Schrödinger and Heisenberg pictures

Intuition [Beginner]

A quantum system changes with time. The question is: what carries that change? In quantum mechanics there are three equivalent answers, called the Schrödinger picture, the Heisenberg picture, and the interaction picture. They give exactly the same predictions for every experiment. The difference is purely computational — which bookkeeping convention makes the algebra easiest for the problem at hand.

In the Schrödinger picture, the state vector $|\psi (t)⟩$ rotates in Hilbert space while the operators (the things you measure with) stay fixed. Think of a spinning arrow: the arrow moves, the detector sits still. The equation that governs this rotation is the Schrödinger equation, and the operator that drives it is the Hamiltonian $H$ — the total-energy operator. For a time-independent Hamiltonian the solution has a clean form: start with $|\psi (0)⟩$ and rotate it by the unitary operator $e^{-iHt/\hslash }$ to get $|\psi (t)⟩$ at any later time.

In the Heisenberg picture, the state stays frozen at its initial value $|\psi (0)⟩$ and the operators carry the time dependence instead. An observable $A$ becomes $A(t)=e^{iHt/\hslash }A{e}^{-iHt/\hslash }$. The physical predictions — expectation values, probabilities — come out identical because the rotation that was applied to the state has been absorbed into the operators. This is a change of perspective, not a change of physics.

Why would you freeze the state and move the operators? Because for some problems the operator equations are simpler. The Heisenberg equation of motion $\frac{dA}{dt}=\frac{i}{\hslash }[H,A]$ (with an extra term if $A$ depends explicitly on time) looks like Hamilton's equation from classical mechanics with the commutator $[\cdot ,\cdot ]$ replacing the Poisson bracket. That structural analogy is not a coincidence — it is the reason canonical quantisation works at all.

The interaction picture (also called the Dirac picture) splits the Hamiltonian into two pieces: $H=H_0+V$. The "free" part $H_0$ gets absorbed into the states (they evolve with $H_0$ alone) and the "interaction" part $V$ generates a time-dependent operator $V_I(t)$ that carries the remaining dynamics. This split is useful whenever you can solve the $H_0$ problem exactly and treat $V$ as a correction — which is the setup for time-dependent perturbation theory, scattering theory, and most of quantum field theory.

The key relation connecting all three pictures is a single identity for expectation values:

$$⟨\psi (t)|A|\psi (t){⟩}_S=⟨\psi (0)|A(t)|\psi (0){⟩}_H.$$

Same number on both sides. The left side is the Schrödinger-picture computation: evolve the state, apply the fixed operator. The right side is the Heisenberg-picture computation: hold the state fixed, evolve the operator. The unitary rotation that moves the state forward in time is the same rotation that moves the operators forward — just conjugated onto the other side of the sandwich.

Visual [Beginner]

Picture a record on a turntable. In the Schrödinger picture the record spins and the needle is fixed — the groove passes under the needle, producing sound that changes with time. In the Heisenberg picture the record sits still and the needle traces the groove — same sound, same physics, different accounting. The angle between needle and groove at every instant is identical in both descriptions.

Worked example [Beginner]

A spin-1/2 particle sits in a constant magnetic field $\mathbf{B}=B_0\hat{z}$. The Hamiltonian is $H=-\gamma B_0{S}_z=-\frac{\hslash \omega }{2}{\sigma }_z$ where $\omega =\gamma B_0$ is the Larmor frequency and ${\sigma }_z$ is the Pauli $z$-matrix. Take the initial state $|\psi (0)⟩=|{\uparrow }_x⟩=\frac{1}{\sqrt{2}}(|\uparrow ⟩+|\downarrow ⟩)$, spin-up along $x$.

Schrödinger picture. The evolution operator is $U(t)=e^{-iHt/\hslash }=e^{i\omega t{\sigma }_z/2}$. Since ${\sigma }_z$ is diagonal, this acts by multiplying each basis ket by a phase:

$$|\psi (t)⟩=\frac{1}{\sqrt{2}}\left(e^{i\omega t/2}|\uparrow ⟩+e^{-i\omega t/2}|\downarrow ⟩\right).$$

The expectation of $S_x=\frac{\hslash }{2}{\sigma }_x$ at time $t$ is $⟨\psi (t)|S_x|\psi (t)⟩=\frac{\hslash }{2}\cos \omega t$. The spin precesses in the $xy$-plane at angular frequency $\omega$.

Heisenberg picture. The state is frozen at $|\psi (0)⟩=|{\uparrow }_x⟩$. The operator evolves: $S_x(t)=e^{iHt/\hslash }{S}_x{e}^{-iHt/\hslash }$. Computing directly gives $S_x(t)=\frac{\hslash }{2}({\sigma }_x\cos \omega t-{\sigma }_y\sin \omega t)$. The expectation is $⟨{\uparrow }_x|S_x(t)|{\uparrow }_x⟩=\frac{\hslash }{2}\cos \omega t$ — the same answer.

Both pictures predict spin precession at the Larmor frequency. The Schrödinger picture tracks a rotating state; the Heisenberg picture tracks a rotating operator. Either way, the physical spin vector traces a cone around the magnetic field direction.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $\mathcal{H}$ be the Hilbert space of the system and let $H$ be the Hamiltonian, a self-adjoint operator on $\mathcal{H}$. By Stone's theorem, $H$ generates a strongly continuous one-parameter unitary group $U(t)=e^{-iHt/\hslash }$ satisfying $U(t)U(s)=U(t+s)$, $U(0)=1$, and the group identity $U(-t)=U(t{)}^{†}$.

The Schrödinger picture. States evolve according to the Schrödinger equation:

$$\boxed{i\hslash \frac{d}{dt}|\psi (t){⟩}_S=H|\psi (t){⟩}_S.}$$

For a time-independent Hamiltonian the solution with initial condition $|\psi (0)⟩$ is

$$|\psi (t){⟩}_S=e^{-iHt/\hslash }|\psi (0)⟩.$$

Operators $A_S$ are constant in time unless they carry explicit time dependence: $A_S(t)=A_S(0)$.

