---
title: Time Evolution, Propagators, and the Heisenberg Picture
module: The Formalism of Quantum Mechanics
moduleNumber: 4
lessonNumber: 6
order: 406
summary: >
  Time evolution is generated by the Hamiltonian and implemented by a unitary
  operator that preserves probability. We build that operator, expand a state in
  stationary states to see why probability densities freeze while phases wind,
  introduce the propagator, transfer the time dependence onto operators in the
  Heisenberg picture, and derive Ehrenfest's theorem — which recovers classical
  equations of motion for expectation values and identifies conserved quantities
  as observables commuting with the Hamiltonian.
topics: [The Formalism of Quantum Mechanics]
sources:
  - book: Sakurai & Napolitano
    ref: "Ch. 2; §2.1 Time Evolution and the Schrödinger Equation, §2.2 The Schrödinger vs. Heisenberg Picture"
  - book: Shankar
    ref: "Ch. 4 §4.3 The Postulates; Ch. 6 The Classical Limit"
  - book: Griffiths & Schroeter
    ref: "Ch. 2 §2.1 Stationary States; Ch. 3 §3.5.3 Ehrenfest's Theorem"
draft: false
---

Between measurements a quantum state evolves deterministically, and the
[fifth postulate](/quantum-mechanics/formalism/the-postulates-and-quantum-measurement)
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,
$\ket{\psi(t)}=\hat U(t,t_0)\ket{\psi(t_0)}$. 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 $i\hbar\,\partial_t\ket{\psi}=\hat H\ket{\psi}$ gives
the operator equation

$$
i\hbar\,\frac{\partial}{\partial t}\hat U(t,t_0) = \hat H\,\hat U(t,t_0),
\qquad \hat U(t_0,t_0)=\mathbb{1}.
$$

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

$$
\;\hat U(t,t_0) = \exp\!\left[-\frac{i}{\hbar}\hat H\,(t-t_0)\right]\;
$$

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

- **Unitarity.** Since $\hat H$ is Hermitian, $\hat U^\dagger=\exp[+i\hat H(t-t_0)/\hbar]=\hat U^{-1}$,
  so $\hat U^\dagger\hat U=\mathbb{1}$. Probability is conserved:
  $\braket{\psi(t)|\psi(t)}=\braket{\psi(t_0)|\hat U^\dagger\hat U|\psi(t_0)}=\braket{\psi(t_0)|\psi(t_0)}$.
- **Generation by a Hermitian operator.** The Hamiltonian is the generator of time
  translation exactly as momentum generates space translation: both are
  $\hat U=e^{-i\hat G\theta}$ with $\hat G$ Hermitian, the general relationship
  between [symmetries and their generators](/quantum-mechanics/oscillator-and-symmetry/symmetries-generators-and-conservation-laws).

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**,

$$
\hat U(t,t_0) = \mathcal{T}\exp\!\left[-\frac{i}{\hbar}\int_{t_0}^{t}\hat H(t')\d t'\right]
= \mathbb{1} - \frac{i}{\hbar}\int_{t_0}^t\hat H(t')\d t'
+ \left(\frac{-i}{\hbar}\right)^2\!\!\int_{t_0}^t\!\!\d t'\!\int_{t_0}^{t'}\!\!\d t''\,\hat H(t')\hat H(t'') + \cdots,
$$

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](/atomic-physics/radiative-transitions-and-line-shapes/time-dependent-perturbation-golden-rule).

$$
% caption: Unitary evolution is a rigid rotation of the state vector in Hilbert
% space: the vector turns but its length is fixed, so total probability is
% conserved at every instant.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black] (0,0) circle (2.4);
\draw[black, ->] (0,0) -- (2.9,0) node[right, black!70] {basis 1};
\draw[black, ->] (0,0) -- (0,2.9) node[above, black!70] {basis 2};
\draw[->, black, very thick] (0,0) -- (60:2.4) node[above right, black!70] {$t_0$};
\draw[->, acc, very thick] (0,0) -- (20:2.4) node[right, acc] {$t$};
\draw[acc, ->] (60:2.4) arc (60:20:2.4);
\node[acc, anchor=west] at (38:2.65) {U rotates};
\end{tikzpicture}
$$

## Stationary states

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

$$
\hat U(t,0)\ket{n} = e^{-iE_n t/\hbar}\ket{n},
$$

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 $\ket{\psi(t)}=e^{-iE_n t/\hbar}\ket{n}$ gives
$\braket{\psi(t)|\hat A|\psi(t)}=\braket{n|\hat A|n}$, 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
$\ket{\psi(0)}=\sum_n c_n\ket{n}$ with $c_n=\braket{n|\psi(0)}$; then

$$
\ket{\psi(t)} = \sum_n c_n\,e^{-iE_n t/\hbar}\ket{n}.
$$

Each amplitude rotates at its own frequency $\omega_n=E_n/\hbar$. The probabilities
$\lvert c_n\rvert^2$ 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 $e^{-i(E_n-E_m)t/\hbar}$ between different energy components. Time
dependence in quantum mechanics is interference between stationary states beating at
their difference frequencies.

