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In physics, an observable is a
physical quantity A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
that can be measured. Examples include
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In
quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
, it is an
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
, or gauge, where the property of the quantum state can be determined by some sequence of
operations Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
. For example, these operations might involve submitting the system to various
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
s and eventually reading a value. Physically meaningful observables must also satisfy
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Trans ...
laws that relate observations performed by different observers in different frames of reference. These transformation laws are
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s of the state space, that is bijective
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Trans ...
s that preserve certain mathematical properties of the space in question.


Quantum mechanics

In
quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
, observables manifest as linear operators on a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
representing the state space of quantum states. The eigenvalues of observables are
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
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, up to a phase constant, pure states are given by non-zero vectors in a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
''V''. Two vectors v and w are considered to specify the same state if and only if \mathbf = c\mathbf for some non-zero c \in \Complex. Observables are given by self-adjoint operators on ''V''. Not every self-adjoint operator corresponds to a physically meaningful observable. Also, not all physical observables are associated with non-trivial self-adjoint operators. For example, in quantum theory, mass appears as a parameter in the Hamiltonian, not as a non-trivial operator. 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 Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigroup ...
(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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, 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 Irreversible may refer to: * Irreversible process, in thermodynamics, a process that is not reversible *'' Irréversible'', a 2002 film * ''Irréversible'' (soundtrack), soundtrack to the film ''Irréversible'' * An album recorded by hip-hop artis ...
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 the
relative state interpretation The many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is objectively real, and that there is no wave function collapse. This implies that all possible outcomes of quantum me ...
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 such as position, translational (linear)
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
, orbital angular momentum,
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
, and total angular momentum are each associated with a Hermitian operator \hat that acts on the state of the quantum system. The
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of operator \hat correspond to the possible values that the dynamical variable can be observed as having. For example, suppose , \psi_\rangle is an eigenket ( eigenvector) of the observable \hat, with eigenvalue a, and exists in a
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. Then \hat, \psi_a\rangle = a, \psi_a\rangle. This eigenket equation says that if a
measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
of the observable \hat 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, then the eigenvalue a is returned with probability , \langle \psi_a, \phi\rangle, ^2, by the Born rule. The above definition is somewhat dependent upon our convention of choosing real numbers to represent real
physical quantities A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
. 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 self-adjoint operator 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
symmetric operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
, which may not be defined everywhere. The reason for such a change is that in an infinite-dimensional Hilbert space, the observable operator can become unbounded, 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 dimension of the state it acts upon, and by the well-ordering property, 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
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
is uncountably infinite. 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 complementarity. This is mathematically expressed by non-
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
of the corresponding operators, to the effect that the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
\left hat, \hat\right:= \hat\hat - \hat\hat \neq \hat. This inequality expresses a dependence of measurement results on the order in which measurements of observables \hat and \hat are performed. Observables corresponding to non-commuting operators are called ''incompatible observables''. Incompatible observables cannot have a complete set of common eigenfunctions. Note that there can be some simultaneous eigenvectors of \hat and \hat, but not enough in number to constitute a complete basis.


See also

*
Measure (physics) The measure in quantum physics is the integration measure used for performing a path integral. In quantum field theory, one must sum over all possible histories of a system. When summing over possible histories, which may be very similar to each ...
* Observable universe * Observer (quantum physics) * Table of QM operators * Unobservable


References


Further reading

* * * * * * {{Quantum mechanics topics Quantum mechanics