The Heisenberg picture. States are frozen at their initial values: $|\psi {⟩}_H=|\psi (0)⟩$. Operators evolve according to the Heisenberg equation of motion:

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

The relation between the two pictures is the unitary conjugation

$$A_H(t)=e^{iHt/\hslash }{A}_S{e}^{-iHt/\hslash },|\psi (t){⟩}_S=e^{-iHt/\hslash }|\psi {⟩}_H.$$

Expectation values are picture-independent:

$$⟨\psi (t)|A_S|\psi (t){⟩}_S=⟨\psi |e^{iHt/\hslash }{A}_S{e}^{-iHt/\hslash }|\psi {⟩}_H=⟨\psi |A_H(t)|\psi {⟩}_H.$$

The interaction (Dirac) picture. Split the Hamiltonian as $H=H_0+V$ where $H_0$ is a solvable "free" part. States evolve with $V$ only, operators evolve with $H_0$ only:

$$i\hslash \frac{d}{dt}|\psi (t){⟩}_I=V_I(t)|\psi (t){⟩}_I,A_I(t)=e^{i{H}_{0}t/\hslash }{A}_S{e}^{-i{H}_{0}t/\hslash }.$$

The interaction-picture potential is $V_I(t)=e^{i{H}_{0}t/\hslash }V{e}^{-i{H}_{0}t/\hslash }$. This splitting is useful when $H_0$ is diagonalisable and $V$ is small — the setup for the Dyson series and time-dependent perturbation theory.

Time-ordered exponential. For a time-dependent Hamiltonian $H(t)$, the evolution operator is the Dyson time-ordered exponential:

$$U(t,t_0)=\mathcal{T}\exp \left(-\frac{i}{\hslash }{\int }_{t_0}^{t}H(t^{\prime })d{t}^{\prime }\right),$$

where $\mathcal{T}$ orders operators with later times to the left. For time-independent $H$ this reduces to $e^{-iH(t-t_0)/\hslash }$.

Counterexamples to common slips

• The Schrödinger and Heisenberg pictures are not different theories. They are related by a fixed unitary transformation and produce identical physical predictions. Confusing the Heisenberg picture (a choice of time-dependence convention) with Heisenberg's 1925 matrix mechanics (a historically separate formulation of quantum theory, prior to the Hilbert-space unification) is a category error.
• The interaction picture requires $H_0$ to be self-adjoint so that $e^{i{H}_{0}t/\hslash }$ is unitary. If $H_0$ is not self-adjoint, the interaction-picture transformation does not preserve the norm of states.
• The Heisenberg equation $\frac{dA}{dt}=\frac{i}{\hslash }[H,A]$ requires $A_H(t)$ to be well-defined as a self-adjoint operator for all $t$. For unbounded operators this demands attention to domain issues: $e^{iHt/\hslash }$ must map the domain of $A$ into itself. In practice, physics texts assume this holds and the rigorous treatment is deferred to functional analysis.

Key theorem with proof [Intermediate+]

Theorem (Equivalence of the Schrödinger and Heisenberg pictures). Let $H$ be a self-adjoint operator on $\mathcal{H}$ and let $U(t)=e^{-iHt/\hslash }$. Define the Schrödinger-picture state $|\psi (t){⟩}_S=U(t)|\psi (0)⟩$ and Heisenberg-picture operator $A_H(t)=U(t{)}^{†}{A}_{S}U(t)$. Then for any state $|\psi (0)⟩$ and any operator $A_S$:

$$⟨\psi (t)|A_S|\psi (t){⟩}_S=⟨\psi (0)|A_H(t)|\psi (0)⟩\text{for all }t.$$

Furthermore, $A_H(t)$ satisfies the Heisenberg equation of motion.

Proof.

Step 1: Expectation-value identity. Substitute $|\psi (t){⟩}_S=U(t)|\psi (0)⟩$:

$$⟨\psi (t)|A_S|\psi (t){⟩}_S=⟨\psi (0)|U(t{)}^{†}{A}_{S}U(t)|\psi (0)⟩=⟨\psi (0)|A_H(t)|\psi (0)⟩.$$

This is a single algebraic step: $U^{†}{A}_{S}U$ is the definition of $A_H(t)$, and the $U$'s cancel between the bra and the operator. The identity holds for all $t$, all states, and all operators — no approximations, no special cases.

Step 2: Heisenberg equation from the conjugation. Differentiate $A_H(t)=e^{iHt/\hslash }{A}_S{e}^{-iHt/\hslash }$ with respect to $t$:

$$\frac{d{A}_H}{dt}=\frac{iH}{\hslash }{e}^{iHt/\hslash }{A}_S{e}^{-iHt/\hslash }-e^{iHt/\hslash }{A}_S{e}^{-iHt/\hslash }\frac{iH}{\hslash }.$$

Factor out $e^{iHt/\hslash }$ and $e^{-iHt/\hslash }$:

$$\frac{d{A}_H}{dt}=\frac{i}{\hslash }\left(H{A}_H(t)-A_H(t)H\right)=\frac{i}{\hslash }[H,A_H(t)].$$

This is the Heisenberg equation of motion. Note: $H$ commutes with itself, so $H$ has the same form in both pictures: $H_H=U^{†}HU=H$.

Step 3: Conservation of energy. Set $A=H$ in the Heisenberg equation: $\frac{d{H}_H}{dt}=\frac{i}{\hslash }[H,H]=0$. The Hamiltonian is conserved in the Heisenberg picture — energy conservation is automatic from the structure of the equation, not an additional postulate.

