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Measurement in quantum mechanics

Overview

The framework of quantum mechanics

Quantum mechanics

Quantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...

requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics

Interpretation of quantum mechanics

An interpretation of quantum mechanics is a statement which attempts to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has received thorough experimental testing, many of these experiments are open to different interpretations...

, for which there is currently no consensus.

Measurement is viewed in different ways in the many interpretations of quantum mechanics; however, despite the considerable philosophical differences, they almost universally agree on the practical question of what results from a routine quantum-physics laboratory measurement.

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Encyclopedia

The framework of quantum mechanics

Quantum mechanics

requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics

Interpretation of quantum mechanics

, for which there is currently no consensus.

Measurement from a practical point of view

Measurement is viewed in different ways in the many interpretations of quantum mechanics; however, despite the considerable philosophical differences, they almost universally agree on the practical question of what results from a routine quantum-physics laboratory measurement. To describe this, a simple framework to use is the Copenhagen interpretation

Copenhagen interpretation

The Copenhagen interpretation is an interpretation of quantum mechanics. A key feature of quantum mechanics is that the state of every particle is described by a wavefunction, which is a mathematical representation used to calculate the probability for it to be found in a location or a state of...

, and it will be implicitly used in this section; the utility of this approach has been verified countless times, and all other interpretations are necessarily constructed so as to give the same quantitative predictions as this in almost every case.

Qualitative overview

The quantum state

Quantum state

In quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be...

of a system is a mathematical object that fully describes the quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Once the quantum state has been prepared, some aspect of it is measured (for example, its position or energy). If the experiment is repeated, so as to measure the same aspect of the same quantum state prepared in the same way, the result of the measurement will often be different.

The expected result of the measurement is in general described by a probability distribution

Probability distribution

In probability theory and statistics, a probability distribution identifies either the probability of each value of an unidentified random variable , or the probability of the value falling within a particular interval...

that specifies the likelihoods that the various possible results will be obtained. (This distribution can be either discrete

Discrete probability distribution

Discrete probability distributions arise in the mathematical description of probabilistic and statistical problems in which the values that might be observed are restricted to being within a pre-defined list of possible values...

or continuous, depending on what is being measured.)

The measurement process is often said to be random

Stochastic process

In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process...

and indeterministic

Indeterminism

Indeterminism is a philosophical position that maintains that some form of determinism is incorrect: that there are events which do not correspond with determinism .While the ontological determinism rules out the chance, theorizing that the becoming is only by necessity, the...

. (However, there is considerable dispute over this issue; in some interpretations of quantum mechanics, the result merely appears random and indeterministic, in other interpretations the indeterminism is core and irreducible.) This because an important aspect of measurement is wavefunction collapse

Wavefunction collapse

In quantum mechanics, wave function collapse is the process by which a wave function, initially in a superposition of different eigenstates, appears to reduce to a single one of the states after interaction with an observer...

, the nature of which varies according the interpretation adopted.

What is universally agreed, however, is that if the measurement is repeated, without re-preparing the state, one finds the same result as the first measurement. As a result, after measuring some aspect of the quantum state, we normally update the quantum state to reflect the result of the measurement; it is this updating that ensures that if an immediate re-measurement is repeated without re-preparing the state, one finds the same result as the first measurement. The updating of the quantum state model is called wavefunction collapse

Wavefunction collapse

Quantitative details

The mathematical relationship between the quantum state and the probability distribution is, again, widely accepted among physicists, and has been experimentally confirmed countless times. This section summarizes this relationship, which is stated in terms of the mathematical formulation of quantum mechanics

Mathematical formulation of quantum mechanics

The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. It is distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such...

Measurable quantities ("observables") as operators

It is a postulate of quantum mechanics that all measurements have an associated operator (called an observable operator, or just an observable), with the following properties:

The observable is a Hermitian (self-adjoint
Self-adjoint operator
In mathematics, 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...

