Time Evolution, Propagators, and the Heisenberg Picture

Between measurements a quantum state evolves deterministically, and the fifth postulate names the generator: the Hamiltonian. This lesson develops the structure of that evolution. The state is propagated by a unitary operator built from $\hat{H}$ , which preserves total probability exactly because it is unitary. Expanding in stationary states shows what physically moves and what stays fixed. Shifting the time dependence from states to operators gives the Heisenberg picture, in which the equations of motion take the form of classical mechanics with commutators in place of Poisson brackets, and conserved quantities appear as operators that commute with $\hat{H}$ .

The time-evolution operator

Define the operator that carries a state forward from an initial time, $∣ ψ (t) ⟩ = \hat{U} (t, t_{0}) ∣ ψ (t_{0}) ⟩$ . Three requirements fix it. It must reduce to the identity at $t = t_{0}$ , it must compose, $\hat{U} (t_{2}, t_{1}) \hat{U} (t_{1}, t_{0}) = \hat{U} (t_{2}, t_{0})$ , and it must preserve the norm so probability is conserved. Substituting the ansatz into the Schrödinger equation $i ℏ \partial_{t} ∣ ψ ⟩ = \hat{H} ∣ ψ ⟩$ gives the operator equation

i ℏ \frac{\partial}{\partial t} \hat{U} (t, t_{0}) = \hat{H} \hat{U} (t, t_{0}), \hat{U} (t_{0}, t_{0}) = 1.

For a time-independent Hamiltonian the solution is the operator exponential

\hat{U} (t, t_{0}) = exp [- \frac{i}{ℏ} \hat{H} (t - t_{0})]

defined through its power series, or equivalently through the spectral decomposition of $\hat{H}$ . Two properties are immediate.

Unitarity. Since $\hat{H}$ is Hermitian, $\hat{U}^{†} = exp [+ i \hat{H} (t - t_{0}) /ℏ] = \hat{U}^{- 1}$ , so $\hat{U}^{†} \hat{U} = 1$ . Probability is conserved: $⟨ ψ (t) ∣ ψ (t) ⟩ = ⟨ ψ (t_{0}) ∣ \hat{U}^{†} \hat{U} ∣ ψ (t_{0}) ⟩ = ⟨ ψ (t_{0}) ∣ ψ (t_{0}) ⟩$ .
Generation by a Hermitian operator. The Hamiltonian is the generator of time translation exactly as momentum generates space translation: both are $\hat{U} = e^{- i \hat{G} θ}$ with $\hat{G}$ Hermitian, the general relationship between symmetries and their generators.

When $\hat{H}$ depends on time the naive exponential fails, because $\hat{H} (t_{1})$ and $\hat{H} (t_{2})$ need not commute. The correct solution is the time-ordered exponential,

\hat{U} (t, t_{0}) = T exp - \frac{i}{ℏ} t_{0} \int t \hat{H} (t^{'}) d t^{'} = 1 - \frac{i}{ℏ} t_{0} \int t \hat{H} (t^{'}) d t^{'} + (\frac{- i}{ℏ})^{2} t_{0} \int t d t^{'} t_{0} \int t^{'} d t^{''} \hat{H} (t^{'}) \hat{H} (t^{''}) + \dots,

the Dyson series, in which later times stand to the left. The unitarity argument survives unchanged because $\hat{H} (t)$ stays Hermitian at each instant, so probability is conserved throughout. This series is the starting point of time-dependent perturbation theory.

Unitary evolution is a rigid rotation of the state vector in Hilbert space: the vector turns but its length is fixed, so total probability is conserved at every instant.

