Conserved Observables and Symmetry

A constant of motion is an observable whose value is preserved by the dynamics in a specified sense. The most useful operator criterion is:

$$\frac{\partial A}{\partial t} + \frac{i}{\hbar}[H,A] =0.$$

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

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

The explicit derivative matters: a moving observable can remain constant in the Heisenberg picture even when its Schrödinger-picture expression changes. The Core Formalism derivation is Conservation Laws; the symmetry interpretation is Commutators and Conservation Laws and Quantum Noether Principle.

Conservation in one state or in every state

The phrase “conserved” can mean several different things.

Statement	Meaning	What it establishes
$\frac{d}{dt}\langle A\rangle=0$ in one state	one expectation value is stationary	weak
$\frac{d}{dt}\langle A\rangle=0$ in all states	every expectation is preserved	equivalent to operator constancy for bounded observables
$\frac{dA_H}{dt}=0$	the Heisenberg operator is time independent	all spectral probabilities are preserved
spectral projectors of $A$ are preserved	the full measurement distribution is fixed	equivalent spectral formulation of operator constancy

The last three rows express the same conservation law under appropriate regularity assumptions; they are not three successively stronger laws. For unbounded operators, the spectral formulation avoids unproved differentiations on vectors outside the domain. A state need not be an eigenstate of $A$ for $A$ to be conserved.

Schrödinger-Picture Criterion

Let the state obey

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle = H(t)\lvert\psi(t)\rangle,$$

and let $A(t)$ be a possibly time-dependent observable in the Schrödinger picture. Then

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

If the operator identity

$$\frac{\partial A}{\partial t} + \frac{i}{\hbar}[H,A] =0$$

holds, then $\langle A\rangle$ is conserved for every state evolving under $H(t)$.

For a time-independent observable, this becomes $[H(t),A]=0$ at every time. In particular, for a time-independent Hamiltonian,

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

This is the most common constants-of-motion test.

Heisenberg-Picture Criterion

In the Heisenberg picture,

$$A_H(t) = U^\dagger(t,t_0)A_S(t)U(t,t_0).$$

The equation of motion is

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

Thus $A$ is an operator constant of motion when

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

This is often the cleanest definition because it says that the observable itself, not only one of its expectation values, is unchanged by time evolution.

Explicit Time Dependence

A quantity can be conserved even when the Schrödinger-picture operator contains explicit time dependence. The explicit derivative can cancel the commutator term.

For a free particle,

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

Momentum is conserved because $[H,P]=0$. Position is not conserved because

$$\frac{i}{\hbar}[H,X] = \frac{P}{m}.$$

However, the explicitly time-dependent operator

$$A(t)=X-\frac{t}{m}P$$

is a constant of motion:

$$\frac{\partial A}{\partial t} = -\frac{P}{m}, \qquad \frac{i}{\hbar}[H,A] = \frac{P}{m}.$$

Therefore

$$\frac{\partial A}{\partial t} + \frac{i}{\hbar}[H,A] =0.$$

This example is a useful warning: commuting with $H$ is sufficient only when the observable has no explicit time dependence.

Energy and Time-Dependent Hamiltonians

For $A=H(t)$,

$$\frac{d}{dt}\langle H(t)\rangle = \left\langle \frac{\partial H}{\partial t} \right\rangle,$$

because $[H(t),H(t)]=0$ at equal times. Thus energy is conserved for a closed system with a time-independent Hamiltonian, but not generally for a driven system.

If $H(t)$ depends on an externally controlled parameter, the changing expectation value of $H$ usually represents work done by or on that external drive. It is not a failure of unitary quantum mechanics.

Constants from Symmetry

The most common source of constants of motion is symmetry. If

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

is a continuous unitary symmetry of a time-independent Hamiltonian, then

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

When $G$ has no explicit time dependence, $G$ is a constant of motion. This is the ordinary quantum-mechanical Noether pattern:

• translation symmetry gives momentum conservation;
• rotational symmetry gives angular-momentum conservation;
• time-translation symmetry gives energy conservation;
• global phase symmetry gives conservation of the associated charge or number.