Step 4: Classical limit. The Heisenberg equation $\frac{dA}{dt}=\frac{i}{\hslash }[H,A]$ has the same form as Hamilton's classical equation $\frac{df}{dt}=\{f,H\}$ under the substitution $\frac{i}{\hslash }[\cdot ,\cdot ]\leftrightarrow \{\cdot ,\cdot \}$. This is the canonical-quantisation correspondence: commutator divided by $i\hslash$ replaces the Poisson bracket. The structural identity is not approximate — it is exact for the algebraic forms, and the approximation enters only when operator ordering (the $O({\hslash }^2)$ corrections) becomes relevant for specific operator classes. $\square$

Corollary (Energy eigenstates are stationary). If $|\psi (0)⟩=|E⟩$ is an eigenstate of $H$ with eigenvalue $E$, then $|\psi (t){⟩}_S=e^{-iEt/\hslash }|E⟩$ and all expectation values of time-independent observables are constant: $⟨\psi (t)|A|\psi (t)⟩=⟨E|A|E⟩$. This is why energy eigenstates are called stationary states — the time dependence is a global phase that cancels in every expectation value.

Bridge. The picture-equivalence theorem builds toward the path-integral formulation 12.10.01 pending, where the time-evolution operator is re-expressed as a sum over classical trajectories weighted by the action, and appears again in scattering theory 12.11.01 pending, where the S-matrix is defined as a limit of $U(t,t_0)$ with $t_0\to -\infty$ and $t\to +\infty$. The foundational reason the equivalence holds is that unitary conjugation preserves every algebraic invariant — inner products, eigenvalues, commutators — and this is exactly the structure that the density-matrix formalism 12.09.01 pending extends to mixed states, where the picture equivalence generalises from pure states to the full convex set of density operators.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

The equivalence of the Schrödinger and Heisenberg pictures can be stated as a theorem about unitary conjugation in a Hilbert space. Let $H$ be a self-adjoint operator and $U(t)=\exp (-iHt/\hslash )$ the associated unitary group. The claim $⟨\psi |U^{†}AU|\psi ⟩=⟨U\psi |A|U\psi ⟩$ for all $|\psi ⟩$ reduces to unitarity of $U$, which Mathlib's Analysis.InnerProductSpace and Topology.ContinuousFunction.Basic can handle for bounded operators. The Heisenberg equation $\frac{d{A}_H}{dt}=\frac{i}{\hslash }[H,A_H]$ requires differentiating the map $t\mapsto U(t{)}^{†}AU(t)$, which is the derivative of a strongly continuous one-parameter group applied to an operator — Stone's theorem (Mathlib.Analysis.SpecialFunctions.Pow and related) covers the generator-side. The physics-layer postulate (that time evolution is generated by the Hamiltonian) is outside Mathlib's scope and would need to be assumed as an axiom.

Advanced results [Master]

Stone's theorem and the rigorous time-evolution group

Stone's theorem is the rigorous foundation for everything in this unit. Let $\mathcal{H}$ be a Hilbert space and let $H$ be a (possibly unbounded) self-adjoint operator with domain $\mathcal{D}(H)\subset \mathcal{H}$. By the spectral theorem, there exists a unique projection-valued measure $E$ on $\mathbb{R}$ such that $H=\int \lambda dE(\lambda )$. The unitary group is defined by the functional calculus:

$$U(t)=e^{-iHt/\hslash }=\int e^{-i\lambda t/\hslash }dE(\lambda ).$$

Stone's theorem. The map $t\mapsto U(t)$ is a strongly continuous one-parameter unitary group: $U(t+s)=U(t)U(s)$, $U(0)=1$, and ${\lim }_{t\to 0}\Vert U(t)\psi -\psi \Vert =0$ for all $\psi \in \mathcal{H}$. Conversely, every strongly continuous one-parameter unitary group has a unique self-adjoint generator $H$ such that $U(t)=e^{-iHt/\hslash }$. The generator satisfies $\mathcal{D}(H)=\{\psi \in \mathcal{H}:{\lim }_{t\to 0}\frac{U(t)-1}{t}\psi \text{ exists}\}$ and $H\psi =i\hslash {\lim }_{t\to 0}\frac{U(t)-1}{-t}\psi$.

The Schrödinger equation $i\hslash \frac{d}{dt}|\psi (t)⟩=H|\psi (t)⟩$ holds on $\mathcal{D}(H)$ in the strong sense: for $\psi (0)\in \mathcal{D}(H)$, $|\psi (t)⟩=U(t)|\psi (0)⟩$ is strongly differentiable and satisfies the equation. For $\psi (0)\notin \mathcal{D}(H)$, $U(t)\psi (0)$ is still well-defined (the group is bounded) but is not differentiable in $t$ — the state still evolves but does not satisfy the differential equation pointwise.

Domain issues. The Heisenberg-picture conjugation $A_H(t)=U(t{)}^{†}{A}_{S}U(t)$ requires care when $A_S$ is unbounded. The operator $U(t)$ maps $\mathcal{D}(A_S)$ into itself only if $A_S$ commutes with $U(t)$ in a suitable sense (e.g., $A_S$ commutes with the spectral projections of $H$). When $[A_S,H]\ne 0$, the domain of $A_H(t)$ may differ from $\mathcal{D}(A_S)$ at different times, and the Heisenberg equation holds on the intersection of these domains. This technicality is invisible in finite dimensions but matters for quantum field theory, where all interesting operators are unbounded. The rigorous framework for handling these domain questions is the theory of self-adjoint extensions (Reed and Simon 1975, Vol. II Fourier Analysis, Self-Adjointness), which provides criteria (e.g., the Kato-Rellich theorem for relative boundedness) under which $H_0+V$ remains self-adjoint and the associated unitary group is well-defined.

