The Heisenberg Picture

The Heisenberg picture is one of three mathematically equivalent formulations of the time evolution prescribed by Postulate 5 of quantum mechanics — the others being the Schrödinger picture (used implicitly in the postulates page) and the interaction (Dirac) picture. The three pictures are related by a unitary change of basis in time and predict identical physical results; they differ only in what carries the time dependence.

1. The Three Pictures at a Glance

Take a closed quantum system with time-independent Hamiltonian $\hat{H}$. The unitary time-evolution operator (see postulates §5) is

$$\hat{U}(t,t_0)=e^{-i\hat{H}(t-t_0)/\hslash }.$$

All three reproduce identical matrix elements:

$$⟨{\psi }_S(t)|{\hat{A}}_S|{\psi }_S(t)⟩=⟨{\psi }_H|{\hat{A}}_H(t)|{\psi }_H⟩.$$

2. Definition

Choose a reference time $t_0$ at which the Heisenberg and Schrödinger pictures coincide. For all later times define:

• States are frozen: $$|{\psi }_H⟩\equiv |{\psi }_S(t_0)⟩.$$
• Operators carry all time dependence: $$\boxed{{\hat{A}}_H(t)\equiv {\hat{U}}^{†}(t,t_0){\hat{A}}_S\hat{U}(t,t_0).}$$ At the reference time, ${\hat{A}}_H(t_0)={\hat{A}}_S$.

The Hamiltonian itself is the same in both pictures whenever it is time-independent, since $[\hat{H},\hat{U}]=0$:

$${\hat{H}}_H(t)={\hat{U}}^{†}{\hat{H}}_S\hat{U}={\hat{H}}_S\equiv \hat{H}.$$

3. The Heisenberg Equation of Motion

Differentiating the definition with respect to $t$ (and using $i\hslash {\partial }_t\hat{U}=\hat{H}\hat{U}$) gives the Heisenberg equation:

$$\boxed{i\hslash \frac{d{\hat{A}}_H(t)}{dt}=[{\hat{A}}_H(t),\hat{H}]+i\hslash {\left(\frac{\partial \hat{A}}{\partial t}\right)}_H.}$$

The last term is present only if ${\hat{A}}_S$ has explicit time dependence (e.g. a time-dependent external field); for most observables ($\hat{x}$, $\hat{p}$, the components of angular momentum, etc.) it vanishes and the equation reduces to

$$\frac{d{\hat{A}}_H}{dt}=\frac{i}{\hslash }[\hat{H},{\hat{A}}_H].$$

This is the direct quantum analogue of the classical Hamilton equation $df/dt=\{f,H\}$ with the Poisson bracket $\{,\}$ replaced by $(i\hslash {)}^{-1}[,]$ — a manifestation of the canonical quantization correspondence

$$\{f,g{\}}_{\text{PB}}⟷\frac{1}{i\hslash }[\hat{f},\hat{g}].$$

4. Conserved Quantities

If $\hat{A}$ has no explicit time dependence and $[\hat{A},\hat{H}]=0$, then ${\hat{A}}_H$ is constant in time:

$$\frac{d{\hat{A}}_H}{dt}=0⟺[\hat{A},\hat{H}]=0.$$

Symmetries of the Hamiltonian thus give rise to conserved Heisenberg-picture observables, just as in classical mechanics. The energy itself is always conserved (since $[\hat{H},\hat{H}]=0$) for time-independent $\hat{H}$.

5. Example: Harmonic Oscillator

For $\hat{H}=\frac{{\hat{p}}^2}{2m}+\frac{1}{2}m{\omega }^2{\hat{x}}^2$, the Heisenberg equations are

$$\frac{d{\hat{x}}_H}{dt}=\frac{{\hat{p}}_H}{m},\frac{d{\hat{p}}_H}{dt}=-m{\omega }^2{\hat{x}}_H,$$

which are formally identical to the classical equations and have solutions

$${\hat{x}}_H(t)={\hat{x}}_H(0)\cos \omega t+\frac{{\hat{p}}_H(0)}{m\omega }\sin \omega t,$$ $${\hat{p}}_H(t)=-m\omega {\hat{x}}_H(0)\sin \omega t+{\hat{p}}_H(0)\cos \omega t.$$

Equivalently, in terms of ladder operators $\hat{a},{\hat{a}}^{†}$ (which satisfy $[\hat{a},{\hat{a}}^{†}]=1$, $\hat{H}=\hslash \omega ({\hat{a}}^{†}\hat{a}+\frac{1}{2})$),

$${\hat{a}}_H(t)=\hat{a}(0)e^{-i\omega t},{\hat{a}}_H^{†}(t)={\hat{a}}^{†}(0)e^{+i\omega t}.$$

The Heisenberg picture makes the quantum-classical correspondence completely transparent here: operators evolve along orbits indistinguishable from classical phase-space trajectories.

6. Example: Free Particle

For $\hat{H}={\hat{p}}^2/(2m)$,

$$\frac{d{\hat{p}}_H}{dt}=0,\frac{d{\hat{x}}_H}{dt}=\frac{{\hat{p}}_H}{m},$$

so ${\hat{p}}_H(t)=\hat{p}(0)$ and ${\hat{x}}_H(t)=\hat{x}(0)+\hat{p}(0)t/m$. The unequal-time commutator is non-trivial:

$$[{\hat{x}}_H(t),{\hat{x}}_H(0)]=\frac{i\hslash t}{m}.$$

This kind of unequal-time commutator becomes central in QFT, where it underlies microcausality.

7. Equivalence with the Schrödinger Picture

Inserting $\hat{U}{\hat{U}}^{†}=1$ on either side of the Schrödinger expectation value,

$$⟨{\psi }_S(t)|{\hat{A}}_S|{\psi }_S(t)⟩=⟨\psi (t_0)|{\hat{U}}^{†}{\hat{A}}_S\hat{U}|\psi (t_0)⟩=⟨{\psi }_H|{\hat{A}}_H(t)|{\psi }_H⟩,$$

so all measurable quantities (probabilities, expectation values, transition amplitudes) coincide. Eigenvalues of ${\hat{A}}_H(t)$ at any fixed $t$ are the same as those of ${\hat{A}}_S$ — only the eigenvectors shift in time:

$${\hat{A}}_H(t)|a;t{⟩}_H=a|a;t{⟩}_H\text{with}|a;t{⟩}_H={\hat{U}}^{†}(t,t_0)|a{⟩}_S.$$

The Heisenberg-picture eigenstates evolve backward in time relative to the Schrödinger states they coincide with at $t_0$.

8. When to Use Which Picture

9. Generalization to QFT

In quantum field theory the Heisenberg picture is the standard choice: a field operator $\hat{\varphi }(x)=\hat{\varphi }(\mathbf{x},t)$ carries the full spacetime label, and Lorentz covariance is manifest. The state space is fixed (typically the vacuum $|0⟩$ or an asymptotic Fock state), and dynamics live entirely in the time evolution of the operators:

$$\hat{\varphi }(\mathbf{x},t)=e^{i\hat{H}t/\hslash }\hat{\varphi }(\mathbf{x},0)e^{-i\hat{H}t/\hslash }.$$

Vacuum correlation functions $⟨0|T\hat{\varphi }(x_1)\cdots \hat{\varphi }(x_n)|0⟩$ — the central objects of QFT — are inherently Heisenberg-picture quantities. See QFT/preliminaries.md § States vs. Fields.