$$
% caption: A stationary state winds its phase uniformly while its probability
% density stays fixed; only relative phases between different energy components,
% beating at their difference frequency, produce observable motion.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% phase circle winding
\draw[black] (0,0) circle (1.6);
\draw[black, ->] (-2.0,0) -- (2.0,0);
\draw[black, ->] (0,-2.0) -- (0,2.0);
\draw[->, acc, very thick] (0,0) -- (50:1.6);
\node[acc, anchor=south west] at (50:1.6) {phase};
\node[anchor=north, black!70] at (0,-2.1) {phase winds};
% fixed density
\begin{scope}[xshift=5.6cm]
\draw[black, ->] (0,0) -- (3.4,0) node[right, black!70] {$x$};
\draw[black, ->] (0,0) -- (0,2.6);
\draw[acc, very thick] plot[domain=0.15:3.3, samples=110]
  (\x, {2.0*exp(-2.0*(\x-1.6)^2)});
\node[anchor=north, black!70] at (1.7,-0.15) {density steady};
\end{scope}
\end{tikzpicture}
$$

## The propagator

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

$$
\psi(x,t) = \braket{x|\psi(t)} = \int \braket{x|\hat U(t,0)|x'}\,\psi(x',0)\d x'
= \int K(x,t;x',0)\,\psi(x',0)\d x',
$$

which defines the **propagator** $K(x,t;x',0)=\braket{x|\hat U(t,0)|x'}$, 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,

$$
K(x,t;x',0) = \sum_n \psi_n(x)\,\psi_n^\ast(x')\,e^{-iE_n t/\hbar},
$$

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,

$$
K(x,t;x',0) = \int \frac{\d p}{2\pi\hbar}\,e^{ip(x-x')/\hbar}\,e^{-ip^2 t/2m\hbar}
= \sqrt{\frac{m}{2\pi i\hbar t}}\;\exp\!\left[\frac{im(x-x')^2}{2\hbar t}\right],
$$

a Gaussian kernel (evaluated by completing the square in the Gaussian integral) that
spreads any initial packet, the mechanism behind
[wave-packet spreading](/quantum-mechanics/wave-mechanics-1d/the-free-particle-and-wave-packet-dynamics).
The phase $m(x-x')^2/2\hbar t$ is the classical action of a free particle traveling
from $x'$ to $x$ in time $t$, divided by $\hbar$ — the first hint of the
path-integral formulation, in which the propagator is a sum of $e^{iS/\hbar}$ 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 $\hat U^\dagger\hat U=\mathbb{1}$,

$$
\langle\hat A\rangle_t = \braket{\psi(t)|\hat A|\psi(t)}
= \braket{\psi(0)|\hat U^\dagger\hat A\,\hat U|\psi(0)},
$$

the time dependence can be read as belonging to the operator
$\hat A_H(t)=\hat U^\dagger\hat A\,\hat U$ acting in the fixed state $\ket{\psi(0)}$.

> **Definition (Heisenberg picture).** States are frozen at their initial value
> $\ket{\psi}_H=\ket{\psi(0)}$ and observables carry the time dependence,
> $$\hat A_H(t) = \hat U^\dagger(t,0)\,\hat A_S\,\hat U(t,0).$$
> Expectation values, spectra, and all physical predictions are identical to the
> Schrödinger picture. The two coincide at $t=0$.

Differentiating $\hat A_H(t)$ and using $i\hbar\,\partial_t\hat U=\hat H\hat U$ (with
its adjoint) gives the equation of motion. For an observable with no explicit time
dependence,

$$
\;\frac{\d\hat A_H}{\d t} = \frac{i}{\hbar}\,[\hat H,\hat A_H]\;
$$

the **Heisenberg equation of motion**. It is the quantum image of Hamilton's
equation $\dot A=\{A,H\}$ with the Poisson bracket replaced by
$\tfrac{1}{i\hbar}[\ ,\ ]$. 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.

$$
% caption: Schrödinger versus Heisenberg picture: the same physics with the time
% dependence assigned either to the state (left, operators fixed) or to the
% operators (right, state fixed); expectation values agree at every instant.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
% Schrodinger
\node[anchor=south, black!75] at (1.1,2.5) {Schrodinger};
\draw[black, ->] (0,0) -- (2.4,0) node[right, black!70] {op still};
\draw[->, acc, very thick] (0,0) -- (55:2.0) node[above, acc] {state};
\draw[acc, ->] (25:1.4) arc (25:55:1.4);
% Heisenberg
\begin{scope}[xshift=6.2cm]
\node[anchor=south, black!75] at (1.1,2.5) {Heisenberg};
\draw[->, acc, very thick] (0,0) -- (25:2.4) node[right, acc] {op};
\draw[acc, ->] (55:1.4) arc (55:25:1.4);
\draw[->, black, very thick] (0,0) -- (0,2.0) node[above, black!70] {state still};
\end{scope}
\end{tikzpicture}
$$

