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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 quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value.

- Quantum mechanics
- Operators on finite and infinite dimensional Hilbert spaces
- Incompatibility of observables in quantum mechanics
- See also
- Further reading

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 for some non-zero . Observables are given by self-adjoint operators on *V*. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable^{[ citation needed ]}. 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 such as position, translational (linear) momentum, orbital angular momentum, spin, and total angular momentum are each associated with a Hermitian operator that acts on the state of the quantum system. The eigenvalues of operator correspond to the possible values that the dynamical variable can be observed as having. For example, suppose is an eigenket (eigenvector) of the observable , with eigenvalue , and exists in a d-dimensional Hilbert space. Then

This eigenket equation says that if a measurement of the observable is made while the system of interest is in the state , then the observed value of that particular measurement must return the eigenvalue with certainty. However, if the system of interest is in the general state , then the eigenvalue is returned with probability , 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.^{[ citation needed ]}

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

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, 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 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.

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 of the corresponding operators, to the effect that the commutator

This inequality expresses a dependence of measurement results on the order in which measurements of observables and are performed. Observables corresponding to non-commutative operators are called as *incompatible observables*. Incompatible observables cannot have a complete set of common eigenfunctions. Note that there can be some simultaneous eigenvectors of and , but not enough in number to constitute a complete basis.^{ [1] }^{ [2] }

- Auyang, Sunny Y. (1995).
*How is quantum field theory possible?*. New York, N.Y.: Oxford University Press. ISBN 978-0195093452. - Ballentine, Leslie E. (2014).
*Quantum mechanics : a modern development*(Repr. ed.). World Scientific Publishing Co. ISBN 9789814578608. - von Neumann, John (1996).
*Mathematical foundations of quantum mechanics*. Translated by Robert T. Beyer (12. print., 1. paperback print. ed.). Princeton, N.J.: Princeton Univ. Press. ISBN 978-0691028934. - Varadarajan, V.S. (2007).
*Geometry of quantum theory*(2nd ed.). New York: Springer. ISBN 9780387493862. - Weyl, Hermann (2009). "Appendix C: Quantum physics and causality".
*Philosophy of mathematics and natural science*. Revised and augmented English edition based on a translation by Olaf Helmer. Princeton, N.J.: Princeton University Press. pp. 253–265. ISBN 9780691141206. - Claude Cohen-Tannoudji; Bernard Diu; Franck Laloë (4 December 2019).
*Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications*. Wiley. ISBN 978-3-527-34553-3. - David J. Griffiths (2017).
*Introduction to Quantum Mechanics*. Cambridge University Press. ISBN 978-1-107-17986-8.

- ↑ Griffiths, David J. (2017).
*Introduction to Quantum Mechanics*. Cambridge University Press. p. 111. ISBN 978-1-107-17986-8. - ↑ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2019-12-04).
*Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications*. Wiley. p. 232. ISBN 978-3-527-34553-3.

In quantum mechanics, **bra–ket notation,** or **Dirac notation**, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets".

The **mathematical formulations of quantum mechanics** are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert space which is a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces, and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.

The **Schrödinger equation** is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In quantum mechanics, a **density matrix** is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefuntions: while those can only represent pure states, density matrices can also represent *mixed states*. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a part of an quantum entanglement, which cannot be described by pure states.

In quantum mechanics, **wave function collapse** occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an "observation". It is the essence of a measurement in quantum mechanics which connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is the continuous evolution via the Schrödinger equation. Collapse is a black box for a thermodynamically irreversible interaction with a classical environment. Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions *apparently* reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation. More importantly, this is not enough to explain wave function collapse, as decoherence does not reduce it to a single eigenstate.

A **wave function** in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters *ψ* and Ψ.

**Quantum indeterminacy** is the apparent *necessary* incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics. Prior to quantum physics, it was thought that

In physics, an **operator** is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

In quantum mechanics, a **probability amplitude** is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability density.

In quantum physics, a **measurement** is the testing or manipulation of a physical system in order to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.

In physics, **canonical quantization** is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory, to the greatest extent possible.

In quantum mechanics, a **complete set of commuting observables** (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system.

In quantum mechanics, an energy level is **degenerate** if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.

The **Born rule** is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a particle at a given point, when measured, is proportional to the square of the magnitude of the particle's wavefunction at that point. It was formulated by German physicist Max Born in 1926.

In quantum mechanics, the **position operator** is the operator that corresponds to the position observable of a particle.

In quantum mechanics, the **expectation value** is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the *most* probable value of a measurement; indeed the expectation value may have zero probability of occurring. It is a fundamental concept in all areas of quantum physics.

In quantum physics, a **quantum state** is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called **pure quantum states**, while all other states are called **mixed quantum states**. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

The **Koopman–von Neumann mechanics** is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively.

In mathematical physics, the **Dirac–von Neumann axioms** give a mathematical formulation of quantum mechanics in terms of operators on a Hilbert space. They were introduced by Paul Dirac in 1930 and John von Neumann in 1932.

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