) operator mapping a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

(namely, the state space
State space (physics)
In physics, a state space is a complex Hilbert space within which the possible instantaneous states of the system may be described by a unit vector. These state vectors, using Dirac's bra-ket notation, can often be treated as vectors and operated on using the rules of linear algebra...

, which consists of all possible quantum states) into itself.
The observable's eigenvalues are real
Real number
In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real...

. The possible outcomes of the measurement are precisely the eigenvalues of the given observable.
For each eigenvalue there are one or more corresponding eigenvectors (which in this context are called eigenstates), which will make up the state of the system after the measurement.
The observable has a set of eigenvectors which span
Linear span
In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set...

the state space. It follows that each observable generates an orthonormal basis
Basis (linear algebra)
In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others...

of eigenvectors (called an eigenbasis). Physically, this is the statement that any quantum state can always be represented as a superposition
Quantum superposition
Quantum superposition is the fundamental law of quantum mechanics. It defines the collection of all possible states that an object can have.In probability theory, every possible event has a non-negative real number between zero and one associated to it, the probability, which gives the chance that...

of the eigenstates of an observable.

Important examples of observables are:

The Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

operator, representing the total energy
Energy
In physics, energy is a scalar physical quantity that describes the amount of work that can be performed by a force, an attribute of objects and systems that is subject to a conservation law...

of the system; with the special case of the nonrelativistic Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

operator: .
The momentum
Momentum
In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section "modern definitions of momentum" on this page...

operator: (in the position basis).
The position operator
Position operator
In quantum mechanics, the position operator corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L², the Hilbert space of complex-valued and square-integrable functions on...

: , where (in the momentum basis).

Operators can be noncommuting

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:...

. Two Hermitian operators commute if (and only if) there is at least one basis of vectors, each of which is an eigenvector of both operators (this is sometimes called a simultaneous eigenbasis). Noncommuting observables are said to be incompatible and cannot in general be measured simultaneously. In fact, they are related by an uncertainty principle

Uncertainty principle

In quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known...

, as a consequence of the Robertson-Schrödinger relation.

Measurement probabilities and wavefunction collapse

There are a few possible ways to mathematically describe the measurement process (both the probability distribution and the collapsed wavefunction). The most convenient description depends on the spectrum (i.e., set of eigenvalues) of the observable.

Discrete, nondegenerate spectrum

Let be an observable, and suppose that it has discrete eigenstates (in bra-ket notation

Bra-ket notation

Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in pure mathematics...

) for and corresponding eigenvalues , no two of which are equal.

Assume the system is prepared in state . Since the eigenstates of an observable form a basis

Basis (linear algebra)

In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others...

(the eigenbasis), it follows that can be written in terms of the eigenstates as
(where are complex numbers). Then measuring can yield any of the results , with corresponding probabilities given by
Usually is assumed to be normalized, in which case this expression reduces to

If the result of the measurement is , then the system's quantum state after the measurement is
so any repeated measurement of will yield the same result . (This phenomenon is called wavefunction collapse

Wavefunction collapse

Continuous, nondegenerate spectrum

Let be an observable, and suppose that it has a continuous spectrum

Continuous spectrum

In physics, continuous spectrum refers to a range of values which may be graphed to fill a range with closely-spaced or overlapping intervals. The term is derived from the use of the word spectrum to describe the 'ghost-like' rainbow which appears when white light is shone through a clear...

of eigenvalues filling the interval

Interval (mathematics)

In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

(a,b). Assume further that each eigenvalue x in this range is associated with a unique eigenstate .

Assume the system is prepared in state , which can be written in terms of the eigenbasis as
(where is a complex-valued function). Then measuring can yield a result anywhere in the interval (a,b), with probability density function

Probability density function

In probability theory, a probability density function —often referred to as a probability distribution function—or density, of a random variable is a function that describes the density of probability at each point in the sample space...