Stationary states

The evolution is simplest in the energy eigenbasis. Let $\hat{H} ∣ n ⟩ = E_{n} ∣ n ⟩$ . An energy eigenstate evolves by a pure phase,

\hat{U} (t, 0) ∣ n ⟩ = e^{- i E_{n} t /ℏ} ∣ n ⟩,

because the exponential of $\hat{H}$ acts on its own eigenvector by exponentiating the eigenvalue. These are the stationary states: the phase is unobservable in any expectation value, since $∣ ψ (t) ⟩ = e^{- i E_{n} t /ℏ} ∣ n ⟩$ gives $⟨ ψ (t) ∣ \hat{A} ∣ ψ (t) ⟩ = ⟨ n ∣ \hat{A} ∣ n ⟩$ , independent of time. A state of definite energy is physically frozen — its probability density and every expectation value are constant.

A general state is a superposition, and its evolution follows term by term. Expand $∣ ψ (0) ⟩ = \sum_{n} c_{n} ∣ n ⟩$ with $c_{n} = ⟨ n ∣ ψ (0) ⟩$ ; then

∣ ψ (t) ⟩ = n \sum c_{n} e^{- i E_{n} t /ℏ} ∣ n ⟩ .

Each amplitude rotates at its own frequency $ω_{n} = E_{n} /ℏ$ . The probabilities $∣ c_{n} ∣^{2}$ of the individual energies never change — energy is conserved — but observables that do not commute with $\hat{H}$ acquire time dependence through the relative phases $e^{- i (E_{n} - E_{m}) t /ℏ}$ between different energy components. Time dependence in quantum mechanics is interference between stationary states beating at their difference frequencies.

A stationary state winds its phase uniformly while its probability density stays fixed; only relative phases between different energy components, beating at their difference frequency, produce observable motion.

The propagator

In the position representation the evolution operator becomes an integral kernel. Insert position completeness on both sides of $\hat{U}$ ,

ψ (x, t) = ⟨ x ∣ ψ (t) ⟩ = \int ⟨ x ∣ \hat{U} (t, 0) ∣ x^{'} ⟩ ψ (x^{'}, 0) d x^{'} = \int K (x, t; x^{'}, 0) ψ (x^{'}, 0) d x^{'},

which defines the propagator $K (x, t; x^{'}, 0) = ⟨ x ∣ \hat{U} (t, 0) ∣ x^{'} ⟩$ , the amplitude for a particle at $x^{'}$ to be found at $x$ after time $t$ . Expanding $\hat{U}$ in the energy eigenbasis gives its spectral form,

K (x, t; x^{'}, 0) = n \sum ψ_{n} (x) ψ_{n}^{*} (x^{'}) e^{- i E_{n} t /ℏ},

a sum over stationary states weighted by their phases. The propagator contains the full dynamics: once it is known, any initial wavefunction is evolved by a single integration. For the free particle the energy sum becomes an integral over plane waves,

K (x, t; x^{'}, 0) = \int \frac{d p}{2 π ℏ} e^{i p (x - x^{'}) /ℏ} e^{- i p^{2} t /2 m ℏ} = \frac{m}{2 π i ℏ t} exp [\frac{im ( x - x ^{'} ) ^{2}}{2ℏ t}],

a Gaussian kernel (evaluated by completing the square in the Gaussian integral) that spreads any initial packet, the mechanism behind wave-packet spreading. The phase $m (x - x^{'})^{2} /2ℏ t$ is the classical action of a free particle traveling from $x^{'}$ to $x$ in time $t$ , divided by $ℏ$ — the first hint of the path-integral formulation, in which the propagator is a sum of $e^{i S /ℏ}$ over every trajectory, not only the classical one.

The Heisenberg picture

The description so far puts all time dependence in the state and holds operators fixed — the Schrödinger picture. An equivalent description holds states fixed and moves the time dependence onto operators. The two agree on every expectation value, which is all that is physical. Starting from the Schrödinger-picture expectation and inserting $\hat{U}^{†} \hat{U} = 1$ ,

⟨ \hat{A} ⟩_{t} = ⟨ ψ (t) ∣ \hat{A} ∣ ψ (t) ⟩ = ⟨ ψ (0) ∣ \hat{U}^{†} \hat{A} \hat{U} ∣ ψ (0) ⟩,

the time dependence can be read as belonging to the operator $\hat{A}_{H} (t) = \hat{U}^{†} \hat{A} \hat{U}$ acting in the fixed state $∣ ψ (0) ⟩$ .