Time-dependent Hamiltonians and the time-ordered exponential

When $H=H(t)$ depends explicitly on time, the operators $H(t_1)$ and $H(t_2)$ need not commute for $t_1\ne t_2$. The evolution operator is the Dyson series (also called the time-ordered exponential):

$$U(t,t_0)=\mathcal{T}\exp \left(-\frac{i}{\hslash }{\int }_{t_0}^{t}H(t^{\prime })d{t}^{\prime }\right)=1+\sum _{n=1}^{\infty }{\left(\frac{-i}{\hslash }\right)}^n{\int }_{t_0}^{t}d{t}_1{\int }_{t_0}^{t_1}d{t}_2\cdots {\int }_{t_0}^{t_{n-1}}d{t}_{n}H(t_1)H(t_2)\cdots H(t_n).$$

The integrals are nested with $t_1>t_2>\cdots >t_n$, which is the meaning of the time-ordering symbol $\mathcal{T}$. When $[H(t_1),H(t_2)]=0$ for all $t_1,t_2$, the time ordering is unnecessary and $U(t,t_0)=\exp \left(-\frac{i}{\hslash }{\int }_{t_0}^{t}H(t^{\prime })d{t}^{\prime }\right)$.

The time-ordered exponential satisfies $U(t,t_0)=U(t,s)U(s,t_0)$ for any $t_0<s<t$ (the composition law), but it does not satisfy $U(t,t_0)=U(t-t_0)$ — there is no single-parameter group structure because the Hamiltonian breaks time-translation invariance.

An important special case is the adiabatic Hamiltonian $H(t)=H_0+\lambda (t)V$ where $\lambda (t)$ varies slowly. The adiabatic theorem (Born and Fock 1928 Z. Phys. 51; Kato 1950 J. Phys. Soc. Japan 5) states that a system starting in a non-degenerate eigenstate of $H(0)$ evolves into the corresponding eigenstate of $H(t)$ at later times, up to a dynamical phase $e^{-i{\int }_0^{t}E(t^{\prime })d{t}^{\prime }/\hslash }$ and a geometric (Berry) phase. The adiabatic limit is the regime where the rate of change $\dot{\lambda }$ is small compared to the spectral gap of $H(t)$ — the picture formalism extends to this time-dependent setting by replacing $U(t)=e^{-iHt/\hslash }$ with the time-ordered exponential throughout.

The interaction picture and the Dyson series for perturbation theory

The interaction picture sets up perturbation theory as follows. Given $H=H_0+V$, define the interaction-picture state by $|\psi (t){⟩}_I=e^{i{H}_{0}t/\hslash }|\psi (t){⟩}_S$. The equation of motion is

$$i\hslash \frac{d}{dt}|\psi (t){⟩}_I=V_I(t)|\psi (t){⟩}_I,V_I(t)=e^{i{H}_{0}t/\hslash }V{e}^{-i{H}_{0}t/\hslash }.$$

The formal solution is the Dyson time-evolution operator in the interaction picture:

$$U_I(t,t_0)=\mathcal{T}\exp \left(-\frac{i}{\hslash }{\int }_{t_0}^t{V}_I(t^{\prime })d{t}^{\prime }\right).$$

Expanding $U_I$ in powers of $V$ produces the Dyson series, and truncating at order $V^n$ gives time-dependent perturbation theory at $n$-th order. The first-order transition amplitude from state $|i⟩$ to state $|f⟩$ (both eigenstates of $H_0$) is

$$c_f^{(1)}(t)=-\frac{i}{\hslash }{\int }_{t_0}^t{e}^{i(E_f-E_i)t^{\prime }/\hslash }⟨f|V|i⟩d{t}^{\prime }.$$

This is Fermi's golden rule in integral form. The square $|c_f^{(1)}{|}^2$ gives the transition probability, and taking $t-t_0\to \infty$ with $V$ time-independent yields the delta-function energy-conservation factor $2\pi \delta (E_f-E_i)$ that appears in Fermi's golden rule.

The Dyson series in the interaction picture converges for a bounded interaction $V_I(t)$ on any finite time interval. For unbounded $V$ (the physically relevant case in most QFT applications), convergence has been established rigorously only for specific models (e.g., the Nelson model with an ultraviolet cutoff). In renormalisable quantum field theories, individual terms of the Dyson series are ultraviolet-divergent and must be regularised and renormalised order by order; the renormalised perturbation series is expected to be asymptotic rather than convergent (Dyson 1952 Phys. Rev. 85).

The Magnus expansion

An alternative to the Dyson series that preserves unitarity order by order is the Magnus expansion. Write $U(t,t_0)=e^{\Omega (t,t_0)}$ where $\Omega$ is an anti-Hermitian operator-valued function. The series for $\Omega$ is:

$$\Omega (t,t_0)=\sum _{n=1}^{\infty }{\Omega }_n(t,t_0),$$

with ${\Omega }_1=-\frac{i}{\hslash }{\int }_{t_0}^{t}H(t_1)d{t}_1$ and higher terms involving commutators of $H$ at different times. The second-order term is ${\Omega }_2=-\frac{1}{2{\hslash }^2}{\int }_{t_0}^{t}d{t}_1{\int }_{t_0}^{t_1}d{t}_2[H(t_1),H(t_2)]$, which vanishes when the Hamiltonian commutes with itself at different times. The third-order term involves double commutators $[H(t_1),[H(t_2),H(t_3)]]$ and is responsible for the leading correction beyond the time-averaged Hamiltonian approximation.

The Magnus expansion converges when $\frac{1}{\hslash }{\int }_{t_0}^t\Vert H(t^{\prime })\Vert d{t}^{\prime }<\pi$ (a sufficient condition due to Blanes et al. 2009 Phys. Rep. 478). Its advantage over the Dyson series is that $e^{\Omega }$ is unitary at every truncation order, whereas the truncated Dyson series is not. This matters for numerical time propagation: Magnus-expansion integrators are symplectic (in the classical limit) and preserve probability exactly at each step.

Wigner's theorem and the geometry of time evolution

Wigner's theorem states that any symmetry transformation that preserves transition probabilities ($|⟨\varphi |\psi ⟩|\mapsto |⟨{\varphi }^{\prime }|{\psi }^{\prime }⟩|$) is implemented by a unitary or antiunitary operator. Time evolution preserves transition probabilities (Exercise 11), so it must be unitary (it is connected to the identity and continuous, ruling out antiunitarity). Wigner's theorem is the deep reason why time evolution is unitary: it is not assumed but derived from the weaker postulate that probabilities are preserved.

