NationMaster - Encyclopedia: Measurement in quantum mechanics

The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. Fig. ...

1 The mathematical formalism of measurement
2 Philosophical problems of quantum measurements
3 See also
4 External links

The mathematical formalism of measurement

The goal of a particular measurement of a particular system, in any experimental trial, is to obtain a characterization of the system in mutual agreement between all members of this system, and therefore by a particular method which is reproducible by all members of the system, at least in principle. In classical physics and engineering, measurement generally refers to the process of estimating or determining the ratio of a magnitude of a quantitative property or relation to a unit of the same type of quantitative property or relation. ... In the scientific method, an experiment is a set of actions and observations, performed to support or falsify a hypothesis or research concerning phenomena. ... This article is in need of attention from an expert on the subject. ...

Measurable quantities ("observables") as operators

Observable quantities are represented mathematically by a Hermitian operator, with its eigenvalues representing any definite result value which might be obtained as a result of the measurement, and the corresponding eigenstate being the state of the system during the trial. This representation is possible and appropriate because A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

Its eigenvalues are real, and the possible result values of a measurement are correspondingly real numbers.
It can be unitarily diagonalized (See Spectral theorem). In other words, it has a basis of eigenvectors which spans the entire space of system states corresponding to any possible outcome of the measurement with a definite result value. Distinct system states in distinct trials, resulting in distinct definite result values, are thereby guaranteed to be represented by distinct eigenvectors, and the state of a system can be represented as a linear combination of eigenvectors of any suitable operator.
Its trace is real, corresponding to the (appropriately weighted) real average of definite result values which may be obtained from an ensemble of trials.

Important examples are: In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In aircraft design, see Wingspan. ... In mathematics, a bounded linear operator A over a Hilbert space H is said to be in the trace class if for some (and hence all) orthonormal bases {ek}k of H the sum of positive terms is finite. ... The word ensemble can refer to a musical ensemble an ensemble cast (drama) a statistical ensemble in mathematical physics, for example a thermodynamic ensemble a quantum ensemble a fluid mechanical ensemble a Climate Ensemble ensemble forecasting (meteorology) ensemble averaging a distribution ensemble (maths) a neural ensemble a DAB ensemble Ensemble...

The Hamiltonian operator, ${hat E} = {i hbar}{partial over partial t}$ , representing the measurable quantity called "energy"; with the special case of
The nonrelativistic Hamiltonian operator: ${hat H} = {hat p^2 over 2m} + V(r)$ .
The momentum operator: ${hat p} = {hbar over i}{partial over partial x}$ .
The distance operator: ${hat x}$ , where ${hat x} = {-hbar over i}{partial over partial p}$ .

Many operators are pairwise noncommuting; that is, for a given set of observational data, from a particular trial, one may obtain a definite real result value for one quantity, but not for the other, or even for neither. Even if the state of the system in one particular trial corresponds to one particular eigenstate of one operator, this state is then a nontrivial linear combination of eigenstates of the other operator. The Hamiltonian, denoted H, has two distinct but closely related meanings. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ... In physics, momentum is the product of the mass and velocity of an object. ... Proximity (2001) is also a movie with Rob Lowe, Fred Ward and James Coburn. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ...

Eigenstates and projection

In Quantum mechanics, when you take a measurement of a system with state vector (wave function) $|psirang$ where the corresponding measurement operator ${hat O}$ has eigenstates $|nrang$ for $n = 1,2,3,...$ , and if you found one definite result value $O N$ , the system will later be found "collapsed" to the state $|Nrang$ . Fig. ... Quite literally, quantum state describes the state of a quantum system. ... 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 linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

The case of a continuous spectrum is more problematic, since the basis has uncountably many eigenvectors. These can be represented by a set of Delta functions. Since the delta function is in fact not a function, and moreover, doesn't belong to the Hilbert space of square-integrable functions, this can cause difficulties such as singularities and infinite values. In all practical cases, the resolution of any given measurement is finite, and therefore the continuous space may be divided into discrete segments. Another solution is to approximate any lab experiments by a "box" potential (which bounds the volume in which the particle can be found, and thus ensures a countable spectrum). In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Spectrum (plural spectra) can refer to: // Common-noun meanings The spectrum of activity of a drug The political spectrum of opinion The bipolar spectrum, in psychology A spectre In the physical sciences An emission spectrum or absorption spectrum observed in light The energy spectrum of a collection of particles The... In mathematics, an uncountable set is a set which is not countable. ... The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... Singularity has several different meanings: mathematical singularity - a point where a mathematical function goes to infinity or is in certain other ways ill-behaved gravitational singularity - an infinity occurring in an astrophysical model, involving infinite curvature (a mathematical singularity) in the space/time continuum technological singularity - a predicted point in... Infinity is a term with very distinct, separate meanings which arise in theology, philosophy, mathematics and everyday life. ... The word resolution has several meanings, depending on context. ... In physics the particle in a box refers to a simple mathematical exercise performed in quantum mechanics. ... In mathematics the term countable set is used to describe the size of a set, e. ...

