Observables

In, an observable is a that can be measured. Examples include and. In systems governed by, it is a -valued "function" on the set of all possible system states. In, it is an operator, or , where the property of the  can be determined by some sequence of. For example, these operations might involve submitting the system to various s and eventually reading a value.

Physically meaningful observables must also satisfy laws which relate observations performed by different s in different. These transformation laws are s of the state space, that is s which preserve certain mathematical properties of the space in question.

Quantum mechanics
In, observables manifest as on a  representing the  of quantum states. The eigenvalues of observables are that correspond to possible values the dynamical variable represented by the observable can be measured as having. That is, observables in quantum mechanics assign real numbers to outcomes of particular measurements, corresponding to the of the operator with respect to the system's measured. As a consequence, 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.

The relation between the state of a quantum system and the value of an observable requires some for its description. In the, states are given by non-zero s in a V. Two vectors v and w are considered to specify the same state if and only if $$\mathbf{w} = c\mathbf{v}$$ for some non-zero $$c \in \Complex$$. Observables are given by s on V. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable. For the case of a system of s, the space V consists of functions called s or.

In the case of transformation laws in quantum mechanics, the requisite automorphisms are (or ) linear transformations of the Hilbert space V. Under or, the mathematics of frames of reference is particularly simple, considerably restricting 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, 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. The nature of measurement operations in quantum physics is sometimes referred to as the  and is described mathematically by s. By the structure of quantum operations, this description is mathematically equivalent to that offered by  where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the  of the state of the larger system.

In quantum mechanics, dynamical variables $$A$$ such as position, translational (linear), , , and are each associated with a  $$\hat{A}$$ that acts on the  of the quantum system. The of operator $$\hat{A}$$ correspond to the possible values that the dynamical variable can be observed as having. For example, suppose $$|\psi_{a}\rangle$$ is an eigenket of the observable $$\mathbf{A}$$, with eigenvalue $$a$$, and exists in a d-dimensional. Then


 * $$\mathbf{A}|\psi_a\rangle = a|\psi_a\rangle.$$

This eigenket equation says that if a of the observable $$\mathbf{A}$$ is made while the system of interest is in the state $$|\psi_a\rangle$$, 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 $$|\phi\rangle \in \mathcal{H}$$, then the eigenvalue $$a$$ is returned with probability $$|\langle \psi_a|\phi\rangle|^2$$, by the.

To be more precise, the dynamical variable/observable is a in a Hilbert space.

Operators on finite and infinite dimensional Hilbert spaces
Observables can be represented by a Hermitian matrix if the Hilbert space is finite-dimensional. In an infinite-dimensional Hilbert space, the observable is represented by a, which. The reason for such a change is that in an infinite-dimensional Hilbert space, the observable operator can become, which means that it no longer has a largest eigenvalue. This is not the case in a finite-dimensional Hilbert space: an operator can have no more eigenvalues than the of the state it acts upon, and by the, any finite set of real numbers has a largest element. For example, the position of a point particle moving along a line can take any real number as its value, and the set of is. Since the eigenvalue of an observable represents a possible physical quantity that its corresponding dynamical variable can take, we must conclude that there is no largest eigenvalue for the position observable in this uncountably infinite-dimensional Hilbert space.

Incompatibility of observables in quantum mechanics
A crucial difference between classical quantities and quantum mechanical observables is that the latter may not be simultaneously measurable, a property referred to as. This is mathematically expressed by non- of the corresponding operators, to the effect that the


 * $$\left[\mathbf{A}, \mathbf{B}\right] := \mathbf{A}\mathbf{B} - \mathbf{B}\mathbf{A} \neq \mathbf{0}.$$

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