The Time-Evolution Operator and Energy Basis Dynamics

Derivation

The Time-Evolution Operator and Energy Basis Dynamics

D-148 Home PU-202 Threads energy · symmetry · waves Depends on Separation of Variables and Stationary States, Hermitian Operators: Real Eigenvalues and Orthogonal Eigenstates

Statement

For a system whose Hamiltonian Ĥ is Hermitian and time-independent, the state at time t is obtained from the state at t₀ by a unitary operator Û(t, t₀) = exp[−i Ĥ (t − t₀) / ħ], so that if |ψ(t₀)⟩ = Σ_n c_n |E_n⟩ then |ψ(t)⟩ = Σ_n c_n e^−iE_nt/ħ |E_n⟩: each energy-eigenstate amplitude merely accumulates a phase at rate E_n/ħ.

Why it matters

The time-dependent Schrödinger equation is a first-order differential equation in time, and its formal solution is not merely a function but an operator that propagates any initial state forward. Recognising this operator as a single object — the time-evolution operator — converts the problem of quantum dynamics into the algebra of one exponentiated Hermitian generator, and immediately exposes the deep structure: energy is the generator of time translations, exactly as momentum generates spatial translations.

The energy basis is special precisely because it diagonalises this generator. In that basis the dynamics decouples completely: an N-dimensional coupled problem becomes N independent phases rotating at their own frequencies ω_n = E_n/ħ. Every interference phenomenon in quantum mechanics — beats, Rabi oscillations, wave-packet spreading, revivals — is ultimately the reassembly of these independently precessing phases.

Assumptions

The Hamiltonian is time-independent.If Ĥ = Ĥ(t) the simple exponential fails; the propagator becomes a time-ordered exponential and successive Hamiltonians at different times need not commute, so Û no longer factorises into a single exp[−iĤt/ħ].

Ĥ is Hermitian.Hermiticity is what makes Û unitary and the eigenvalues E_n real; drop it and probability is not conserved and the "phases" acquire real exponents that grow or decay.

The energy eigenstates form a complete orthonormal basis.Assumed via the spectral theorem for the (self-adjoint) Hamiltonian; if the spectrum has a continuous part the sum becomes an integral, and if Ĥ is not essentially self-adjoint (e.g. a particle reaching a boundary in finite time) no unitary propagator exists without a choice of self-adjoint extension.

Derivation

iħ ∂_t |ψ(t)⟩ = Ĥ |ψ(t)⟩

The time-dependent Schrödinger equation — the postulated law of motion; we seek an operator that generates its solution. A

|ψ(t)⟩ = Û(t, t₀) |ψ(t₀)⟩

Define the propagator by requiring linearity in the initial state and Û(t₀, t₀) = 1. Because the equation is linear, one operator serves for all initial states. A

iħ ∂_t Û(t, t₀) = Ĥ Û(t, t₀)

Substitute step 2 into step 1; since |ψ(t₀)⟩ is arbitrary the operator equation must hold as an identity. A

∂_t Û = −(i/ħ) Ĥ Û

Divide by iħ. With Ĥ constant in time this is a linear operator ODE with constant coefficient. A

Û(t, t₀) = exp[ −i Ĥ (t − t₀) / ħ ]

Integrate the constant-coefficient equation. The operator exponential is defined by its power series Σ_k (1/k!)(−i(t−t₀)/ħ)^k Ĥ^k, which is legitimate because every term is a power of the same operator and hence self-commuting; term-by-term differentiation reproduces step 4. B

Û^† Û = exp[+iĤ(t−t₀)/ħ] exp[−iĤ(t−t₀)/ħ] = 1

Since Ĥ^† = Ĥ, taking the adjoint flips the sign in the exponent. The two exponentials share the operator Ĥ, so they commute and their exponents add to zero — Û is unitary, and probability ⟨ψ|ψ⟩ is conserved. C

Ĥ |E_n⟩ = E_n |E_n⟩, ⟨E_m|E_n⟩ = δ_mn, Σ_n |E_n⟩⟨E_n| = 1

Invoke the prior results: the time-independent Schrödinger equation supplies the eigenpairs, and Hermiticity guarantees real E_n with a complete orthonormal eigenbasis (spectral theorem). A

Û |E_n⟩ = Σ_k (1/k!)(−i(t−t₀)/ħ)^k Ĥ^k |E_n⟩ = e^{−iE_n(t−t₀)/ħ} |E_n⟩

Act term by term: Ĥ^k|E_n⟩ = E_n^k|E_n⟩, so the operator series collapses to the ordinary scalar exponential of the eigenvalue. Energy eigenstates are the eigenvectors of Û. B

|ψ(t₀)⟩ = Σ_n c_n |E_n⟩, c_n = ⟨E_n|ψ(t₀)⟩

Expand the initial state on the complete basis (insert the resolution of identity from step 7). The coefficients are fixed by orthonormality. A

|ψ(t)⟩ = Û Σ_n c_n |E_n⟩ = Σ_n c_n e^{−iE_n(t−t₀)/ħ} |E_n⟩

Apply Û using linearity and step 8. The evolution is diagonal in the energy basis: each amplitude simply rotates in phase. A

