Compatibility, Commutators, and Uncertainty

Operator order matters in quantum mechanics. This chapter connects the commutator to compatible sharp measurements, common eigenstates, and bounds on the spreads of outcomes. An uncertainty bound describes a prepared state; a claim about the errors or disturbance of a detector requires a separate measurement model.

Commutators and spread

The commutator is $[A,B]=AB-BA$. For finite-dimensional self-adjoint matrices, $[A,B]=0$ means that a common orthonormal eigenbasis exists. It does not mean that their outcomes are statistically independent, or that every vector in a degenerate eigenspace of one is already an eigenvector of the other.

For a normalized state with the required finite moments and operator products defined, the Robertson bound is

$$\Delta A\,\Delta B\ge\frac12\lvert\langle[A,B]\rangle\rvert.$$

The right side depends on the state. For Pauli observables, $[\sigma_x,\sigma_y]=2i\sigma_z$. In $\lvert0\rangle$, the two standard deviations are both one and the bound is saturated. In $\lvert+\rangle$, the $\sigma_x$ deviation and the commutator expectation both vanish, although the operators still do not commute.

For position and momentum on the line, suitable states satisfy $\Delta X\Delta P\ge\hbar/2$. Unbounded operators require domain care; strong commutation means that their spectral projectors commute, which is stronger than a formal zero commutator on an arbitrary chosen domain.

Reading sequence

Question	Article	Main distinction
When are sharp observables compatible?	Compatible Observables	joint spectral structure versus independence
What does operator order measure?	Commutators	antisymmetric product versus physical interpretation
What algebra relates position and momentum?	Canonical Commutation Relations	formal relation versus domain-qualified operator statement
What is the symmetric operator product?	Anticommutators	covariance role versus fermionic preview
What are the consequences of noncommutation?	Noncommuting Observables	incompatibility versus disturbance
What is the general spread bound?	General Uncertainty Relations	Robertson bound versus covariance-strengthened bound
Why must position and momentum spreads trade off?	Position-Momentum Uncertainty	preparation spread versus instrument resolution
Which energy–time statements are valid?	Energy-Time Uncertainty	dynamical timescale versus nonexistent universal time operator
How can several quantities be sharp together?	Simultaneous Eigenstates	common eigenvector versus complete joint basis

Follow the distinction you need

For a first pass, read compatible observables, commutators, and noncommuting observables, then work through the general uncertainty derivation. Its covariance term explains why the simple Robertson bound does not capture every correlation.

Position–momentum uncertainty leads to Gaussian packets. Energy–time uncertainty needs its own treatment because time is an evolution parameter in the usual formulation. It does not permit energy conservation to be violated briefly.

Try it. Could finite-dimensional matrices satisfy $[X,P]=i\hbar I$ exactly?

Solution

No. The trace of a finite matrix commutator is zero, while $\operatorname{Tr}(i\hbar I)=i\hbar d$ in dimension $d$. A numerical truncation can approximate the relation on selected states, but cannot satisfy it throughout its finite space.

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, rev. ed., Academic Press, 1980.
• H. P. Robertson, “The Uncertainty Principle,” Physical Review 34, 163–164, 1929, doi:10.1103/PhysRev.34.163.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• E. Schrödinger, “Zum Heisenbergschen Unschärfeprinzip,” Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 296–303, 1930.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.