Not every constant of motion is obviously tied to a manifest geometric symmetry. Some are hidden or accidental, such as the additional conserved structure behind the Coulomb problem.

Compatible Constants

Several quantities may each commute with the Hamiltonian without commuting with one another. If

$$[H,A]=0, \qquad [H,B]=0,$$

it does not follow that $[A,B]=0$.

This matters because only mutually commuting observables can generally be used simultaneously to label stationary states. In a rotationally invariant system,

$$[H,J_x]=[H,J_y]=[H,J_z]=0,$$

but

$$[J_x,J_y]=i\hbar J_z.$$

One usually chooses a compatible set such as $H$, $J^2$, and $J_z$, not all three components $J_x,J_y,J_z$.

Complete Sets and Integrability Preview

A complete commuting set has one-dimensional joint eigenspaces in a discrete spectrum: its joint eigenvalues distinguish states up to phase. If a degeneracy remains, further labels are needed and the set is not complete in that sector. In ordinary central-potential problems, the labels $E$, $l$, and $m$ come from a commuting set involving $H$, $L^2$, and $L_z$.

In classical mechanics, integrability is tied to having enough independent constants of motion in involution. There is no single equally universal definition of quantum integrability. Merely listing all spectral projectors of $H$ is not useful: that can be done for any finite Hamiltonian. Many-body integrability generally requires additional structure, such as an extensive family of local or quasilocal commuting charges.

This is only a preview. The practical state-labeling story is developed in Simultaneous Eigenstates and Good Quantum Numbers. The important practical point here is modest: conserved operators are most useful as labels when they are mutually compatible.

Exercises

1. For a free particle, show that $A(t)=X-tP/m$ is a constant of motion.

Solution

With $H=P^2/(2m)$,

$$\frac{\partial A}{\partial t} = -\frac{P}{m}.$$

Also,

$$\frac{i}{\hbar}[H,A] = \frac{i}{\hbar}[H,X] -\frac{t}{m}\frac{i}{\hbar}[H,P].$$

Since $[H,P]=0$ and $(i/\hbar)[H,X]=P/m$,

$$\frac{i}{\hbar}[H,A] = \frac{P}{m}.$$

The two terms cancel, so

$$\frac{\partial A}{\partial t} + \frac{i}{\hbar}[H,A] =0.$$

2. Suppose $[H,A]=0$ and $[H,B]=0$. Must $A$ and $B$ commute?

Solution

No. Angular momentum in a rotationally invariant Hamiltonian is the standard example:

$$[H,J_x]=[H,J_y]=[H,J_z]=0,$$

but

$$[J_x,J_y]=i\hbar J_z.$$

Each component is conserved, but the components cannot all be diagonalized simultaneously.

3. Let $H(t)=H_0+\lambda(t)B$. Compute $d\langle H(t)\rangle/dt$ for closed evolution.

Solution

Use the expectation-value equation with $A=H(t)$:

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

The commutator is zero, and

$$\frac{\partial H}{\partial t} = \dot\lambda(t)B.$$

Therefore

$$\frac{d}{dt}\langle H(t)\rangle = \dot\lambda(t)\langle B\rangle.$$

4. In a central potential, why can one label states by $L^2$ and $L_z$ but not by all three components $L_x,L_y,L_z$?

Solution

For a central potential, $H$ commutes with each component of $\mathbf L$ and with $L^2$. However, the angular momentum components do not commute with one another:

$$[L_i,L_j] = i\hbar\sum_k\epsilon_{ijk}L_k.$$

Thus one can choose a mutually commuting set such as $H$, $L^2$, and $L_z$, but not $H$, $L_x$, $L_y$, and $L_z$ all at once.

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.
• 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.