Result

Û(t, t₀) = e^{−iĤ(t−t₀)/ħ} ⟹ |ψ(t)⟩ = Σ_n c_n e^{−iE_n(t−t₀)/ħ} |E_n⟩

Reading. Time evolution is generated by the Hamiltonian: Û is the exponential of −i/ħ times energy times elapsed time. In the energy basis it is diagonal, so each eigenstate contribution keeps its magnitude |c_n| fixed and rotates its phase at angular frequency ω_n = E_n/ħ. A stationary state (single n) has trivial dynamics up to a global phase; all interesting motion comes from relative phases between different energies.

Units check. The exponent must be dimensionless. [E·t/ħ] = (J)(s)/(J·s) = 1. Equivalently ω_n = E_n/ħ has units J/(J·s) = s⁻¹, a genuine angular frequency, and ω_nt is a pure angle in radians. Û is dimensionless, as an operator on states must be.

Limiting cases

Stationary state: if only one c_n ≠ 0, then |ψ(t)⟩ = e^−iE_nt/ħ|E_n⟩ — the density |ψ|² and every expectation value ⟨Â⟩ are constant; hence the name "stationary".
Short times: to first order Û ≈ 1 − (i/ħ)ĤΔt, recovering the infinitesimal Schrödinger step and identifying Ĥ as the generator of time translations.
Group property: Û(t₂, t₁)Û(t₁, t₀) = Û(t₂, t₀) and Û(t, t₀)⁻¹ = Û(t₀, t): a one-parameter unitary group, reflecting time-translation symmetry of a static Hamiltonian.
Classical / large action: for a state peaked at large quantum numbers the phase E_nt/ħ varies enormously fast between neighbouring n; stationary-phase reassembly reproduces classical trajectories (Ehrenfest/WKB limit).

Breaks when

Time-dependent Hamiltonian. If Ĥ = Ĥ(t) with [Ĥ(t), Ĥ(t′)] ≠ 0, the single exponential is wrong; one must use the Dyson time-ordered exponential Û = T exp[−(i/ħ)∫Ĥ(t′)dt′]. Even the "energy basis" is ill-defined because instantaneous eigenstates change in time.
Open / non-Hermitian systems. For a subsystem coupled to an environment, or an effective non-Hermitian Ĥ with complex eigenvalues, Û is no longer unitary: amplitudes decay (|c_n|e^−Γ_nt/2) rather than merely rotating, and total probability leaks away.
Continuous or unbounded spectrum without self-adjointness. If Ĥ is only symmetric but not self-adjoint (deficiency indices nonzero), no unitary Û exists until a self-adjoint extension is chosen; the naive exponential is undefined.

Failure modes

Evolving a non-eigenstate by one phase. Writing |ψ(t)⟩ = e^−iEt/ħ|ψ(0)⟩ with a single E for a superposition. Only exact eigenstates carry one phase; a superposition needs each c_n multiplied by its own e^−iE_nt/ħ.
Applying the phase to |c_n|². Attaching e^−iE_nt/ħ to the probabilities. Probabilities in the energy basis are strictly constant; the phase lives on the amplitude only.
Sign error in the exponent. Using e^+iĤt/ħ. The physical convention with iħ∂_t = Ĥ forces a minus sign; the plus sign evolves backward in time.
Exponentiating a non-diagonal matrix element-wise. Computing e^−iĤt/ħ by exponentiating each entry of Ĥ in a non-eigenbasis. One must diagonalise first (or sum the matrix power series); the exponential of a matrix is not the matrix of exponentials.
Forgetting ħ / mixing E and ω. Writing e^−iEt with E in joules. The exponent needs E/ħ (or ω) so the argument is dimensionless.

Discussion

The central lesson is that Ĥ is the generator of time translations, in exact parallel with momentum generating spatial translations and angular momentum generating rotations. Each conserved quantity of Noether's theorem appears here as the Hermitian generator of a one-parameter unitary group; the time-evolution operator is that group for the symmetry of time-translation invariance. That a static Hamiltonian commutes with itself at all times — [Ĥ, Û] = 0 — is exactly the statement that energy is conserved.