Wavefunction collapse

Given any quantum state which is a superposition of eigenstates

$| psi rang = c_1 e^{-i E_1 t} | 1 rang + c_2 e^{-i E_2 t} | 2 rang + c_3 e^{-i E_3 t} | 3 rang + cdots ,$

if we measure, for example, the energy of the system and receive E₂

(this result is chosen randomly according to probability given by

$Pr( E_n ) = frac{ | A_n |^2 }{sum_k | A_k |^2}$ ),

then the system's quantum state after the measurement is

$| psi rang = e^{-i E_2 t} | 2 rang$

so any further measurement of energy will always yield E₂.

Figure 1. The process of wavefunction collapse illustrated mathematically. A quantum state is any possible state in which a quantum mechanical system can be. ... The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... A quantum state is any possible state in which a quantum mechanical system can be. ... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ...

The process in which a quantum state becomes one of the eigenstates of the operator corresponding to the measured observable 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. The wavefunction collapse raises serious questions of determinism and locality, as demonstrated in the EPR paradox and later in GHZ entanglement. 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. ... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ... The double-slit experiment consists of letting light diffract through two slits producing fringes on a screen. ... Determinism is the philosophical proposition that every event, including human cognition and action, is causally determined by an unbroken chain of prior occurrences. ... The term locality has different meanings in different disciplines: Geography In geography, a locality is a place. ... In quantum mechanics, the EPR paradox is a thought experiment that demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ... A gigahertz is a billion hertz or a thousand megahertz, a measure of frequency. ... Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ...

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, 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. In quantum mechanics, quantum decoherence is the process by which quantum systems in complex environments exhibit classical behavior. ... Quantum coherence refers to the condition of a quantum system whose constituents are in-phase. ...

There are two major approaches toward the "wavefunction collapse":

Accept it as it is. This approach was supported by Niels Bohr and his Copenhagen interpretation which accepts the collapse as one of the elementary properties of nature (at least, for small enough systems). According to this, there is an inherent randomness embedded in nature, and physical observables exist only after they are measured (for example: as long as a particle's speed isn't measured it doesn't have any defined speed).
Reject it as a physical process and relate to it only as an illusion. This approach says that there is no collapse at all, and we only think there is. Those who support this approach usually offer another interpretation of quantum mechanics, which avoids the wavefunction collapse.

Niels Bohr Niels Henrik David Bohr (October 7, 1885 â€“ November 18, 1962) was a Danish physicist who made essential contributions to understanding atomic structure and quantum mechanics. ... Solvay conference, 1927. ... An interpretation of quantum mechanics is an attempt to answer the question: what exactly is quantum mechanics talking about? Quantum mechanics has been described as the most precisely tested and most successful theory in the history of science (c. ...

