Commutator Dynamics

Commutator dynamics is the viewpoint that a Hamiltonian generates time evolution by commutation. For an observable $A$ with no explicit time dependence,

$$\frac{dA_H}{dt} = \frac{i}{\hbar}[H,A_H].$$

Equivalently,

$$\frac{dA_H}{dt} = \frac{1}{i\hbar}[A_H,H].$$

The same Hamiltonian that generates unitary motion of state vectors also generates a derivation on the algebra of observables. This page is the dynamics-side home for that idea. The picture-specific derivation lives in Heisenberg Equations of Motion, while the basic definition and identities of $[A,B]$ live in Commutators.

Infinitesimal Time Evolution

For a short time step $\Delta t$ and time-independent $H$,

$$U(\Delta t) = e^{-iH\Delta t/\hbar}.$$

The Heisenberg update of an observable is

$$A_H(t+\Delta t) = U^\dagger(\Delta t)A_H(t)U(\Delta t).$$

Expanding to first order gives

$$\begin{aligned} A_H(t+\Delta t) &= \left( I+\frac{iH\Delta t}{\hbar} \right) A_H(t) \left( I-\frac{iH\Delta t}{\hbar} \right) +O(\Delta t^2) \\ &= A_H(t) + \frac{i\Delta t}{\hbar}[H,A_H(t)] +O(\Delta t^2). \end{aligned}$$

Thus

$$\frac{A_H(t+\Delta t)-A_H(t)}{\Delta t} = \frac{i}{\hbar}[H,A_H(t)] +O(\Delta t).$$

Taking the limit $\Delta t\to0$ gives the Heisenberg equation. The commutator is not an analogy for the generator; it is the infinitesimal generator of unitary conjugation.

The Hamiltonian Derivation

Define the Hamiltonian derivation

$$\mathcal D_H(A) = \frac{i}{\hbar}[H,A].$$

Then the Heisenberg equation becomes

$$\dot A_H = \mathcal D_H(A_H).$$

The word derivation is literal: $\mathcal D_H$ obeys the product rule

$$\mathcal D_H(AB) = \mathcal D_H(A)B + A\mathcal D_H(B).$$

This follows from the commutator identity

$$[H,AB] = [H,A]B + A[H,B].$$

It means Hamiltonian dynamics is compatible with operator multiplication. If the dynamics of $A$ and $B$ is known, the dynamics of their product is not arbitrary; it is fixed by the same derivation.

The derivation also respects adjoints when $H=H^\dagger$:

$$\mathcal D_H(A^\dagger) = \mathcal D_H(A)^\dagger .$$

So self-adjoint observables remain self-adjoint under Heisenberg evolution.

Lie Algebra Structure

The map $A\mapsto[H,A]$ is an inner derivation of the operator algebra. Commutators of such derivations are again inner derivations. If

$$\mathcal D_A(X) = \frac{i}{\hbar}[A,X], \qquad \mathcal D_B(X) = \frac{i}{\hbar}[B,X],$$

then the Jacobi identity gives

$$[\mathcal D_A,\mathcal D_B] = \mathcal D_{(i/\hbar)[A,B]}.$$

This compact formula is one reason commutators appear in dynamics, symmetry, and perturbation theory at the same time. The Hamiltonian is not merely an energy observable; it is the generator of a one-parameter family of automorphisms.

For a time-independent closed system,

$$\alpha_t(A) = e^{iHt/\hbar}Ae^{-iHt/\hbar}$$

satisfies

$$\alpha_{t+s} = \alpha_t\circ\alpha_s, \qquad \alpha_0(A)=A.$$

The derivative of $\alpha_t$ at $t=0$ is $\mathcal D_H$.