The energy basis earns its privileged status because it simultaneously diagonalises Ĥ and Û. In any other basis the propagator mixes components and the equations of motion couple; in the energy basis they uncouple into independent oscillators, one per eigenvalue. This is the quantum analogue of normal-mode decomposition of a classical coupled system: the E_n/ħ are the normal frequencies and the |E_n⟩ are the normal modes.

Physically, everything that moves in a static quantum system is interference between different energies. The probability of finding the system in some state |φ⟩ is |Σ_n c_n⟨φ|E_n⟩e^−iE_nt/ħ|², whose cross terms oscillate at the Bohr frequencies ω_mn = (E_m − E_n)/ħ. These beat frequencies — not the absolute energies — are what spectroscopy measures, which is why only energy differences are observable and the zero of energy is a free choice (it contributes only a global, unobservable phase).

At the level of rigour, the clean exponential relies on Ĥ being self-adjoint (not merely symmetric): Stone's theorem states that every strongly-continuous one-parameter unitary group Û(t) has a unique self-adjoint generator Ĥ with Û(t) = e^−iĤt/ħ, and conversely. The distinction matters for unbounded operators: the domain of Ĥ is a dense subspace, the exponential is defined for all states via the spectral measure dE_λ, and Û(t) = ∫ e^−iλt/ħ dE_λ covers continuous spectra where no normalisable eigenstates exist.

Common misconceptions. The overall phase e^−iE₀t/ħ is not physical — only relative phases between energies produce observable dynamics, so the choice of energy zero cannot matter. A "stationary state" is not motionless in every sense; its amplitude still rotates, but all observables are constant. And the energy basis is not simply "the eigenbasis of position or momentum" — it is the eigenbasis of whatever Ĥ happens to be, which is what makes it the correct basis for dynamics.

Worked examples

Two-level system: phase difference after a fixed time.

A spin-½ in a magnetic field has Ĥ with eigenvalues E₀ = 0 and E₁ = 4.0 × 10⁻²⁴ J. Start in |ψ(0)⟩ = (|E₀⟩ + |E₁⟩)/√2. Find the state and the relative phase at t = 5.0 ns.

|ψ(t)⟩ = (1/√2)[ e^−iE₀t/ħ|E₀⟩ + e^−iE₁t/ħ|E₁⟩ ]

Apply the result: each amplitude picks up its own phase. A

Δφ = (E₁ − E₀)t/ħ = E₁t/ħ

Only the difference is observable; factor out the (here zero) common phase. A

Δφ = (4.0 × 10⁻²⁴ J)(5.0 × 10⁻⁹ s) / (1.055 × 10⁻³⁴ J·s)

Insert numbers with ħ = 1.055 × 10⁻³⁴ J·s. A

Δφ = 2.0 × 10⁻³² / 1.055 × 10⁻³⁴ ≈ 1.9 × 10² rad ≈ 190 rad

Reading. The two amplitudes have precessed 190 rad apart — about 30 full turns — so the superposition has cycled through its interference pattern roughly 30 times. Reduced mod 2π, Δφ ≈ 190 − 30(2π) ≈ 1.5 rad. The state is (1/√2)[|E₀⟩ + e^−i(190)|E₁⟩], magnitudes unchanged.

Rabi-type oscillation of a survival probability.

Same two-level system, energies E₀ = 0 and E₁ = 4.0 × 10⁻²⁴ J, initial state |ψ(0)⟩ = (|E₀⟩ + |E₁⟩)/√2. Find the probability of returning to the exact initial state, P(t) = |⟨ψ(0)|ψ(t)⟩|², and its period.

⟨ψ(0)|ψ(t)⟩ = ½( e^−iE₀t/ħ + e^−iE₁t/ħ )

Take the overlap using orthonormality ⟨E_m|E_n⟩ = δ_mn. A

P(t) = ¼ |1 + e^−iΔφ|² = ¼ · 2(1 + cos Δφ) = cos²(Δφ/2)

Factor the common phase and use |1 + e^−iθ|² = 2 + 2cosθ. Here Δφ = E₁t/ħ. B

T = 2πħ/E₁ = 2π(1.055 × 10⁻³⁴ J·s)/(4.0 × 10⁻²⁴ J)

P = cos²(E₁t/2ħ) returns to 1 when E₁t/ħ = 2π, i.e. period T = 2πħ/E₁. A

P(t) = cos²(E₁t/2ħ), T ≈ 1.66 × 10⁻¹⁰ s ≈ 0.17 ns

Reading. The survival probability oscillates fully between 1 and 0 at the Bohr frequency ω₁₀ = E₁/ħ ≈ 3.8 × 10¹⁰ rad/s, period ≈ 0.17 ns. Absolute energies set the phases; only the energy difference sets this measurable period. Units check: [ħ/E] = (J·s)/J = s. ✓

