Observables and Operators

An observable describes the possible values of a measured quantity and the subspaces associated with them. Operators also describe transformations, so learning operator algebra includes learning which physical role a particular operator plays. This chapter connects eigenvalues, projectors, and spectral decompositions to concrete measurement questions.

Required background. States and representations explains how to interpret a quantum state independently of its coordinates.

Helpful background. Bra–ket notation explains bras, kets, and operator matrix elements; linear maps introduces the linear action of operators.

Values and their eigenspaces

In a finite-dimensional system a sharp observable is a self-adjoint matrix. Its spectral decomposition is

$$A=\sum_a aP_a, \qquad P_aP_b=\delta_{ab}P_a, \qquad \sum_aP_a=I.$$

The number $a$ is a possible outcome; $P_a$ projects onto the entire space of states with that value. A degenerate outcome can therefore correspond to several independent eigenvectors.

For $A=\operatorname{diag}(1,1,2)$, the event “the result is 1” has projector $P_1=\operatorname{diag}(1,1,0)$. The normalized state $(1,1,0)^T/\sqrt2$ gives that result with certainty. Replacing $P_1$ by a projector onto only the first coordinate would describe a different, more restrictive event.

In infinite dimensions, a differential formula and its boundary conditions together define an operator. Symmetry under integration by parts is not by itself self-adjointness. Position and momentum also have continuous spectra, where spectral projectors for intervals replace sums over normalizable eigenvectors.

Reading sequence

Question	Article	Main distinction
What does an observable represent?	Observables	physical quantity versus apparatus
What is an operator?	Operators	linear map versus observable role
What makes a value definite?	Eigenvalues and Eigenstates	state versus eigenspace label
How do discrete and continuous outcomes differ?	Discrete and Continuous Spectra	normalizable eigenvectors versus generalized spectral labels
Why are domains important?	Hermitian vs Self-Adjoint Operators	matrix Hermiticity versus operator self-adjointness
What represents a yes-no alternative?	Projectors	subspace versus chosen vector
How is an observable reconstructed from outcomes?	Spectral Decomposition	eigenvalues together with spectral projectors
How can degeneracy be resolved?	Complete Sets of Commuting Observables	one label versus joint labels
What do exponentials and square roots of operators mean?	Functions of Operators	spectral calculus versus entrywise operations
Why can one operator have many concrete forms?	Operator Representations	abstract map versus basis-dependent representation

Learn the algebra through a measurement

Begin with observables, operators, eigenvalues, and projectors. Use spectral decomposition to reconstruct a small Hermitian matrix from its outcomes. Then study degeneracy and complete commuting labels.

The spectrum and self-adjointness articles become essential when you move to differential operators in wave mechanics. Functions of operators explains why $f(A)$ acts on eigenvalues, rather than on individual matrix entries.

Try it. What is $A^2$ for the example above, and which projector describes outcome 4?

Solution

$A^2=\operatorname{diag}(1,1,4)$, and outcome 4 has projector $\operatorname{diag}(0,0,1)$. The eigenspaces are unchanged here; more generally, distinct eigenvalues can merge when a function takes the same value on them.

References

• P. Busch, P. J. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum Measurement, Springer, 2016, doi:10.1007/978-3-319-43389-9.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, rev. ed., Academic Press, 1980.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.