The antiunitary case arises for discrete symmetries: time reversal $T$ is antiunitary (it complex-conjugates the amplitudes, sending $i\to -i$ and thereby reversing the sign of the Schrödinger equation), as is charge conjugation in relativistic QFT. The continuity argument that excludes antiunitarity for time evolution is physical: at $t=0$ the transformation is the identity, which is unitary; by continuity the transformation remains unitary for all finite $t$.

Combined with Stone's theorem, the chain of reasoning is: probability conservation (a physical requirement) implies unitarity (by Wigner), which implies a self-adjoint generator exists (by Stone), which is identified with the Hamiltonian (the energy observable). The entire structure of time evolution in quantum mechanics flows from the single physical input that probabilities must sum to one at all times.

Coherent states and the classical limit of the Heisenberg picture

The Heisenberg equations for the harmonic oscillator, $a_H(t)=a{e}^{-i\omega t}$ and $a_H^{†}(t)=a^{†}{e}^{i\omega t}$, show that the operator-level dynamics reproduces the classical complex-amplitude rotation $A(t)=A(0)e^{-i\omega t}$. For coherent states $|\alpha ⟩$ with $a|\alpha ⟩=\alpha |\alpha ⟩$, the Heisenberg-picture expectation is $⟨\alpha |a_H(t)|\alpha ⟩=\alpha e^{-i\omega t}$ — the complex amplitude follows the classical trajectory. The quantum fluctuations around this trajectory are constant and equal to the ground-state uncertainty $\Delta x=\sqrt{\hslash /(2m\omega )}$.

In the limit $\hslash \to 0$ (or equivalently, for states with $|\alpha |\gg 1$), the relative uncertainty $\Delta x/⟨x⟩\sim 1/|\alpha |\to 0$ and the Heisenberg-picture operators reduce to classical observables evolving under Hamilton's equations. This is the precise statement that the Heisenberg picture has the classical limit built into its structure: the Heisenberg equation $\frac{dA}{dt}=\frac{i}{\hslash }[H,A]$ becomes Hamilton's equation $\frac{df}{dt}=\{f,H\}$ under the substitution $\frac{i}{\hslash }[\cdot ,\cdot ]\to \{\cdot ,\cdot \}$.

Ehrenfest's theorem and the semiclassical correspondence

Ehrenfest's theorem (1927) provides the quantitative link between Heisenberg-picture operator equations and classical equations of motion at the level of expectation values. For a particle with Hamiltonian $H=p^2/(2m)+V(x)$ and any observable $A$:

$$\frac{d}{dt}⟨A⟩=\frac{i}{\hslash }⟨[H,A]⟩+⟨\frac{\partial A}{\partial t}⟩.$$

Setting $A=x$ and $A=p$ separately:

$$\frac{d}{dt}⟨x⟩=\frac{⟨p⟩}{m},\frac{d}{dt}⟨p⟩=-⟨V^{\prime }(x)⟩.$$

These are the quantum analogues of Hamilton's equations. When $V^{\prime }(x)$ is linear (the harmonic oscillator, the free particle), $⟨V^{\prime }(x)⟩=V^{\prime }(⟨x⟩)$ and the expectation values obey the exact classical trajectory. For a general potential, Taylor expansion around $⟨x⟩$ gives $⟨V^{\prime }(x)⟩=V^{\prime }(⟨x⟩)+\frac{1}{2}{V}^{\prime \prime \prime }(⟨x⟩)(\Delta x{)}^2+\cdots$. The classical equations are recovered when the wavepacket is sharply peaked relative to the scale over which $V^{\prime }$ varies — the quantum corrections are controlled by the ratio $(\Delta x{)}^2/{\ell }_V^2$ where ${\ell }_V$ is the length scale on which $V^{\prime }$ varies.

Ehrenfest's theorem is exact and holds for all states, not only wavepackets. It makes no reference to $\hslash \to 0$; the classical limit emerges from the state (narrow wavepackets in potentials without higher-order structure), not from the dynamics. The theorem is a direct consequence of the Heisenberg equation of motion and the identification $d⟨A⟩/dt=⟨d{A}_H/dt⟩$, which identifies the expectation value of the Heisenberg equation with the time derivative of the expectation. This is the foundational reason that Heisenberg-picture operator equations have the same algebraic form as classical equations: the commutator $[H,A]/(i\hslash )$ replaces the Poisson bracket $\{H,A\}$, and the expectation value passes through the derivative by linearity.

The Trotter product formula and numerical time evolution

For a Hamiltonian $H=H_0+V$ where the exponentials $e^{-i{H}_{0}t/\hslash }$ and $e^{-iVt/\hslash }$ are individually computable but $e^{-iHt/\hslash }$ is not, the Trotter product formula gives a constructive approximation:

$$e^{-iHt/\hslash }=\underset{N\to \infty }{\lim }{\left(e^{-i{H}_{0}t/(N\hslash )}{e}^{-iVt/(N\hslash )}\right)}^N.$$

The formula holds when $H_0$ and $V$ are self-adjoint and $H_0+V$ is essentially self-adjoint on the intersection of their domains (Trotter 1959 Proc. Amer. Math. Soc. 10; Kato 1978 Perturbation Theory for Linear Operators). For bounded operators the convergence is in operator norm; for unbounded operators the convergence is strong (pointwise on the domain).

The Trotter splitting is the foundation of numerical quantum dynamics. The first-order split $S_1(\Delta t)=e^{-i{H}_0\Delta t/\hslash }{e}^{-iV\Delta t/\hslash }$ has error $O(\Delta t^2)$ per step. The symmetric (Strang) splitting $S_2(\Delta t)=e^{-i{H}_0\Delta t/(2\hslash )}{e}^{-iV\Delta t/\hslash }{e}^{-i{H}_0\Delta t/(2\hslash )}$ achieves $O(\Delta t^3)$ per step and is time-reversible, a property connected to the symplectic structure of the classical limit.

