask a narrower question — are there any solutions of the special product form Ψ(x, t) = ψ(x)φ(t), in which the x-dependence and the t-dependence never mix? — and the answer comes in three lines. The narrower question is worth asking because the special solutions turn out to form a complete set: once you have all of them, every other solution is a superposition of them. But keep the logic straight. The ansatz is a guess about the form of a solution, justified only by substituting it and finding that it works. Nothing so far promises that such a solution exists, and nothing claims a general Ψ has this form.

Put Ψ = ψφ into the equation. The time derivative touches only φ and the space derivative only ψ, so iħ ψ φ′ = −(ħ²/2m) ψ″ φ + V ψφ. Divide every term by the product ψφ — legitimate wherever it is non-zero — and you are left with iħ φ′/φ = −(ħ²/2m) ψ″/ψ + V. Now read the two sides. The left is a function of t and of nothing else. The right is a function of x and of nothing else, provided V = V(x). A function of t alone can equal a function of x alone only if neither actually varies: hold x fixed and the right side is a fixed number, so the left must equal that number for every t, and conversely. Both sides equal one constant. If instead V = V(x, t) — a laser field switched on, a well wall in motion — the right-hand side still contains t after the division, the argument collapses at this exact line, and the product ansatz buys nothing. That is the entire restriction on the method, and it is worth noticing that it shows up here, at the start, rather than at the end.

Call the constant E. Nothing in the algebra says what it is; the identification comes from outside. Two arguments agree. First, the time equation is iħφ′ = Eφ, so φ ∝ e^(−iEt/ħ), and comparing this with the free-particle plane wave e^{(i(kx − ωt))} gives E = ħω — the Planck relation the whole equation was built on. Second, the spatial equation reads Ĥψ = Eψ with Ĥ = −(ħ²/2m)d²/dx² + V the operator for total energy, so for normalised ψ the mean energy is ⟨H⟩ = ∫ψ*Ĥψ dx = E∫|ψ|² dx = E. The variance goes the same way: ⟨H²⟩ = ∫ψ*Ĥ(Eψ) dx = E², so σ_H² = E² − E² = 0. A separable state has a sharp energy, not a spread of energies, and E is its value. That is the strongest justification for the name — not that the units work out, but that an energy measurement on such a state returns E every single time.

Integrating iħφ′ = Eφ gives φ(t) = e^(−iEt/ħ), any constant of integration being absorbed into ψ. For real E this has modulus exactly one: e^(−iEt/ħ) × e^(+iEt/ħ) = 1. So |Ψ(x, t)|² = |ψ(x)|² |φ(t)|² = |ψ(x)|², frozen for all time. The same cancellation runs through every expectation value, since in ∫Ψ*ÂΨ dx the two phases meet as e^(+iEt/ħ)e^(−iEt/ħ) = 1, so ⟨x⟩, ⟨p⟩ and the mean of any operator carrying no explicit time dependence are constants as well. This is why the separable solutions are called stationary states, and it is a strong physical claim: an electron in the n = 2 level of an atom is not orbiting, and its probability cloud is not sloshing. The phase, meanwhile, is very much alive. It turns through 2π in a time h/E — ten femtoseconds or so for an electron confined to a nanometre — and no measurement on this state can see that rotation, because a global phase cancels out of every expectation value. Only the relative phase between two terms of a superposition is observable, which is exactly what makes superpositions the interesting case.

What remains is −(ħ²/2m)ψ″ + Vψ = Eψ, an ordinary differential equation in x, second order and linear. As an ODE it is unfussy: choose any real E and any V and solutions exist. Quantisation therefore does not come from the equation. It comes from insisting that ψ be an admissible wavefunction — normalisable, continuous, with ψ′ continuous wherever V is finite, and satisfying whatever the boundaries demand. In an infinite well of width L those conditions force ψ(0) = ψ(L) = 0, which kills every solution except sin(nπx/L) and leaves Eₙ = n²π²ħ²/2mL² = n²h²/8mL². For an electron in L = 1.00 nm the ladder is 0.376 eV, 1.504 eV, 3.384 eV — discrete because the box refuses everything else, not because the differential equation did. Ask for E = 0.5 eV and a mathematical solution still exists; it simply fails to vanish at both walls. Reading the spatial part as an eigenvalue problem is what makes the finite well, the harmonic oscillator and the hydrogen atom one technique instead of three.

Because Ĥ is linear, any sum Ψ(x, t) = Σ cₙ ψₙ(x) e^{(−iEₙt/ħ)} also solves the time-dependent equation, with the coefficients fixed once and for all by the initial state through cₙ = ∫ψₙ*(x)Ψ(x,0) dx. That sum is the general solution, and it is the reason separating was worth the trouble. But it is not an eigenstate. Feed it to Ĥψ = Eψ and the left side returns Σ cₙ Eₙ ψₙ e^{(−iEₙt/ħ)}, which is not E times the state for any single E unless exactly one cₙ is non-zero. Two occupied levels leave a cross term in |Ψ|² that beats at (Eₘ − Eₙ)/h — in the hundreds of terahertz for the lowest pairs of that 1.00 nm well — so the density genuinely moves and no time-independent equation can describe it. The rule to carry away is to write Ĥψ = Eψ only for a state you have already committed to being an energy eigenstate. Everything else evolves.