Observable

In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off some gauge. In systems governed by classical mechanics, any experimentally observable value can be shown to be given by a real-valued function on the set of all possible system states.

Physically meaningful observables must also satisfy transformation laws which relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations which preserve some mathematical property.

Quantum mechanics

In the case of transformation laws in quantum mechanics, the requisite automorphisms are unitary (or antiunitary) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.

In quantum mechanics, measurement of observables exhibits some seemingly unintuitive properties. Specifically, if a system is in a state described by a vector in a Hilbert space, the measurement process affects the state in a non-deterministic, but statistically predictable way. In particular, after a measurement is applied, the state description by a single vector may be destroyed, being replaced by a statistical ensemble. The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problem and is described mathematically by quantum operations. By the structure of quantum operations, this description is mathematically equivalent to that offered by relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system.

In quantum mechanics each dynamical variable (e.g. position, translational momentum, orbital angular momentum, spin, total angular momentum, energy, etc.) is associated with a Hermitian operator that acts on the state of the quantum system and whose eigenvalues correspond to the possible values of the dynamical variable. For example, suppose $|a\backslash rangle$ is an eigenket (eigenvector) of the observable $\backslash mathbf\{A\}$, with eigenvalue $a$, and exists in a d-dimensional Hilbert space, then

:$\backslash mathbf\{A\}$$|a\backslash rangle$ = $a$ $|a\backslash rangle.$

This eigenket equation says that if a measurement of the observable $\backslash scriptstyle\; \backslash mathbf\{A\}$ is made while the system of interest is in the state $\backslash scriptstyle\; |a\backslash rangle$, then the observed value of that particular measurement must return the eigenvalue $a$ with certainty. However, if the system of interest is in the general state $\backslash scriptstyle\; |\backslash phi\backslash rangle\backslash in\backslash mathcal\{H\}$, then the eigenvalue $a$ is returned with probability $\backslash scriptstyle\; |\backslash langle\; a|\backslash phi\backslash rangle|^2$ (Born rule). One must note that the above definition is somewhat dependent upon our convention of choosing real numbers to represent real physical quantities. Indeed, just because dynamical variables are "real" and not "unreal" in the metaphysical sense does not mean that they must correspond to real numbers in the mathematical sense.

To be more precise, the dynamical variable/observable is a (not necessarily bounded) Hermitian operator in a Hilbert Space and thus is represented by a Hermitian matrix if the space is finite-dimensional. In an infinite-dimensional Hilbert space, the observable is represented by a symmetric operator, which may not be defined everywhere (i.e. its domain is not the whole space - there exist some states that are not in the domain of the operator). The reason for such a change is that in an infinite-dimensional Hilbert space, the operator becomes unbounded, which means that it no longer has a largest eigenvalue. This is not the case in a finite-dimensional Hilbert space, where every operator is bounded - it has a largest eigenvalue. For example, if we consider the position of a point particle moving along a line, this particle's position variable can take on any number on the real-line, which is uncountably infinite. Since the eigenvalue of an observable represents a real physical quantity for that particular dynamical variable, then we must conclude that there is no largest eigenvalue for the position observable in this uncountably infinite-dimensional Hilbert space, since the field we're working over consists of the real-line. Nonetheless, whether we are working in an infinite-dimensional or finite-dimensional Hilbert space, the role of an observable in quantum mechanics is to assign real numbers to outcomes of particular measurements; this means that only certain measurements can determine the value of an observable for some state of a quantum system. In classical mechanics, any measurement can be made to determine the value of an observable.

Incompatibility of observables in quantum mechanics

:$\backslash mathbf\{A\}\backslash mathbf\{B\}\; -\; \backslash mathbf\{B\}\backslash mathbf\{A\}\; \backslash neq\; \backslash mathbf\{0\}.$

This inequality expresses a dependence of measurement results on the order in which measurements of observables $\backslash scriptstyle\; \backslash mathbf\{A\}$ and $\backslash scriptstyle\; \backslash mathbf\{B\}$ are performed. Observables corresponding to non-commutative operators are called incompatible.

See also

Further reading

Category:Quantum mechanics