The Trotter formula also provides the rigorous starting point for the Feynman path integral 12.10.01 pending. Inserting complete sets of position eigenstates between each Trotter factor and taking $N\to \infty$ converts the operator product into an integral over piecewise-classical trajectories — the Trotter discretisation is the lattice regularisation that makes the path integral mathematically meaningful before the continuum limit is taken. The central insight is that the Trotter formula decomposes quantum evolution into a sequence of free-propagation and potential-kick steps; this is exactly the structure that the interaction picture exploits analytically and that split-operator methods exploit numerically.

Baker-Campbell-Hausdorff and operator exponentials

The relation $e^A{e}^B=e^{A+B+\frac{1}{2}[A,B]+\cdots }$ for noncommuting operators is the Baker-Campbell-Hausdorff (BCH) formula. For the time-evolution operator with a time-dependent Hamiltonian, the BCH expansion is the algebraic engine behind both the Dyson series and the Magnus expansion. The first few terms are:

$$\ln (e^A{e}^B)=A+B+\frac{1}{2}[A,B]+\frac{1}{12}[A,[A,B]]-\frac{1}{12}[B,[A,B]]+\cdots$$

When $[A,B]$ commutes with both $A$ and $B$ (as for the Pauli matrices in a fixed direction), the series terminates at the first commutator: $e^A{e}^B=e^{A+B+\frac{1}{2}[A,B]}$. This is the case for spin in a constant magnetic field, where $[{\sigma }_z,{\sigma }_z]=0$ ensures the time-evolution exponential closes exactly.

For the Magnus expansion, the BCH formula applied to $e^{{\Omega }_1}{e}^{{\Omega }_2}=e^{{\Omega }_1+{\Omega }_2+\frac{1}{2}[{\Omega }_1,{\Omega }_2]+\cdots }$ generates the higher-order terms ${\Omega }_n$ as nested commutators of the Hamiltonian at different times. The convergence condition for the Magnus series (the operator norm of the integrated Hamiltonian bounded by $\pi \hslash$) is a sufficient condition for the BCH series to converge when applied to the time-ordered product of infinitesimal evolution operators.

The BCH formula identifies the Lie-algebra structure of quantum mechanics with the group structure of unitary evolution: the commutator $[H,A]/(i\hslash )$ at the algebra level becomes the conjugation $e^{iHt/\hslash }A{e}^{-iHt/\hslash }$ at the group level. This is exactly the exponential map from the Lie algebra of skew-Hermitian operators to the unitary group, and the BCH series measures the failure of this map to be a group homomorphism when the algebra is non-abelian.

The virial theorem from Heisenberg equations

The quantum virial theorem is a direct application of the Heisenberg equation to the dilatation generator $G=\frac{1}{2}(xp+px)$, which generates scale transformations on phase space. For a Hamiltonian $H=p^2/(2m)+V(x)$ with a homogeneous potential $V(\lambda x)={\lambda }^{n}V(x)$, the Heisenberg equation gives:

$$\frac{dG}{dt}=\frac{i}{\hslash }[H,G]=\frac{p^2}{m}-x{V}^{\prime }(x).$$

In a stationary state $|E⟩$, $⟨E|dG/dt|E⟩=0$ (the expectation of any time derivative vanishes in an energy eigenstate), which yields $2⟨T⟩=n⟨V⟩$ where $T=p^2/(2m)$ is the kinetic energy. For a Coulomb potential ($n=-1$), $⟨T⟩=-E$ and $⟨V⟩=2E$; for a harmonic oscillator ($n=2$), $⟨T⟩=⟨V⟩=E/2$.

The virial theorem illustrates a general pattern: conserved quantities in the Heisenberg picture generate transformations that leave the Hamiltonian invariant. The dilatation generator $G$ is conserved when $V$ is scale-invariant; the angular momentum $L$ is conserved when $V$ is rotationally invariant; the linear momentum $p$ is conserved when $V$ is translationally invariant. Each conservation law follows from $d{A}_H/dt=\frac{i}{\hslash }[H,A_H]=0$, which is the Heisenberg-picture version of Noether's theorem 09.04.02 pending. The pattern recurs across all of quantum mechanics: symmetry of $H$ under a transformation implies conservation of the corresponding generator, and the proof is a two-line computation in the Heisenberg equation.

Synthesis. The advanced results in this section assemble into a coherent picture of quantum time evolution at three levels of rigour. Stone's theorem provides the functional-analytic foundation — the existence and uniqueness of the unitary group generated by a self-adjoint Hamiltonian — and this is exactly the structure that the Schrödinger, Heisenberg, and interaction pictures exploit via different unitary factorisations. The Dyson series and the Magnus expansion build toward the perturbative and non-perturbative machinery of quantum field theory 12.11.01 pending, while Ehrenfest's theorem and the coherent-state analysis identify the semiclassical bridge back to classical mechanics 09.04.02 pending. The Trotter product formula is the bridge between the operator formalism and the path integral 12.10.01 pending, and Wigner's theorem provides the foundational reason that time evolution must be unitary — probability conservation, not an additional postulate. Putting these together, the entire apparatus of quantum dynamics flows from two inputs: a self-adjoint Hamiltonian (Stone) and probability conservation (Wigner). The pattern recurs throughout physics: the choice of picture is a choice of computational convenience, and the physical content is invariant under the resulting unitary transformations.

Full proof set [Master]

Proposition 1 (Ehrenfest's equations for position and momentum). Let $H=p^2/(2m)+V(x)$ where $V$ is a smooth potential and $x,p$ satisfy $[x,p]=i\hslash$. In the Heisenberg picture:

$$\frac{d{x}_H}{dt}=\frac{p_H}{m},\frac{d{p}_H}{dt}=-V^{\prime }(x_H),$$

where $V^{\prime }(x_H)$ denotes the operator $dV/dx$ evaluated at the Heisenberg-picture position operator.