Problems

(A) A particle is in a stationary state |E₃⟩ with E₃ = 6.0 eV. By how much has the wavefunction's phase advanced after 1.0 fs? Is the probability density time-dependent?
Solution
Δφ = E₃t/ħ. Convert E₃ = 6.0 × 1.602 × 10⁻¹⁹ J = 9.6 × 10⁻¹⁹ J. Then Δφ = (9.6 × 10⁻¹⁹)(1.0 × 10⁻¹⁵)/(1.055 × 10⁻³⁴) ≈ 9.1 × 10⁰ ≈ 9.1 rad (about 1.45 turns). The probability density |ψ|² = |e^−iE₃t/ħ|²|E₃⟩⟨E₃| is constant: the phase is global and cancels in |ψ|². That is exactly why it is called stationary.
(A) Show explicitly that Û(t₂, t₁) Û(t₁, t₀) = Û(t₂, t₀) for a time-independent Hamiltonian, and state where time-independence is essential.
Solution
Û(t₂, t₁)Û(t₁, t₀) = e^{−iĤ(t₂−t₁)/ħ} e^{−iĤ(t₁−t₀)/ħ}. Both exponentials are functions of the same operator Ĥ, so they commute and the exponents add: = e^{−iĤ[(t₂−t₁)+(t₁−t₀)]/ħ} = e^{−iĤ(t₂−t₀)/ħ} = Û(t₂, t₀). Time-independence is essential in the step "same operator ⟹ exponents add": if Ĥ(t) varied, the generators at different times would generally not commute (Baker–Campbell–Hausdorff terms survive) and only the time-ordered product would compose correctly.
(B) A system starts in |ψ(0)⟩ = √0.3 |E₁⟩ + √0.7 |E₂⟩ with E₁ = 1.0 eV, E₂ = 3.0 eV. Find the probabilities of measuring E₁ and E₂ at t = 2.0 fs, and the Bohr frequency of any observable oscillation.
Solution
Energy-measurement probabilities are |c_n|² and are time-independent: P(E₁) = 0.3, P(E₂) = 0.7 at all times, since |c_ne^−iE_nt/ħ|² = |c_n|². Observable oscillation (e.g. in ⟨x̂⟩) occurs at the Bohr frequency ω₂₁ = (E₂ − E₁)/ħ = (2.0 eV)(1.602 × 10⁻¹⁹)/(1.055 × 10⁻³⁴) ≈ 3.0 × 10¹⁵ rad/s (period ≈ 2.1 fs). The energy populations are frozen; only cross-basis observables move.
(B) For a two-level system with energies E₀, E₁, verify by matrix exponentiation in the energy basis that Û(t) = diag(e^−iE₀t/ħ, e^−iE₁t/ħ) is unitary, and confirm Û^†(t) = Û(−t).
Solution
In the energy basis Ĥ = diag(E₀, E₁), so Û = e^−iĤt/ħ = diag(e^−iE₀t/ħ, e^−iE₁t/ħ) (a diagonal matrix exponentiates entry-wise). Its adjoint is Û^† = diag(e^+iE₀t/ħ, e^+iE₁t/ħ). Then Û^†Û = diag(|e^−iE₀t/ħ|², |e^−iE₁t/ħ|²) = diag(1, 1) = 1: unitary. And Û^†(t) has entries e^+iE_nt/ħ = e^{−iE_n(−t)/ħ} = Û(−t): the adjoint runs evolution backward, consistent with Û⁻¹ = Û^†.
(C) The energy eigenstates carry a global phase freedom |E_n⟩ → e^iα_n|E_n⟩. Show that no physical prediction of the dynamics depends on the α_n. Then explain why a uniform shift Ĥ → Ĥ + V₀1 is likewise unobservable, but a relative shift of one level is not.
Solution
Rephasing the basis multiplies each expansion coefficient c_n = ⟨E_n|ψ⟩ → e^−iα_nc_n while |E_n⟩ → e^iα_n|E_n⟩; the product c_n|E_n⟩ in |ψ⟩ is invariant, so |ψ(t)⟩ and all |⟨φ|ψ(t)⟩|² are unchanged. For a uniform shift, Û → e^{−i(Ĥ+V₀)t/ħ} = e^−iV₀t/ħ e^−iĤt/ħ (the identity commutes with Ĥ). The extra factor is a global phase common to every term, so it cancels in |ψ|² and in all expectation values — energy zero is a free choice. A relative shift of just one level, however, changes a genuine difference E_m − E_n, hence a Bohr frequency ω_mn, which appears in the cross terms of |⟨φ|ψ(t)⟩|² and is measurable (e.g. a Stark or Zeeman shift of a spectral line).