; i.e., a result between y and z will occur with probability
Again, is often assumed to be normalized, in which case this expression reduces to
If the result of the measurement is x, then the new wave function will be

Alternatively, it is often possible and convenient to analyze a continuous-spectrum measurement by taking it to be the limit of a different measurement with a discrete spectrum. For example, an analysis of scattering

Scattering

Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

involves a continuous spectrum of energies, but by adding a "box" potential

Particle in a box

In quantum mechanics, the particle in a box model describes a particle, which is free to move in a space surrounded by impenetrable barriers...

(which bounds the volume in which the particle can be found), the spectrum becomes discrete

Discrete spectrum

In physics, discrete spectrum is a finite set or a countable set of eigenvalues of an operator. An operator acting on a Hilbert space is said to have a discrete spectrum if its spectrum consists of isolated points. If the spectrum of an operator is not discrete, we say that it is a continuous...

. By considering larger and larger boxes, this approach need not involve any approximation, but rather can be regarded as an equally valid formalism in which this problem can be analyzed.

Degenerate spectra

If there are multiple eigenstates with the same eigenvalue (called degeneracies), the analysis is a bit less simple to state, but not essentially different. In the discrete case, for example, instead of finding a complete eigenbasis, it is a bit more convenient to write the Hilbert space as a direct sum of eigenspaces. The probability of measuring a particular eigenvalue is the squared component of the state vector

Quantum state

in the corresponding eigenspace, and the new state after measurement is the projection

Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P² = P. It leaves its image unchanged....

of the original state vector into the appropriate eigenspace.

Density matrix formulation

Instead of performing quantum-mechanics computations in terms of wavefunction

Wavefunction

A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics describe how the wave function evolves over time...

s (kets

Bra-ket notation

), it is sometimes necessary to describe a quantum-mechanical system in terms of a density matrix

Density matrix

In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix, , of trace one, that describes the statistical state of a quantum system...

. The analysis in this case is formally slightly different, but the physical content is the same, and indeed this case can be derived from the wavefunction formulation above. The result for the discrete, degenerate case, for example, is as follows:

Let be an observable, and suppose that it has discrete eigenvalues , associated with eigenspaces respectively. Let be the projection operator

Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P² = P. It leaves its image unchanged....

into the space .

Assume the system is prepared in the state described by the density matrix ρ. Then measuring can yield any of the results , with corresponding probabilities given by
where Tr denotes trace

Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...

. If the result of the measurement is n, then the new density matrix will be
Alternatively, one can say that the measurement process results in the new density matrix
where the difference is that ρ ' ' is the density matrix describing the entire ensemble, whereas ρ ' is the density matrix describing the sub-ensemble whose measurement result was n.

Statistics of measurement

As detailed above, the result of measuring a quantum-mechanical system is described by a probability distribution. Some properties of this distribution are as follows:

Suppose we take a measurement corresponding to observable , on a state whose quantum state is .

The mean
Expected value
In probability theory and statistics, the expected value of a random variable is the integral of the random variable with respect to its probability measure....

(average) value of the measurement is (see Expectation value (quantum mechanics)
Expectation value (quantum mechanics)
In quantum mechanics, the expectation value is the predicted mean value of the result of an experiment. It is a fundamental concept in all areas of quantum physics.- Operational definition :...

).
The variance
Variance
In probability theory and statistics, the variance of a random variable or distribution is the expected square deviation of that variable from its expected value or mean, or to put it another way: variance is the measure of the amount of variation of all the scores for a variable...

of the measurement is
The standard deviation
Standard deviation
In probability theory and statistics, the standard deviation of a statistical population, a data set, or a probability distribution is the square root of its variance. Standard deviation is a widely used measure of the variability or dispersion, being algebraically more tractable though...

of the measurement is

These are direct consequences of the above formulas for measurement probabilities.