Proof. Apply the Heisenberg equation $\frac{d{A}_H}{dt}=\frac{i}{\hslash }[H,A_H]$ to $A=x$:

$$\frac{d{x}_H}{dt}=\frac{i}{\hslash }[H,x_H]=\frac{i}{\hslash }\left[\frac{p_H^2}{2m},x_H\right]+\frac{i}{\hslash }[V(x_H),x_H].$$

The potential term vanishes because $V(x_H)$ is a function of $x_H$ alone: $[V(x_H),x_H]=0$. For the kinetic term, use $[p^2,x]=p[p,x]+[p,x]p=-2i\hslash p$:

$$\frac{d{x}_H}{dt}=\frac{i}{\hslash }\frac{1}{2m}(-2i\hslash p_H)=\frac{p_H}{m}.$$

For $A=p$:

$$\frac{d{p}_H}{dt}=\frac{i}{\hslash }[H,p_H]=\frac{i}{\hslash }\left[\frac{p_H^2}{2m},p_H\right]+\frac{i}{\hslash }[V(x_H),p_H].$$

The kinetic term vanishes: $[p^2,p]=0$. For the potential term, express $V(x_H)$ as a power series and use $[x^n,p]=i\hslash n{x}^{n-1}$, which gives $[V(x),p]=i\hslash V^{\prime }(x)$:

$$\frac{d{p}_H}{dt}=\frac{i}{\hslash }(i\hslash V^{\prime }(x_H))=-V^{\prime }(x_H).\square$$

Proposition 2 (Trotter product formula, bounded case). Let $A$ and $B$ be bounded self-adjoint operators on a Hilbert space $\mathcal{H}$. Then:

$$e^{i(A+B)t}=\underset{N\to \infty }{\lim }{\left(e^{iAt/N}{e}^{iBt/N}\right)}^N,$$

where the limit is in operator norm.

Proof. Define $S_N=e^{i(A+B)t/N}$ and $T_N=e^{iAt/N}{e}^{iBt/N}$. Expand both to second order in $1/N$:

$$S_N=1+\frac{i(A+B)t}{N}-\frac{(A+B{)}^2{t}^2}{2N^2}+O(N^{-3}),$$

$$T_N=\left(1+\frac{iAt}{N}-\frac{A^2{t}^2}{2N^2}+O(N^{-3})\right)\left(1+\frac{iBt}{N}-\frac{B^2{t}^2}{2N^2}+O(N^{-3})\right)=1+\frac{i(A+B)t}{N}-\frac{(A^2+2AB+B^2)t^2}{2N^2}+O(N^{-3}).$$

The difference is:

$$T_N-S_N=\frac{t^2}{2N^2}\left((A+B{)}^2-(A^2+2AB+B^2)\right)+O(N^{-3})=\frac{t^2}{2N^2}[B,A]+O(N^{-3}).$$

Since $A$ and $B$ are bounded, $\Vert T_N-S_N\Vert \le C/N^2$ for some constant $C$ depending on $\Vert A\Vert$, $\Vert B\Vert$, and $t$. The telescoping identity $T_N^N-S_N^N={\sum }_{k=0}^{N-1}{T}_N^k(T_N-S_N)S_N^{N-1-k}$ gives:

$$\Vert T_N^N-S_N^N\Vert \le N\Vert T_N-S_N\Vert \cdot \underset{k}{\max }(\Vert T_N^k\Vert \Vert S_N^{N-1-k}\Vert )\le \frac{C}{N}{e}^{(\Vert A\Vert +\Vert B\Vert )|t|}.$$

Each factor $\Vert T_N^k\Vert$ and $\Vert S_N^{N-1-k}\Vert$ is bounded by $e^{(\Vert A\Vert +\Vert B\Vert )|t|}$ for large $N$, so the product $N\cdot C/N^2\cdot e^{2(\Vert A\Vert +\Vert B\Vert )|t|}\to 0$ as $N\to \infty$. $\square$

Proposition 3 (Preservation of the canonical commutation relation). If $[x,p]=i\hslash 1$ and $U(t)$ is unitary, then $[x_H(t),p_H(t)]=i\hslash 1$ for all $t$, where $x_H(t)=U^{†}xU$ and $p_H(t)=U^{†}pU$.

Proof. Expand the commutator of the Heisenberg-picture operators:

$$[x_H(t),p_H(t)]=U^{†}xU{U}^{†}pU-U^{†}pU{U}^{†}xU=U^{†}(xp-px)U=U^{†}[x,p]U.$$

Substituting $[x,p]=i\hslash 1$ and using $U^{†}1U=1$:

$$[x_H(t),p_H(t)]=i\hslash U^{†}U=i\hslash 1.$$

The key step is $U{U}^{†}=1$ appearing between the position and momentum factors — unitarity cancels the conjugation. The result holds for any unitary $U$, not only time evolution, and generalises: all algebraic relations among operators are preserved by simultaneous unitary conjugation. $\square$

Connections [Master]

The Schrödinger and Heisenberg pictures connect to every downstream topic in quantum mechanics and quantum field theory. The Schrödinger picture is the starting point for wave mechanics: the position-space wavefunction $\psi (x,t)$ satisfies the partial differential equation $i\hslash {\partial }_t\psi =-\frac{{\hslash }^2}{2m}{\nabla }^2\psi +V(x)\psi$, whose solutions for specific potentials (particle in a box 12.04.01, hydrogen atom 14.04.01, harmonic oscillator 12.06.01) are the computational core of introductory QM.

The Heisenberg picture is the natural language for quantum field theory. In QFT the fields $\varphi (x)$ are operator-valued distributions satisfying Heisenberg equations of motion derived from a Hamiltonian (or Lagrangian) density. The interaction picture, applied to QFT, gives the Dyson series for the S-matrix, which is the object that encodes all scattering amplitudes. Tomonaga and Schwinger's formulation of QFT is essentially the interaction picture applied to relativistically covariant time evolution.

