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. 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. In quantum physics, on the other hand, the relation between system state and the value of an observable is more subtle, requiring some basic linear algebra to explain. In the mathematical formulation of quantum mechanics, states are given by non-zero vectors in a Hilbert space V (where two vectors are considered to specify the same state if, and only if, they are scalar multiples of each other) and 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. Physics is the science of Nature. ... Fig. ... In physics, the term state is used in several related senses, each of which expresses something about the way a physical system is. ... An operational definition of a quantity is the description of a specific process, or set of validation tests, accessible to more persons than the definer (i. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... hello can you hear me For the card game, see Experiment (game). ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Partial plot of a function f. ... Fig. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... One of the remarkable characteristics of the mathematical formulation of quantum mechanics, which distinguishes it from mathematical formulations of theories developed prior to the early 1900s, is its use of abstract mathematical structures, such as Hilbert spaces and operators on these spaces. ... In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... On a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. ... A particle is Look up Particle in Wiktionary, the free dictionary In particle physics, a basic unit of matter or energy. ... In the most restricted usage in quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued square integrable function ψ defined over a portion of space normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared...

In quantum mechanics, measurement of observables exhibits some seemingly mysterious phenomena. This often leads to many misconceptions about the nature of quantum mechanics itself. The facts of the matter, however, are far more prosaic. Specifically, if a system is in a state described by a wave function, 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 wave function may be destroyed, being replaced by a statistical ensemble of wave functions. 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 physics, a statistical ensemble is a very large set of similar systems, considered all at once. ... In thermodynamics, a reversible process (or reversible cycle if the process is cyclic) is a process that can be reversed by means of infinitesimal changes in some property of the system (Sears and Salinger, 1986). ... The measurement problem is the key set of questions that every interpretation of quantum mechanics must answer. ... In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. ... The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics, based on Hugh Everetts relative-state formulation. ... In linear algebra and functional analysis, the partial trace is a generalization of the trace. ...

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. In the case of quantum mechanics, the requisite automorphisms are unitary (or anti-unitary) 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 general, an observer is any system which receives information from an object. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. ... In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. ... In general, the principle of relativity is the requirement that the laws of physics be the same for all observers. ... For a non-technical introduction to the topic, please see Introduction to Special relativity. ...

References

• S. Auyang, How is Quantum Field Theory Possible, Oxford University Press, 1995.
• G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963.
• V. Varadarajan, The Geometry of Quantum Mechanics vols 1 and 2, Springer-Verlag 1985.