Example

Suppose that we have a particle in a 1-dimensional box

Particle in a box

In quantum mechanics, the particle in a box model describes a particle, which is free to move in a space surrounded by impenetrable barriers...

, set up initially in the ground state . As can be computed from the time-independent Schrödinger equation

Schrödinger equation

In physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...

, the energy of this state is (where m is the particle's mass and L is the box length), and the spatial wavefunction is . If the energy is now measured, the result will always certainly be , and this measurement will not affect the wavefunction.

Next suppose that the particle's position is measured. The position x will be measured with probability density
If the measurement result was x=S, then the wavefunction after measurement will be the position eigenstate . If the particle's position is immediately measured again, the same position will be obtained.

The new wavefunction can, like any wavefunction, be written as a superposition of eigenstates of any observable. In particular, using energy eigenstates, , we have
If we now leave this state alone, it will smoothly evolve in time according to the Schrödinger equation

Schrödinger equation

In physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time...

. But suppose instead that an energy measurement is immediately taken. Then the possible energy values will be measured with relative probabilities:
and moreover if the measurement result is , then the new state will be the energy eigenstate .

So in this example, due to the process of wavefunction collapse

Wavefunction collapse

, a particle initially in the ground state can end up in any energy level, after just two subsequent non-commuting

Commutativity

In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...

measurements are made.

Wavefunction collapse

The process in which a quantum state becomes one of the eigenstates of the operator corresponding to the measured observable

Observable

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. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...

is called "collapse", or "wavefunction collapse". The final eigenstate appears randomly with a probability equal to the square of its overlap with the original state. The process of collapse has been studied in many experiments, most famously in the double-slit experiment

Double-slit experiment

In quantum mechanics, the double-slit experiment demonstrates the inseparability of the wave and particle natures of light and other quantum particles. A coherent light source illuminates a thin plate with two parallel slits cut in it, and the light passing through the slits strikes a screen...

. The wavefunction collapse raises serious questions regarding "the measurement problem", as well as, questions of determinism

Determinism

Determinism is the view that every event, including human cognition, behavior, decision, and action, is causally determined by an unbroken chain of prior occurrences. With numerous historical debates, many varieties and philosophical positions on the subject of determinism exist from...

and locality

Principle of locality

In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual...

, as demonstrated in the EPR paradox

EPR paradox

In quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory...

and later in GHZ entanglement

Greenberger-Horne-Zeilinger state

In physics, in the area of quantum information theory, a Greenberger-Horne-Zeilinger state is a certain type of entangled quantum state which involves at least three subsystems . It was first studied by D. Greenberger, M.A. Horne and Anton Zeilinger in 1989...

. (See below.)

In the last few decades, major advances have been made toward a theoretical understanding of the collapse process. This new theoretical framework, called quantum decoherence

Quantum decoherence

In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior...

, supersedes previous notions of instantaneous collapse and provides an explanation for the absence of quantum coherence after measurement. While this theory correctly predicts the form and probability distribution of the final eigenstates, it does not explain the randomness inherent in the choice of final state.

von Neumann measurement scheme

The von Neumann measurement scheme, an ancestor of quantum decoherence theory, describes measurements by taking into account the measuring apparatus which is also treated as a quantum object.
Let the quantum state be in the superposition , where are eigenstates of the operator that needs to be measured. In order to make the measurement, the measured system described by needs to interact with the measuring apparatus described by the quantum state , so that the total wave function before the interaction is . During the interaction of object and measuring instrument the unitary

Unitary operator

In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfying...

evolution is supposed to realize the following transition from the initial to the final total wave function:
where are orthonormal states of the measuring apparatus. The unitary evolution above is referred to as premeasurement. The relation with wave function collapse is established by calculating from the final total wave function the final density operator of the object as This density operator is interpreted by von Neumann as describing an ensemble of objects being after the measurement with probability in the state

The transition
is often referred to as weak von Neumann projection, the wave function collapse or strong von Neumann projection
being thought to correspond to an additional selection of a subensemble by means of observation.