The classical limit of the Heisenberg picture connects back to Hamiltonian mechanics (09.04.02) via the correspondence $\frac{i}{\hslash }[\cdot ,\cdot ]\leftrightarrow \{\cdot ,\cdot \}$. The path integral (12.10.01) provides a third, independent formulation of quantum dynamics that is equivalent to both pictures but expressed in terms of classical trajectories rather than operators or states.

The tension between the deterministic, unitary Schrödinger evolution and the stochastic, discontinuous collapse postulate is the measurement problem (20.07.01). The two dynamics — unitary evolution between measurements, collapse at measurements — are both postulated, and their coexistence is the deepest unresolved conceptual issue in quantum foundations. The pictures developed in this unit formalise the unitary branch; the collapse branch is addressed in the measurement-theory units (12.09.01).

In chemistry (14.04.01), the time-dependent Schrödinger equation governs molecular dynamics: the Born-Oppenheimer approximation splits the electronic and nuclear Hamiltonians (an interaction-picture-like splitting), and the nuclear wavepacket evolves on the potential energy surface defined by the electronic eigenvalues. Femtochemistry — observing chemical reactions on their natural timescale — is the experimental realisation of watching $|\psi (t)⟩$ evolve under a molecular Hamiltonian.

Historical and philosophical context [Master]

Schrödinger's 1926 series of papers ("Quantisierung als Eigenwertproblem", Annalen der Physik 79) introduced the wave equation that bears his name. Schrödinger started from the de Broglie relation and the Hamilton-Jacobi equation, seeking a wave-mechanical equation whose eigenvalues reproduce the observed atomic spectra. He quickly demonstrated the equivalence of his wave mechanics with Heisenberg's 1925 matrix mechanics, showing that the two formalisms give identical energy spectra for the hydrogen atom.

Heisenberg's 1925 matrix mechanics (developed with Born and Jordan, the "Dreimannerarbeit") formulated quantum theory entirely in terms of observable quantities — arrays of transition amplitudes satisfying noncommutative multiplication. There was no wavefunction, no differential equation, only algebraic relations among observables. The Heisenberg picture of time evolution — operators carrying the dynamics, states fixed — is the direct descendant of this algebraic approach.

Dirac's 1927 transformation theory unified the two. In a series of papers and then in his 1930 textbook The Principles of Quantum Mechanics, Dirac showed that wave mechanics and matrix mechanics are related by a change of basis in Hilbert space — what we now call a unitary transformation. The bra-ket notation, the abstract Hilbert-space formulation, and the concept of "representations" (what we now call pictures) are all Dirac's contributions. The interaction picture appears in Dirac's 1927 paper on quantum electrodynamics ("The Quantum Theory of the Emission and Absorption of Radiation", Proc. Roy. Soc. A 114).

Eckart's 1927 paper ("Operator Calculus and the Solution of the Equations of Dynamic Quantum Mechanics", Proc. Nat. Acad. Sci. 13) gave an independent equivalence proof, and von Neumann's 1932 Mathematische Grundlagen placed the entire discussion on rigorous footing by proving the spectral theorem for unbounded self-adjoint operators and establishing Stone's theorem connecting self-adjoint operators to one-parameter unitary groups.

The philosophical significance of the picture equivalence is this: there is no fact of the matter about whether "the state really moves" or "the operators really move." Physics does not choose between the Schrödinger and Heisenberg pictures. What is observer-independent is the set of expectation values and probabilities — the quantum-mechanical predictions. The choice of picture is a choice of description, not a choice about reality. This is a specific instance of the broader principle that physics is about gauge-invariant, representation-independent quantities.

Bibliography [Master]

• Schrödinger, E. "Quantisierung als Eigenwertproblem." Annalen der Physik 79, 361–376 and 489–527, 1926.

• Heisenberg, W. "Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen." Zeitschrift fur Physik 33, 879–893, 1925.

• Born, M., Heisenberg, W., and Jordan, P. "Zur Quantenmechanik II." Zeitschrift fur Physik 35, 557–615, 1926.

• Dirac, P. A. M. "The Quantum Theory of the Emission and Absorption of Radiation." Proc. Roy. Soc. A 114, 243–265, 1927.

• Dirac, P. A. M. The Principles of Quantum Mechanics, 4th ed. Oxford University Press, 1958. Ch. III.

• Eckart, C. "Operator Calculus and the Solution of the Equations of Dynamic Quantum Mechanics." Proc. Nat. Acad. Sci. 13, 462–466, 1927.

• von Neumann, J. Mathematische Grundlagen der Quantenmechanik. Springer, 1932. Ch. IV. English translation: Princeton University Press, 1955.

• Stone, M. H. "On One-Parameter Unitary Groups in Hilbert Space." Annals of Mathematics 33, 643–649, 1932.

• Sakurai, J. J. and Napolitano, J. Modern Quantum Mechanics, 2nd ed. Cambridge University Press, 2017. Ch. 2.1–2.2.

• Griffiths, D. J. and Schroeter, D. F. Introduction to Quantum Mechanics, 3rd ed. Cambridge University Press, 2018. Ch. 2, 3.

• Susskind, L. and Friedman, A. Quantum Mechanics: The Theoretical Minimum. Basic Books, 2014. Lectures 3–4.

• Cohen-Tannoudji, C., Diu, B., and Laloe, F. Quantum Mechanics, Vol. I. Wiley, 1977. Complement D-III.

• Magnus, W. "On the Exponential Solution of Differential Equations for a Linear Operator." Comm. Pure Appl. Math. 7, 649–673, 1954.

• Wigner, E. P. "Group Theory and its Application to the Quantum Mechanics of Atomic Spectra." Academic Press, 1959. Ch. 26.

• Reed, M. and Simon, B. Methods of Modern Mathematical Physics, Vol. I: Functional Analysis. Academic Press, 1980. Ch. VIII (Stone's theorem).

• Hall, B. C. Quantum Theory for Mathematicians. Springer GTM 267, 2013. Ch. 2, 3, 14.