Separation of Variables and Stationary States

Derivation

Separation of Variables and Stationary States

D-139 Home PU-202 Threads energy · waves · matter Depends on The Time-Dependent Schrödinger Equation

Statement

For a potential that does not depend on time, V = V(r), the time-dependent Schrödinger equation admits separable solutions Ψ(r,t) = ψ(r) T(t). Substituting and separating forces both sides to equal a single real constant E, yielding the time-independent Schrödinger equation Ĥψ = Eψ for the spatial factor and the universal phase evolution T(t) = e^−iEt/ℏ for the temporal factor. The product is a stationary state: its probability density |Ψ|² is independent of time.

Why it matters

Almost every exactly solvable quantum system — the infinite well, harmonic oscillator, hydrogen atom, rigid rotor — is solved by first reducing the full dynamical equation to this eigenvalue problem. Separation of variables converts a partial differential equation in four variables into an ordinary (or lower-dimensional) eigenvalue equation plus a trivial phase, and the eigenvalues are exactly the allowed energies you measure spectroscopically.

The stationary states also form the basis in which all dynamics is expressed: any general solution is a superposition Ψ(r,t) = Σ_n c_n ψ_n(r) e^−iE_nt/ℏ. Understanding one stationary state and its phase clock is therefore the key to the time evolution of everything.

Assumptions

The potential is time-independent, V = V(r).If V depends on t the variables no longer separate; the Hamiltonian is not conserved and one needs time-dependent perturbation theory or a full propagator instead.

The Hamiltonian is the operator Ĥ = −(ℏ²/2m)∇² + V(r).If Ĥ contains explicit time or momentum-dependent gauge terms that do not commute with the split, the clean spatial/temporal factorisation fails.

Ĥ is Hermitian on the physical Hilbert space with suitable boundary conditions.Hermiticity guarantees the separation constant E is real; drop it (e.g. a complex absorbing potential) and eigenvalues become complex, giving decaying "quasi-stationary" resonances rather than true stationary states.

The separation constant is the same constant everywhere in space and time.A function of r alone equalling a function of t alone must be constant; if the domain is disconnected or the wavefunction is not required single-valued this step can fail locally.

Derivation

iℏ ∂Ψ(r,t)/∂t = Ĥ Ψ(r,t), Ĥ = −(ℏ²/2m)∇² + V(r)

Start from the time-dependent Schrödinger equation (prior result); V carries no t. A

Ψ(r,t) = ψ(r) T(t)

Ansatz: seek a separable (product) solution. We are not assuming all solutions have this form, only building a family of them. A

iℏ ψ(r) dT/dt = T(t) Ĥψ(r)

Substitute the product. Because Ĥ acts only on r it passes T(t), and ∂/∂t acts only on T, so ψ factors out of the time derivative. A

iℏ (1/T) dT/dt = (1/ψ) Ĥψ

Divide both sides by ψ(r)T(t) (legal wherever the product is non-zero; zeros are a measure-zero set handled by continuity). B

iℏ (1/T) dT/dt = E = (1/ψ) Ĥψ

The left side depends only on t, the right only on r; an equality of such functions on a connected domain forces both to a common constant E (dimension of energy). C

Ĥψ(r) = E ψ(r)

Read off the spatial equation from the right-hand equality: the time-independent Schrödinger equation, an energy eigenvalue problem. A

dT/dt = −(iE/ℏ) T(t)

Rearrange the temporal equality into a first-order linear ODE for T. A

T(t) = T(0) e^−iEt/ℏ

Integrate the ODE; the integrating factor gives a pure complex exponential. Absorb T(0) into ψ by normalisation, set T(0)=1. A

⟨ψ|Ĥ|ψ⟩ = E⟨ψ|ψ⟩ ⇒ E = E* (E real)

Take the inner product of step 6 with ψ; Hermiticity of Ĥ makes ⟨ψ|Ĥ|ψ⟩ real, so with ⟨ψ|ψ⟩>0 the eigenvalue E is real and the exponential is a phase, not a growth/decay. C

Ψ(r,t) = ψ(r) e^−iEt/ℏ

Recombine. This product solves the time-dependent equation for every eigenpair (E, ψ). A

Result

Ĥψ(r) = Eψ(r), Ψ(r,t) = ψ(r) e^−iEt/ℏ

Reading. The spatial shape ψ is fixed by the eigenvalue equation and does not move; time enters only through a global complex phase turning at angular frequency ω = E/ℏ (the de Broglie–Planck relation E = ℏω). Because the phase is spatially uniform, |Ψ(r,t)|² = |ψ(r)|² and every expectation value of a time-independent operator is constant — hence "stationary".

Units check. [E]/[ℏ] = J/(J·s) = s⁻¹, so Et/ℏ is dimensionless and the exponent is a pure number, as an argument of exp must be. Ĥψ and Eψ both carry units of (energy)×(units of ψ), consistent.