### 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.

> **Theorem (Conserved observables).** An observable $\hat A$ without explicit time
> dependence is a constant of the motion if and only if $[\hat H,\hat A]=0$. Such an
> $\hat A$ shares an eigenbasis with $\hat H$, so its eigenvalues are good quantum
> numbers labeling the stationary states.

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](/quantum-mechanics/oscillator-and-symmetry/symmetries-generators-and-conservation-laws).
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 $\langle\cdot\rangle$ of the
equation of motion,

$$
\frac{\d\langle\hat A\rangle}{\d t} = \frac{i}{\hbar}\braket{[\hat H,\hat A]},
$$

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

$$
\frac{\d\langle\hat x\rangle}{\d t} = \frac{\langle\hat p\rangle}{m},
\qquad
\frac{\d\langle\hat p\rangle}{\d t} = -\bigl\langle V'(\hat x)\bigr\rangle
= \langle\hat F\rangle,
$$

> **Theorem (Ehrenfest).** The expectation values of position and momentum obey
> $$m\,\frac{\d^2\langle\hat x\rangle}{\d t^2} = -\bigl\langle V'(\hat x)\bigr\rangle,$$
> the quantum counterpart of Newton's second law: the mean position accelerates
> according to the mean force.

The theorem is why classical mechanics emerges for macroscopic bodies, but the
correspondence is not exact. Newton's law for the average would require
$\langle V'(\hat x)\rangle=V'(\langle\hat x\rangle)$, 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.

$$
% caption: Ehrenfest's theorem: the centroid of a narrow wave packet tracks the
% classical trajectory, since across a small packet the force is nearly constant so
% the mean force equals the force at the mean position.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (0,0) -- (7.2,0) node[right, black!70] {$x$};
\draw[black, ->] (0,0) -- (0,3.2) node[above, black!70] {$t$};
% classical trajectory (parabola-like curve)
\draw[black, very thick, dashed] plot[domain=0:3.0, samples=80] ({0.6 + 0.6*\x + 0.28*\x*\x}, \x);
\node[black!70, anchor=west] at (5.7,2.7) {classical};
% packet centroids as dots with small widths
\foreach \t in {0.3,1.0,1.7,2.4}{
  \pgfmathsetmacro\xc{0.6 + 0.6*\t + 0.28*\t*\t}
  \fill[acc] (\xc,\t) circle (2.0pt);
  \draw[acc, thick] (\xc-0.28,\t) -- (\xc+0.28,\t);
}
\node[acc, anchor=west] at (1.2,2.5) {packet};
\end{tikzpicture}
$$

> **Worked example.** For the harmonic oscillator $V(x)=\tfrac12 m\omega^2 x^2$ the
> force is linear, $V'(\hat x)=m\omega^2\hat x$, so Ehrenfest's equations close
> exactly on the expectation values:
> $$
> \frac{\d\langle\hat x\rangle}{\d t}=\frac{\langle\hat p\rangle}{m},
> \qquad \frac{\d\langle\hat p\rangle}{\d t}=-m\omega^2\langle\hat x\rangle.
> $$
> These are the classical oscillator equations, with solution
> $\langle\hat x\rangle(t)=\langle\hat x\rangle_0\cos\omega t + \tfrac{\langle\hat p\rangle_0}{m\omega}\sin\omega t$.
> The centroid of any oscillator state — spread out or not — oscillates at exactly
> the classical frequency $\omega$, with no correction, because the potential is
> quadratic. This exact tracking is the reason the
> [coherent state](/quantum-mechanics/oscillator-and-symmetry/coherent-and-squeezed-states)
> behaves as a classical oscillating packet.

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](/quantum-mechanics/oscillator-and-symmetry/ladder-operators-and-the-number-states)
module applies this machinery, solving the harmonic oscillator with ladder
operators and organizing the spectrum by symmetry.

[^sakurai-time]: **Sakurai & Napolitano**, _Modern Quantum Mechanics_ 3rd ed., §2.1–§2.2 — the time-evolution operator, its unitarity and composition, the propagator, and the Schrödinger versus Heisenberg pictures with the Heisenberg equation of motion. Cambridge, 2021.
[^griffiths-ehr]: **Griffiths & Schroeter**, _Introduction to Quantum Mechanics_ 3rd ed., §3.5.3 and §2.1 — Ehrenfest's theorem and stationary states; the emergence of classical motion for expectation values. Cambridge, 2018.