Commutators and Conservation Laws

The basic quantum conservation test is a commutator test. For the Core Formalism derivation of the expectation-value formula, see Conservation Laws. For a closed system with Hamiltonian $H$, an observable $A$ with no explicit time dependence is conserved when

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

This page explains the relation between symmetry, commutators, and constants of motion. For the first-order derivation of the commutator action itself, see Infinitesimal Transformations. For the broader symmetry-conservation framing, see Quantum Noether Principle. For practical tests, see Constants of Motion.

The classical counterpart is the Poisson-bracket test $\{f,H\}=0$, reviewed in Poisson Brackets.

Expectation-Value Equation

For an observable $A(t)$ in closed-system dynamics,

$$\frac{d}{dt}\langle A\rangle = \frac{i}{\hbar}\langle[H,A]\rangle + \left\langle\frac{\partial A}{\partial t}\right\rangle.$$

If $A$ has no explicit time dependence, this reduces to

$$\frac{d}{dt}\langle A\rangle = \frac{i}{\hbar}\langle[H,A]\rangle.$$

Therefore $[H,A]=0$ implies $\langle A\rangle$ is constant in every state evolving under $H$.

Operator Equation

In the Heisenberg picture,

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

If $A$ has no explicit time dependence and commutes with the Hamiltonian, then

$$\frac{dA_H}{dt}=0.$$

This is a stronger statement than conservation of one expectation value in one state: the observable is a constant of motion.

Symmetry Generator Implies Conservation

Suppose a continuous unitary symmetry is generated by $G$:

$$U(\alpha)=e^{-i\alpha G/\hbar}.$$

If the Hamiltonian is invariant under this symmetry, then

$$U(\alpha)HU^\dagger(\alpha)=H$$

for all $\alpha$. Differentiating at $\alpha=0$ gives

$$[G,H]=0.$$

Equivalently, $[H,G]=0$, so $G$ is conserved when it has no explicit time dependence.

Free Particle Momentum Conservation

For a free particle in one dimension,

$$H=\frac{P^2}{2m}.$$

Since $P$ commutes with any function of $P$,

$$[H,P]=0.$$

Momentum is conserved. This is the operator statement behind translation invariance.

Central Potential Angular Momentum Conservation

For a three-dimensional central potential,

$$H=\frac{\mathbf p^2}{2m}+V(r), \qquad r=\lvert\mathbf r\rvert,$$

the Hamiltonian is rotationally invariant. Consequently,

$$[H,L_i]=0$$

for each component of orbital angular momentum. It follows that $L^2$ and a chosen component, usually $L_z$, can be used to label energy eigenstates.

Energy Conservation

If the Hamiltonian itself has no explicit time dependence, then

$$\frac{d}{dt}\langle H\rangle = \frac{i}{\hbar}\langle[H,H]\rangle = 0.$$

Energy conservation in closed quantum mechanics is therefore the conservation law associated with time-translation symmetry.

What Conservation Does Not Mean

Conservation of $A$ does not mean every state has a sharp value of $A$. It means the distribution of $A$ is constant under time evolution when $A$ commutes with $H$. A superposition of $A$ eigenstates can remain a superposition while the probabilities for measuring each $A$ eigenvalue remain fixed.

Common Mistakes

• Checking $\langle[H,A]\rangle=0$ in one state and concluding $[H,A]=0$ as an operator.
• Forgetting the explicit $\partial A/\partial t$ term.
• Assuming conservation means the state is an eigenstate of the conserved quantity.
• Confusing a conserved expectation value in a special state with a symmetry of the Hamiltonian.
• Ignoring domain subtleties for unbounded operators.

Cross-Links

• Commutators
• Poisson Brackets
• Conservation Laws
• Commutator Dynamics
• One-Parameter Unitary Groups
• Generators
• Infinitesimal Transformations
• Quantum Noether Principle
• Constants of Motion
• From Quantum Generators to Noether Currents
• Symmetry Constraints on Hamiltonians
• Exact Symmetry
• Explicit Symmetry Breaking
• Heisenberg Equations of Motion
• Ehrenfest Theorem
• Commutator Table

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.

Exercises

1. Let $H=p^2/(2m)+V(x)$. Use $[p,V(x)]=-i\hbar V'(x)$ to determine when momentum is conserved.

Solution

Since $[p,p^2/(2m)]=0$,

$$[p,H]=[p,V(x)]=-i\hbar V'(x).$$

Momentum is conserved when this vanishes as an operator, which requires $V'(x)=0$ on the region considered. Thus the potential is constant there.

2. If $[H,A]=0$, what can be said about the probabilities of measuring the eigenvalues of $A$?

Solution

When $A$ has no explicit time dependence and commutes with $H$, the projectors onto eigenspaces of $A$ also commute with $H$ under the usual spectral assumptions. The probabilities for the corresponding outcomes are constant under time evolution.