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 $|psirang = sum_n c_n |psi_nrang$ , where $|psi_nrang$ are eigenstates of the operator that needs to be measured. In order to make the measurement, the measured system described by $|psirang$ needs to interact with the measuring apparatus described by the quantum state $|phirang$ , so that the total wave function before the interaction is $|psirang |phirang$ . After the interaction, the total wave function exhibits the unitary evolution $|psirang |phirang rightarrow sum_n c_n |psi_nrang |phi_nrang$ , where $|phi_nrang$ are orthonormal states of the measuring apparatus. The unitary evolution above is referred to as premeasurement. One can also introduce the interaction with the environment $|erang$ , so that, after the interaction, the total wave function takes a form $sum_n c_n |psi_nrang |phi_nrang |e_n rang$ , 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 $|psirangle rightarrow |psi_nrang$ on the level of the measured system, but can also be understood as a process $|phirangle rightarrow |phi_nrang$ on the level of the measuring apparatus, or as a process $|erangle rightarrow |e_nrang$ on the level of the environment. Studying these processes provides considerable insight into the measurement problem 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 ${ |psi_nrang}$ , ${ |phi_nrang}$ , or ${ |e_nrang}$ represents a set of states that do not overlap in space, the collapse can be treated with the Bohm interpretation which, in this case, predicts the same probabilities for collapses to various states as does the conventional interpretation. 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. Recently, some modifications of the Bohm interpretation have been proposed, such that an interpretation of relativistic quantum field theory is included. A separate article covers Saint John Neumann, the American priest. ... Quantum decoherence is the general term for the consequences of irreversible quantum entanglement. ... In quantum mechanics, operators correspond to observable variables, eigenvectors are also called eigenstates, and the eigenvalues of an operator represent those values of the corresponding variable that have non-zero probability of occurring. ... 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. ... Quantum decoherence is the general term for the consequences of irreversible quantum entanglement. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ... The measurement problem is the key set of questions that every interpretation of quantum mechanics must answer. ... The Bohm interpretation of quantum mechanics, sometimes called the Causal interpretation, or Ontological interpretation, is an interpretation postulated by David Bohm in which the existence of a non-local universal wavefunction (SchrÃ¶dinger equation) allows distant particles to interact instantaneously. ... The Bohm interpretation of quantum mechanics, sometimes called the Causal interpretation, or Ontological interpretation, is an interpretation postulated by David Bohm in which the existence of a non-local universal wavefunction (SchrÃ¶dinger equation) allows distant particles to interact instantaneously. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... The Bohm interpretation of quantum mechanics, sometimes called the Causal interpretation, or Ontological interpretation, is an interpretation postulated by David Bohm in which the existence of a non-local universal wavefunction (SchrÃ¶dinger equation) allows distant particles to interact instantaneously. ...

Example

(should be reviewed and cleaned up)

Suppose that we have a particle in a box. If the energy of the particle is measured to be $E_N = frac{N^2pi^2hbar^2}{2mL^2}$ then the corresponding state of the system is $|psi_Nrang = |Nrang = sqrt{ frac{2}{L} }~{rm sin}left(frac{N pi x}{L}right)$ , which is determined by solving the Time-Independent Schrödinger equation for the given potential. In physics the particle in a box refers to a simple mathematical exercise performed in quantum mechanics. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ... It has been suggested that this article or section be merged with Scalar potential. ...

Alternatively, if instead of knowing the energy of the particle the particle's position is determined to be a distance $S$ from the left wall of the box, the corresponding system state is $|psi_Srang = |Srang = delta( S - x )$ .

These two state functions $|psi_Nrang$ and $|psi_Srang$ are distinct functions (of the position $x$ ), but they are in general not orthogonal to each other:

$lang psi_S | psi_Nrang = lang S | Nrang = int_0^L ~delta( S - x)~sqrt{ frac{2}{L} }~{rm sin}left(frac{N pi x}{L}right) dx = sqrt{ frac{2}{L} }~{rm sin}left(frac{N pi S}{L}right)$ .

The two systems are therefore distinct; a position measurement is instantaneous whereas a definite value of energy $E N$ is established only in the limit of an infinitely long observation period.

Completeness of eigenvectors of Hermitian operators guarantees that either system state, being the eigenvector to one measurement operator, can be expressed as a linear combination of eigenvectors of the other measurement operator:

$|Srang = sum_n | n rang leftlangle n | S rightrangle = frac{2}{L}~sum_n {rm sin}left(frac{n pi x}{L}right)~{rm sin}left(frac{n pi S}{L}right) = delta( S - x )$ , and

$|Nrang = int ~|srang leftlangle s | N rightrangle ds = int ~delta( s - x )~sqrt{ frac{2}{L} }~{rm sin}left(frac{N pi s}{L}right) ds = sqrt{ frac{2}{L} }~{rm sin}left(frac{N pi x}{L}right)$ .

The time dependence of the system states is determined by the Time Dependent Schrödinger equation. In the preceding example, with energy eigenvalues $E n$ , it follows that the time dependent solution is In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, describes the time-dependence of quantum mechanical systems. ...

$|psi( t )rang = sum_n |nrang lang n|psi_Srang ~e^{-i t E_n/hbar}$ ,

where $t$ represents the time since the particle's location in space was measured. Consequently

$lang n|psi( t )rang = lang n|psi_Srang ~e^{-i t E_n/hbar} = sqrt{ frac{2}{L} }~{rm sin}left(frac{n pi S}{L}right)~e^{-i t E_n/hbar} ~{not =}~ 0$

at least for several distinct energy eigenstates $|nrang$ , for all values $t$ , and for all $0 < S < L$ .