Nested Commutators

Finite-time Heisenberg evolution can be written as a nested-commutator expansion:

$$\begin{aligned} A_H(t) &= e^{iHt/\hbar}Ae^{-iHt/\hbar} \\ &= A + \frac{it}{\hbar}[H,A] + \frac{1}{2!} \left( \frac{it}{\hbar} \right)^2 [H,[H,A]] + \cdots . \end{aligned}$$

The first commutator gives the instantaneous velocity in operator space. Higher nested commutators encode curvature of the trajectory. If the nested sequence closes after finitely many operators, the dynamics can often be solved algebraically.

This expansion is a special use of the Baker–Campbell–Hausdorff family of identities. For the dynamics treatment, see Baker–Campbell–Hausdorff Formula. For the compact reference formula, see Baker–Campbell–Hausdorff. For finite products that approximate exponentials of sums, see Trotter Product Formula.

Example: Canonical Variables

Let

$$H = \frac{p^2}{2m} + V(x),$$

with $[x,p]=i\hbar$. The position equation is

$$\dot x_H = \frac{i}{\hbar}[H,x_H] = \frac{p_H}{m}.$$

The momentum equation is

$$\dot p_H = \frac{i}{\hbar}[H,p_H] = -V'(x_H).$$

These look like Hamilton’s equations, but they are operator equations. Products, functions of $x_H$, and functions of $p_H$ still require operator care. The expectation-value version is developed in Ehrenfest Theorem.

Example: Harmonic Oscillator Ladder Operator

For

$$H = \hbar\omega \left( a^\dagger a+\frac12 \right),$$

the ladder operator obeys

$$[H,a] = -\hbar\omega a.$$

Therefore

$$\dot a_H = \frac{i}{\hbar}[H,a_H] = -i\omega a_H,$$

and

$$a_H(t) = e^{-i\omega t}a_H(0).$$

The algebra closes on $a$ itself, so the solution is immediate.

Example: Spin Precession

For a spin in a uniform magnetic field, take

$$H = -\gamma\,\mathbf B\cdot\mathbf S,$$

with

$$[S_i,S_j] = i\hbar\epsilon_{ijk}S_k.$$

The Heisenberg equation gives

$$\frac{d\mathbf S}{dt} = \gamma\,\mathbf S\times\mathbf B.$$

The spin operator precesses about the magnetic-field direction. The spin-specific page Larmor Precession develops the physical conventions and sign choices in more detail.

Constants of Motion

If $A$ has no explicit time dependence and

$$[H,A]=0,$$

then

$$\dot A_H=0.$$

This is an operator statement: every spectral projector of $A$ is conserved under the same assumptions. It is stronger than saying one expectation value happens to be constant in one state.

The symmetry-oriented version of this criterion is Commutators and Conservation Laws. The core expectation-value statement is Conservation Laws.

Classical Poisson-Bracket Correspondence

Classical Hamiltonian evolution is

$$\frac{df}{dt} = \{f,H\} + \frac{\partial f}{\partial t}.$$

Quantum Heisenberg evolution is

$$\frac{dA_H}{dt} = \frac{1}{i\hbar}[A_H,H] + \left( \frac{\partial A}{\partial t} \right)_H.$$

The formal correspondence is therefore

$$\{f,g\} \quad \longleftrightarrow \quad \frac{1}{i\hbar}[\hat f,\hat g].$$

This is a correspondence, not a complete quantization algorithm. Operator ordering, domains, representations, and boundary conditions matter. The classical side is reviewed in Poisson Brackets.

Phase-space quantum mechanics sharpens the comparison. Under the Weyl transform, the commutator becomes the Moyal Bracket, whose leading term is the Poisson bracket and whose higher terms encode quantum corrections.

Density-Operator Version

The same commutator controls density operators, but with the dual sign:

$$\dot\rho = -\frac{i}{\hbar}[H,\rho].$$

This is the Liouville–von Neumann Equation. As a linear map on operator space, the same generator is the Hamiltonian Liouvillian described in Liouvillian Superoperators. The sign differs from the observable equation because states and observables transform oppositely under unitary equivalence:

$$A_H = U^\dagger A_SU, \qquad \rho_S = U\rho_0U^\dagger.$$

The two signs are required for expectation values to agree.

