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In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum mechanics, an observable 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.
Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question.
In quantum mechanics, observables manifest as self-adjoint operators on a separable complex Hilbert space representing the quantum state space. [1] Observables assign values to outcomes of particular measurements, corresponding to the eigenvalue of the operator. If these outcomes represent physically allowable states (i.e. those that belong to the Hilbert space) the eigenvalues are real; however, the converse is not necessarily true. [2] [3] [4] 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 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. Not every self-adjoint operator corresponds to a physically meaningful observable. [5] [6] [7] [8] 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. [9]
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 the 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 self-adjoint 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 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 (and and are unit vectors, and the eigenspace of is one-dimensional), then the eigenvalue is returned with probability , by the Born rule.
A crucial difference between classical quantities and quantum mechanical observables is that some pairs of quantum observables may not be simultaneously measurable, a property referred to as complementarity. This is mathematically expressed by non-commutativity of their 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. A measurement of alters the quantum state in a way that is incompatible with the subsequent measurement of and vice versa.
Observables corresponding to commuting operators are called compatible observables. For example, momentum along say the and axis are compatible. Observables corresponding to non-commuting operators are called incompatible observables or complementary variables. For example, the position and momentum along the same axis are incompatible. [10] : 155
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. [11] [12]
Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically designed to ease the types of calculations that frequently come up in quantum mechanics. Its use in quantum mechanics is quite widespread.
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 spaces, which are 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.
Quantum mechanics is a fundamental theory in physics that describes the behavior of nature at and below the scale of atoms. It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science.
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It 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, wave function collapse, also called reduction of the state vector, 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, and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation.
In quantum physics, a wave function is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ. Wave functions are complex-valued. For example, a wave function might assign a complex number to each point in a region of space. The Born rule provides the means to turn these complex probability amplitudes into actual probabilities. In one common form, it says that the squared modulus of a wave function that depends upon position is the probability density of measuring a particle as being at a given place. The integral of a wavefunction's squared modulus over all the system's degrees of freedom must be equal to 1, a condition called normalization. Since the wave function is complex-valued, only its relative phase and relative magnitude can be measured; its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. One has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product is a linear map A that is its own adjoint. If V is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of A is a Hermitian matrix, i.e., equal to its conjugate transpose A∗. By the finite-dimensional spectral theorem, V has an orthonormal basis such that the matrix of A relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.
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 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 for describing the behaviour of systems. The square of the modulus of this quantity represents a probability density.
In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule. For example, a quantum particle like an electron can be described by a quantum state that associates to each point in space a complex number called a probability amplitude. Applying the Born rule to these amplitudes gives the probabilities that the electron will be found in one region or another when an experiment is performed to locate it. This is the best the theory can do; it cannot say for certain where the electron will be found. The same quantum state can also be used to make a prediction of how the electron will be moving, if an experiment is performed to measure its momentum instead of its position. The uncertainty principle implies that, whatever the quantum state, the range of predictions for the electron's position and the range of predictions for its momentum cannot both be narrow. Some quantum states imply a near-certain prediction of the result of a position measurement, but the result of a momentum measurement will be highly unpredictable, and vice versa. Furthermore, the fact that nature violates the statistical conditions known as Bell inequalities indicates that the unpredictability of quantum measurement results cannot be explained away as due to ignorance about "local hidden variables" within quantum systems.
In mathematics, particularly in functional analysis, a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.
In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of commuting observables whose simultaneous eigenspaces span the Hilbert space, so that the eigenvectors are uniquely specified by the corresponding sets of eigenvalues.
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 postulate of quantum mechanics that 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 system in a given state, when measured, is proportional to the square of the amplitude of the system's wavefunction at that state. It was formulated and published by German physicist Max Born in July, 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 embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a quantum-mechanical prediction for the system represented by the state. Knowledge of the quantum state, and the quantum mechanical rules for the system's evolution in time, exhausts all that can be known about a quantum system.
This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.
The Koopman–von Neumann (KvN) theory is a description of classical mechanics as an operatorial theory similar to quantum mechanics, based on a Hilbert space of complex, square-integrable wavefunctions. As its name suggests, the KvN theory is loosely related to work by Bernard Koopman and John von Neumann in 1931 and 1932, respectively. As explained in this entry, however, the historical origins of the theory and its name are complicated.
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.