The particle state $|psi_S rang$ therefore can not have evolved (in the above technical sense) into state $|psi_N rang$ (which is orthogonal to all energy eigenstates, except itself), for any duration $t$ . While this conclusion may be characterized accordingly instead as "the wave function of the particle having been projected, or having collapsed into" the energy eigenstate $|psi_Nrang$ , it is perhaps worth emphasizing that any definite value of energy $E N$ can be established only in the limit of a long-lasting trial and never for any finite value of time. 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... The word projection can mean more than one thing. ... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...

Philosophical problems of quantum measurements

What physical interaction constitutes a measurement?

Until the advent of quantum decoherence theory in the late 20th century, a major conceptual problem of quantum mechanics and especially the 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 best illustrated by the Schrödinger's cat paradox. In quantum mechanics, quantum decoherence is the process by which quantum systems in complex environments exhibit classical behavior. ... Fig. ... Solvay conference, 1927. ... In quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued function Ïˆ defined over a portion of space and normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared of the wavefunction |Ïˆ(x)|2 is the... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ... SchrÃ¶dingers Cat: in one hour, there is a 50% chance that the poisonous gas will be released and kill the cat. ...

Major philosophical and metaphysical questions surround this issue: Metaphysical may refer to: Metaphysics, a branch of philosophy dealing with the ultimate nature of reality; or The Metaphysical poets, a poetic school from seventeenth century England who correspond with baroque period in European literature. ...

The concept of weak measurements.
Macroscopic systems (such as chairs or cats) do not exhibit counterintuitive quantum properties, which can only be observed in microscopic particles such as electrons or photons. This invites the question of when a system is "big enough" to behave classically and not quantum mechanically?

Quantum decoherence theory has successfully addressed other questions that previously haunted quantum measurement theory: Macroscopic means measurable and observable by the naked eye; describes existence as we perceive it. ... Look up chair in Wiktionary, the free dictionary. ... Trinomial name Felis silvestris catus (Linnaeus, 1758) This article is about the domestic cat. ... A microscope (Greek: micron = small and scopos = aim) is an instrument for viewing objects that are too small to be seen by the naked or unaided eye. ... Properties The electron is a fundamental subatomic particle which carries a negative electric charge. ... In physics, the photon (from Greek Ï†Ï‰Ï‚ phos, meaning light) is the quantum of the electromagnetic field, for instance light. ... In quantum mechanics, quantum decoherence is the process by which quantum systems in complex environments exhibit classical behavior. ...

Does a measurement depend on the existence of a self-aware observer?

Answer: No. Coupling an isolated quantum system to another quantum system with many degrees of freedom generically transfers the coherence of the first system into mutual coherence of the two systems. The initially isolated quantum system then appears to "collapse." Interpreting the second system as a measurement apparatus, as in the von Neumann scheme, shows that no consciousness or self-awareness is necessary for collapse of the first system. Self-awareness is the ability to perceive ones own existence, including ones own traits, feelings and behaviours. ... Possible meanings: In general, an observer is any system which receives information from an object. ...

What interactions are strong enough to constitute a measurement?

This question is quantitatively answered by decoherence theory, given a model for the measurement apparatus. The scaling of the measurement effects with the system/apparatus interaction strength usually only weakly depends on the choice of a model for the apparatus, so one can give a generic description of the strength of a measurement induced by a given interaction.

Does measurement actually determine the state?

The question of whether a measurement actually determines the state, is deeply related to the Wavefunction collapse. In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ...

Most versions of the Copenhagen interpretation answer this question with an unqualified "yes". Solvay conference, 1927. ...

The quantum entanglement problem

See EPR paradox. In quantum mechanics, the EPR paradox is a thought experiment that demonstrates that the result of a measurement performed on one part of a quantum system can have an instantaneous effect on the result of a measurement performed on another part, regardless of the distance separating the two parts. ...

External links

Analog: A Farewell to Copenhagen?
"The Double Slit Experiment". (physicsweb.org)
Shahriar S. Afshar, "Waving Copenhagen Good-bye: Were the founders of Quantum Mechanics wrong?" This link presents a view that is not endorsed by the majority of physicists.
"Variation on the similar two-pin-hole "which-way" experiment". (reported in New Scientist; July 24), Reprint at irims.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

Categories: Quantum measurement | Quantum mechanics New Scientist cover - 18 December 2004 New Scientist is a weekly international science magazine covering recent developments in science and technology for a general English-speaking audience. ...

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