In case the measured observable has a degenerate spectrum, weak von Neumann projection is generalized to Lüders projection
in which the vectors for fixed n are the degenerate eigenvectors of the measured observable. For an arbitrary state described by a density operator
Lüders projection is given by

Measurements of the second kind

In a measurement of the second kind the unitary evolution during the interaction of object and measuring instrument is supposed to be given by

in which the states of the object are determined by specific properties of the interaction between object and measuring instrument. They are normalized but not necessarily mutually orthogonal. The relation with wave function collapse is analogous to that obtained for measurements of the first kind, the final state of the object now being with probability Note that many present-day measurement procedures are measurements of the second kind, some even functioning correctly only as a consequence of being of the second kind (for instance, a photon counter, detecting a photon by absorbing and hence annihilating it, thus ideally leaving the electromagnetic field in the vacuum state rather than in the state corresponding to the number of detected photons; also the Stern-Gerlach experiment would not function at all if it really were a measurement of the first kind).

Decoherence in quantum measurement

One can also introduce the interaction with the environment , so that, in a measurement of the first kind, after the interaction the total wave function takes a form
which is related to the phenomenon of decoherence.

The above is completely described by the Schrödinger equation and there are not any interpretational problems with this. Now the problematic wavefunction collapse does not need to be understood as a process on the level of the measured system, but can also be understood as a process on the level of the measuring apparatus, or as a process on the level of the environment. Studying these processes provides considerable insight into the measurement problem

Measurement problem

The measurement problem in quantum mechanics is the unresolved problem of how wavefunction collapse occurs. The inability to observe this process directly has given rise to different interpretations of quantum mechanics, and poses a key set of questions that each interpretation must answer...

by avoiding the arbitrary boundary between the quantum and classical worlds, though it does not explain the presence of randomness in the choice of final eigenstate. If the set of states
, , or
represents a set of states that do not overlap in space, the appearance of collapse can be generated by either the Bohm interpretation

Bohm interpretation

The Bohm or Bohmian interpretation of quantum mechanics, which Bohm called the causal, or later, the ontological interpretation, is an interpretation postulated by David Bohm in 1952 as an alternative to the standard Copenhagen interpretation. The Bohm interpretation grew out of the search for...

or the Everett interpretation which both deny the reality of wavefunction collapse. Both of these are stated to predict the same probabilities for collapses to various states as the conventional interpretation by their supporters. The Bohm interpretation is held to be correct only by a small minority of physicists, since there are difficulties with the generalization for use with relativistic quantum field theory

Quantum field theory

Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically described by fields or of many-body systems. It is widely used in particle physics and condensed matter physics...

. However, there is no proof that the Bohm interpretation is inconsistent with quantum field theory, and work to reconcile the two is ongoing. The Everett interpretation easily accommodates relativistic quantum field theory.

What physical interaction constitutes a measurement?

Until the advent of quantum decoherence

Quantum decoherence

In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior...

theory in the late 20th century, a major conceptual problem of quantum mechanics and especially the Copenhagen interpretation

Copenhagen interpretation

was the lack of a distinctive criterion for a given physical interaction to qualify as "a measurement" and cause a wavefunction to collapse. This is best illustrated by the Schrödinger's cat

Schrödinger's cat

Schrödinger's cat is a thought experiment, often described as a paradox, devised by Austrian physicist Erwin Schrödinger in 1935. It illustrates what he saw as the problem of the Copenhagen interpretation of quantum mechanics applied to everyday objects. The thought experiment presents a cat that...

paradox. Certain aspects of this question are now well understood in the framework of quantum decoherence theory, such as an understanding of weak measurement

Weak measurement

Weak measurements are a type of quantum measurement, where the measured system is very weakly coupled to the measuring device. After the measurement the measuring device pointer is shifted by what is called the "weak value". So that a pointer initially pointing at zero before the measurement would...

s, and quantifying what measurements or interactions are sufficient to destroy quantum coherence. Nevertheless, there remains less than universal agreement among physicists on some aspects of the question of what constitutes a measurement.