Differentiating $\hat{A}_{H} (t)$ and using $i ℏ \partial_{t} \hat{U} = \hat{H} \hat{U}$ (with its adjoint) gives the equation of motion. For an observable with no explicit time dependence,

\frac{d A ^ _{H}}{d t} = \frac{i}{ℏ} [\hat{H}, \hat{A}_{H}]

the Heisenberg equation of motion. It is the quantum image of Hamilton's equation $\dot{A} = {A, H}$ with the Poisson bracket replaced by $\frac{1}{i ℏ} [,]$ . Structurally the two pictures are mirror images: in one the state rotates and operators stand still; in the other the operators rotate and the state stands still.

Schrödinger versus Heisenberg picture: the same physics with the time dependence assigned either to the state (left, operators fixed) or to the operators (right, state fixed); expectation values agree at every instant.

Conservation laws

The Heisenberg equation reads off conserved quantities at a glance. An observable $\hat{A}$ with no explicit time dependence is conserved — $d \hat{A}_{H} / d t = 0$ and its expectation value is constant in every state — precisely when it commutes with the Hamiltonian.

This is the quantum form of Noether's theorem: a symmetry generator that commutes with $\hat{H}$ is conserved, and the conservation of momentum, angular momentum, and parity in symmetric systems all follow this pattern, developed in the symmetries lesson. The energy itself is trivially conserved, $[\hat{H}, \hat{H}] = 0$ .

Ehrenfest's theorem

Applying the expectation of the Heisenberg equation to position and momentum recovers Newtonian mechanics for the averages. Taking $⟨ \cdot ⟩$ of the equation of motion,

\frac{d ⟨ A ^ ⟩}{d t} = \frac{i}{ℏ} ⟨ [\hat{H}, \hat{A}] ⟩,

and specializing to $\hat{H} = \frac{p ^ ^{2}}{2 m} + V (\overset{x}{^})$ . The needed commutators follow from $[\overset{x}{^}, \overset{p}{^}] = i ℏ$ : $[\hat{H}, \overset{x}{^}] = \frac{1}{2 m} [\overset{p}{^}^{2}, \overset{x}{^}] = - \frac{i ℏ}{m} \overset{p}{^}$ and $[\hat{H}, \overset{p}{^}] = [V (\overset{x}{^}), \overset{p}{^}] = i ℏ V^{'} (\overset{x}{^})$ . Substituting,

\frac{d ⟨ x ^ ⟩}{d t} = \frac{⟨ p ^ ⟩}{m}, \frac{d ⟨ p ^ ⟩}{d t} = - ⟨ V^{'} (\overset{x}{^}) ⟩ = ⟨ \hat{F} ⟩,

The theorem is why classical mechanics emerges for macroscopic bodies, but the correspondence is not exact. Newton's law for the average would require $⟨ V^{'} (\overset{x}{^})⟩ = V^{'} (⟨ \overset{x}{^} ⟩)$ , which holds only when $V^{'}$ is linear — free particle, uniform field, harmonic oscillator — or when the packet is narrow enough that the force is effectively constant across its width. For a localized packet in a slowly varying potential the centroid follows the classical trajectory; for a spread-out state in a strongly nonlinear potential the average force and the force at the average diverge, and genuinely quantum behavior persists.

Ehrenfest's theorem: the centroid of a narrow wave packet tracks the classical trajectory, since across a small packet the force is nearly constant so the mean force equals the force at the mean position.

The formalism module is now complete: states in a Hilbert space, observables as Hermitian operators, the Born rule and measurement, the continuous spectra of position and momentum, the uncertainty principle, and unitary time evolution with its classical limit. The algebraic oscillator and symmetry module applies this machinery, solving the harmonic oscillator with ladder operators and organizing the spectrum by symmetry.