Common Mistakes

• Reversing the sign by mixing $[H,A]$ and $[A,H]$ without the compensating minus sign.
• Treating operator equations as ordinary commuting equations after deriving them.
• Forgetting explicit time dependence in $A_S(t)$.
• Checking $\langle[H,A]\rangle=0$ in one state and concluding $[H,A]=0$ as an operator.
• Assuming the Poisson-bracket correspondence removes all ordering and domain questions.
• Expanding nested commutators beyond their useful convergence or domain regime for unbounded operators.
• Confusing Hamiltonian commutator dynamics with dissipative open-system dynamics.

Cross-Links

• Operator Dynamics
• Commutators
• Heisenberg Equations of Motion
• Liouville–von Neumann Equation
• Liouvillian Superoperators
• Baker–Campbell–Hausdorff Formula
• Trotter Product Formula
• Baker–Campbell–Hausdorff
• Commutators and Conservation Laws
• Conservation Laws
• Poisson Brackets
• Moyal Bracket
• Ehrenfest Theorem
• Larmor Precession

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley, 2002.
• C. K. Zachos, D. B. Fairlie, and T. L. Curtright, eds., Quantum Mechanics in Phase Space, World Scientific, 2005.

Exercises

1. Derive the first-order short-time formula

$$A_H(t+\Delta t) = A_H(t) + \frac{i\Delta t}{\hbar}[H,A_H(t)] +O(\Delta t^2).$$

Solution

Use

$$U(\Delta t) = I-\frac{iH\Delta t}{\hbar} +O(\Delta t^2), \qquad U^\dagger(\Delta t) = I+\frac{iH\Delta t}{\hbar} +O(\Delta t^2).$$

Then

$$\begin{aligned} U^\dagger A U &= \left( I+\frac{iH\Delta t}{\hbar} \right) A \left( I-\frac{iH\Delta t}{\hbar} \right) +O(\Delta t^2) \\ &= A + \frac{i\Delta t}{\hbar}(HA-AH) +O(\Delta t^2) \\ &= A + \frac{i\Delta t}{\hbar}[H,A] +O(\Delta t^2). \end{aligned}$$

2. Show that $\mathcal D_H(A)=(i/\hbar)[H,A]$ satisfies the product rule.

Solution

Compute

$$\begin{aligned} \mathcal D_H(AB) &= \frac{i}{\hbar}[H,AB] \\ &= \frac{i}{\hbar}(HAB-ABH). \end{aligned}$$

Insert and subtract $AHB$:

$$HAB-ABH = (HA-AH)B + A(HB-BH).$$

Therefore

$$\mathcal D_H(AB) = \mathcal D_H(A)B + A\mathcal D_H(B).$$

3. For $H=\hbar\omega(a^\dagger a+1/2)$ and $[a,a^\dagger]=1$, derive $\dot a_H=-i\omega a_H$.

Solution

Let $N=a^\dagger a$. Since

$$[N,a] = a^\dagger[a,a]+[a^\dagger,a]a = -a,$$

one has

$$[H,a] = \hbar\omega[N,a] = -\hbar\omega a.$$

The Heisenberg equation gives

$$\dot a_H = \frac{i}{\hbar}[H,a_H] = -i\omega a_H.$$

4. Compare the quantum and classical equations for $H=p^2/(2m)+V(x)$.

Solution

Quantum mechanically,

$$\dot x_H = \frac{i}{\hbar}[H,x_H] = \frac{p_H}{m}, \qquad \dot p_H = \frac{i}{\hbar}[H,p_H] = -V'(x_H).$$

Classically,

$$\dot x = \{x,H\} = \frac{p}{m}, \qquad \dot p = \{p,H\} = -V'(x).$$

The forms match for the canonical variables. The quantum equations remain operator equations, so noncommuting products and functions require the operator interpretation.