(One particularly well-known aspect of this question is whether a conscious observer is necessary for a measurement; see the article Consciousness causes collapse.)

Does measurement actually determine the state?

The question of whether (and in what sense) a measurement actually determines the state is one which differs among the different interpretations of quantum mechanics. (It is also closely related to the understanding of wavefunction collapse.) For example, in most versions of the Copenhagen interpretation

Copenhagen interpretation

, the measurement determines the state, and after measurement the state is definitely what was measured. But according to the Many-worlds interpretation

Many-worlds interpretation

The many-worlds interpretation is an interpretation of quantum mechanics.It is also known as MWI, the relative state formulation, theory of the universal wavefunction, parallel universes, many-universes interpretation or just many worlds.Many-worlds asserts the objective reality of the...

, measurement determines the state in a more restricted sense: In other "worlds", other measurement results were obtained, and the other possible states still exist.

Is the measurement process random or deterministic?

As described above, there is universal agreement that quantum mechanics appears random, in the sense that all experimental results yet uncovered can be predicted and understood in the framework of quantum mechanics measurements being fundamentally random. Nevertheless, it is not settled
whether this is true, fundamental randomness, or merely "emergent" randomness resulting from underlying hidden variables
Hidden variable theory
Historically, in physics, hidden variable theories were espoused by a minority of physicists who argued that the statistical nature of quantum mechanics indicated that quantum mechanics is "incomplete"...

which deterministically cause measurement results to happen a certain way each time. This continues to be an area of active research.

(If there are hidden variables, they would have to be "nonlocal

Principle of locality

", see below.)

Does the measurement process violate locality?

In physics, the Principle of locality is the concept that information cannot travel faster than the speed of light

Speed of light

In physics, the speed of light is a physical constant, the speed at which electromagnetic radiation, such as light, travels in free space . Its value is 299,792,458 metres per second...

(also see special relativity

Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...

). It is known experimentally (see Bell's theorem

Bell's theorem

Bell's theorem is a no-go theorem, loosely stating that:It is the most famous legacy of the late physicist John S. Bell. The theorem has important implications for physics itself and philosophy of science as well.- Overview :...

, which is related to the EPR paradox

EPR paradox

) that if quantum mechanics is deterministic (due to hidden variables, as described above), then it is nonlocal (i.e. violates the principle of locality). Nevertheless, there is not universal agreement among physicists on whether quantum mechanics is nondeterministic, nonlocal, or both.

External links

"The Double Slit Experiment". (physicsweb.org)
"Measurement in Quantum Mechanics" Henry Krips in the Stanford Encyclopedia of Philosophy
Decoherence, the measurement problem, and interpretations of quantum mechanics
Measurements and Decoherence
The conditions for discrimination between quantum states with minimum error
Yonina C. Eldar, Alexandre Megretski, and George C. Verghese. Designing optimal quantum detectors via semidefinite programming. IEEE Transactions on Information Theory, Vol. 49, No. 4, 1007—1012, 2003.

Measurement in quantum mechanics

Measurement from a practical point of view

Qualitative overview

Quantitative details

Measurable quantities ("observables") as operators

Measurement probabilities and wavefunction collapse

Discrete, nondegenerate spectrum

Continuous, nondegenerate spectrum

Degenerate spectra

Density matrix formulation

Statistics of measurement

Example

Wavefunction collapse

von Neumann measurement scheme

Measurements of the second kind

Decoherence in quantum measurement

What physical interaction constitutes a measurement?

Does measurement actually determine the state?

Is the measurement process random or deterministic?

Does the measurement process violate locality?

See also

External links

Further reading