Limiting cases

Ground state, minimal energy. The smallest eigenvalue E₀ gives the slowest phase clock; nothing observable evolves, matching a system sitting in its lowest state.
Free particle, V = 0. ψ becomes a plane wave e^ik·r with continuous E = ℏ²k²/2m; separation still holds but the spectrum is continuous and states are non-normalisable.
Classical / large quantum numbers. For high n the energy spacings E_n+1−E_n shrink relative to E_n and superpositions of nearby stationary states reproduce classical motion (Ehrenfest / correspondence).
Two-state superposition. Combining E₁, E₂ gives density oscillating at the beat frequency (E₂−E₁)/ℏ — the smallest departure from stationarity.

Breaks when

The potential is time-dependent, V = V(r,t). Step 5 fails: the "constant" would have to depend on t, so ψ and T cannot be disentangled. Energy is not conserved and there are no stationary states.
Non-Hermitian or open systems. With an absorbing/complex potential (or coupling to a continuum), E acquires an imaginary part E = E_r − iΓ/2, so |Ψ|² ∝ e^−Γt/ℏ decays — a resonance, not a stationary state.
Relativistic regime. The first-order-in-time, second-order-in-space split assumes the non-relativistic Ĥ; the Dirac/Klein–Gordon equations require a different separation and multicomponent spinors.
Degenerate ill-posed boundary conditions. If no boundary conditions select a discrete spectrum (unbounded, non-confining potentials), the eigenvalue problem has no normalisable solutions and the "quantisation" step is vacuous.

Failure modes

"The phase factor is e^+iEt/ℏ." The sign is fixed by iℏ∂_t on the left; the correct convention is e^−iEt/ℏ. A flipped sign propagates into wrong group velocities and time-reversed dynamics.
Thinking a stationary state is "not moving in time". The state Ψ certainly evolves — its phase rotates continuously; only |Ψ|² and expectation values are static.
Attaching the phase to a superposition as a whole. Each eigenstate carries its own e^−iE_nt/ℏ; you cannot factor one global phase out of a multi-energy superposition.
Assuming every solution is separable. Separation produces a basis; a general Ψ is a sum over such products, not itself a single product.
Forgetting to normalise ψ, then also normalising T. Only the spatial factor is normalised; |T(t)|=1 automatically for real E.
Reusing separation when V depends on t. Students apply Ĥψ=Eψ to driven systems where it is simply invalid.

Discussion

The deep content of this reduction is that time-translation symmetry of the Hamiltonian (no explicit t) is what makes energy a good quantum number. In the language of symmetries, Ĥ generates time translations via the propagator Û(t) = e^−iĤt/ℏ; a stationary state is precisely an eigenvector of that generator, and its evolution is the one-dimensional representation e^−iEt/ℏ. Separation of variables is thus not an algebraic trick but the statement that energy eigenstates diagonalise the time-evolution operator.

The eigenvalue equation Ĥψ = Eψ is where quantisation enters: boundary and normalisability conditions on ψ select a discrete set {E_n} for bound states, while scattering states form a continuum. Nothing about "quantisation" is put in by hand — it emerges from demanding physically admissible (square-integrable, single-valued, continuous) spatial solutions of an otherwise ordinary differential equation.

Because the eigenfunctions of a Hermitian Ĥ are complete and orthogonal, they furnish the natural basis for the full dynamics: expand the initial state, attach each mode's phase clock, and the interference of clocks running at different rates ω_n = E_n/ℏ reproduces all observable time dependence. Spectroscopy measures differences E_m−E_n, never absolute energies, precisely because a common global phase is unobservable.

There is a subtlety about the overall energy zero. Shifting V → V + V₀ shifts every E_n → E_n + V₀ and multiplies every stationary state by the same phase e^{−iV₀t/ℏ}. For a single state this is unobservable, but for superpositions only the differences survive — a concrete illustration that in non-relativistic quantum mechanics the absolute energy is a gauge-like convention, whereas in a gravitating or relativistic context the zero of energy does become physical.

Common misconceptions. A stationary state is not a static wavefunction — it is a rigidly rotating phase; "stationary" refers only to probabilities and expectation values. And the time-independent equation is not a more fundamental law than the time-dependent one; it is a special consequence of it, valid only when V has no explicit time dependence.

Worked examples

Infinite square well, width L: phase-clock period of the ground state

Setup: electron, L = 1.0 nm. Ground energy E₁ = ℏ²π²/(2mL²). A

E₁ = π²ℏ²/(2m_eL²)

Symbolic form first, before inserting numbers. A

E₁ = π²(1.055×10⁻³⁴)² / [2(9.109×10⁻³¹)(1.0×10⁻⁹)²]

Insert ℏ, m_e, L in SI. A

E₁ ≈ 6.02×10⁻²⁰ J ≈ 0.376 eV

Arithmetic; convert with 1 eV = 1.602×10⁻¹⁹ J. A

ω₁ = E₁/ℏ = 6.02×10⁻²⁰/1.055×10⁻³⁴ ≈ 5.71×10¹⁴ s⁻¹; τ = 2π/ω₁

Phase clock period from T(t)=e^−iω₁t. A

E₁ ≈ 0.376 eV, τ = 2πℏ/E₁ ≈ 1.10×10⁻¹⁴ s

Reading. The ground-state phase turns once every ~11 fs; the probability density |ψ₁|² = (2/L)sin²(πx/L) is frozen the whole time.

