Observable

In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued function on the set of all possible system states. In quantum physics, it is an operator, or gauge, where the property of the system state 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. In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued function on the set of all possible system states. In quantum physics, it is an operator, or gauge, where the property of the system state 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. 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 certain mathematical properties of the space in question. In quantum physics, observables manifest as linear operators on a Hilbert space representing the state space of quantum states. The eigenvalues of observables are real numbers 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 eigenvalue of the operator with respect to the system's measured quantum state. 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 linear algebra for its description. In the mathematical formulation of quantum mechanics, states are given by non-zero vectors in a Hilbert space V. Two vectors v and w are considered to specify the same state if and only if w = c v {displaystyle mathbf {w} =cmathbf {v} } for some non-zero c ∈ C {displaystyle cin mathbb {C} } . Observables are given by self-adjoint operators on V. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable. For the case of a system of particles, the space V consists of functions called wave functions or state vectors. 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, 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 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, dynamical variables A {displaystyle A} such as position, translational (linear) momentum, orbital angular momentum, spin, and total angular momentum are each associated with a Hermitian operator A ^ {displaystyle {hat {A}}} that acts on the state of the quantum system. The eigenvalues of operator A ^ {displaystyle {hat {A}}} correspond to the possible values that the dynamical variable can be observed as having. For example, suppose | ψ a ⟩ {displaystyle |psi _{a} angle } is an eigenket (eigenvector) of the observable A {displaystyle mathbf {A} } , with eigenvalue a {displaystyle a} , and exists in a d-dimensional Hilbert space. Then This eigenket equation says that if a measurement of the observable A {displaystyle mathbf {A} } is made while the system of interest is in the state | ψ a ⟩ {displaystyle |psi _{a} angle } , then the observed value of that particular measurement must return the eigenvalue a {displaystyle a} with certainty. However, if the system of interest is in the general state | ϕ ⟩ ∈ H {displaystyle |phi angle in {mathcal {H}}} , then the eigenvalue a {displaystyle a} is returned with probability | ⟨ ψ a | ϕ ⟩ | 2 {displaystyle |langle psi _{a}|phi angle |^{2}} , by the Born rule. 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.