Units check. J/(J·s)=s⁻¹ for ω; 2π/ω gives s. Good.

Two-state beat: superposition of n=1 and n=2 in the same 1 nm well

Setup: Ψ = (1/√2)(ψ₁e^{−iE₁t/ℏ} + ψ₂e^{−iE₂t/ℏ}), with E_n = n²E₁. B

|Ψ|² = ½|ψ₁|² + ½|ψ₂|² + ψ₁ψ₂cos[(E₂−E₁)t/ℏ]

Cross term oscillates at the energy difference; global phases cancel in the modulus. B

ΔE = E₂−E₁ = (4−1)E₁ = 3E₁

Use E_n∝n². A

ΔE = 3(6.02×10⁻²⁰ J) = 1.81×10⁻¹⁹ J ≈ 1.13 eV

Insert E₁ from example 1. A

ω_beat = ΔE/ℏ = 1.71×10¹⁵ s⁻¹, T_beat = 2π/ω_beat

The charge density sloshes across the well at this frequency. A

T_beat = 2πℏ/ΔE ≈ 3.66×10⁻¹⁵ s ≈ 3.7 fs

Reading. A single stationary state is static, but a two-energy mixture oscillates with period set purely by ΔE — the emission wavelength λ = 2πc/ω_beat ≈ 1.1 μm if the electron radiated on this transition.

Units check. J/(J·s)=s⁻¹; T_beat in s. Good.

Problems

(A) Sign of the phase. Show by direct substitution that Ψ = ψe^−iEt/ℏ solves iℏ∂_tΨ=ĤΨ given Ĥψ=Eψ, and that e^+iEt/ℏ does not.
Solution
LHS: iℏ∂_tΨ = iℏ(−iE/ℏ)ψe^−iEt/ℏ = Eψe^−iEt/ℏ. RHS: ĤΨ = (Ĥψ)e^−iEt/ℏ = Eψe^−iEt/ℏ. Equal ✓. With e^+iEt/ℏ the LHS gives iℏ(+iE/ℏ)Ψ = −EΨ while RHS gives +EΨ; equal only if E=0, so it fails.
(A) Stationarity of expectation values. Prove that for a stationary state ⟨Â⟩ is time-independent for any time-independent operator Â.
Solution
⟨Â⟩ = ∫Ψ*ÂΨ = ∫(ψ*e^+iEt/ℏ)Â(ψe^−iEt/ℏ) = e^+iEt/ℏe^−iEt/ℏ∫ψ*Âψ = ∫ψ*Âψ, independent of t. The two phases are complex conjugates and cancel because the same E appears in both factors.
(B) Why E must be real. Suppose E = E_r − iΓ/2 with Γ>0. Show |Ψ|² decays and find the lifetime; explain which assumption of the derivation this violates.
Solution
|Ψ|² = |ψ|²|e^{−i(E_r−iΓ/2)t/ℏ}|² = |ψ|²e^−Γt/ℏ, so probability falls with lifetime τ = ℏ/Γ. A complex E requires a non-Hermitian Ĥ; Hermiticity (step 9) is exactly the assumption forcing E real, so this is a resonance/decaying state, not a true stationary state.
(B) Non-separability under a driving field. For V(x,t)=½mω²x² + qEx cos(Ωt) (driven oscillator), attempt Ψ=ψ(x)T(t) and identify the exact step of the derivation that fails.
Solution
Substituting and dividing gives iℏ(1/T)dT/dt = (1/ψ)[−(ℏ²/2m)ψ'' + ½mω²x²ψ] + qEx cos(Ωt). The last term depends on both x and t and cannot be moved entirely to either side, so the two sides can no longer equal a single constant. Step 5 (the separation-constant step) fails; energy is not conserved and there are no stationary states.
(C) Revival time of a wave packet. In the infinite well E_n=n²E₁. Show that an arbitrary superposition returns exactly to its initial shape after T_rev = 2πℏ/E₁, and compute it for the 1 nm electron well.
Solution
Each term carries phase e^−iE_nt/ℏ = e^{−in²E₁t/ℏ}. After t=T_rev=2πℏ/E₁ the exponent is −in²(2π), an integer multiple of 2πi for every integer n (since n² is an integer), so every phase returns to 1 and Ψ is exactly restored. Numerically T_rev = 2π(1.055×10⁻³⁴)/(6.02×10⁻²⁰) ≈ 1.10×10⁻¹⁴ s ≈ 11 fs. (Note this equals the ground-state phase period because the level spacing is commensurate; in general potentials no exact revival exists.)