Measurement in quantum mechanics
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{{Quantum mechanics|cTopic=Fundamental concepts}}
The framework of [[quantum mechanics]] requires a careful definition of '''measurement''', and a thorough discussion of its practical and philosophical implications.
==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]], 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]] 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, the result of the measurement will often be different.
In fact, the expected result of the measurement is in general described not by a single number, but by a [[probability distribution]] that specifies the likelihoods that the various possible results will be obtained. (This distribution can be either [[discrete probability distribution|discrete]] or [[continuous random variable|continuous]], depending on what is being measured.) The measurement process is often said to be [[stochastic process|random]] and [[indeterminism|indeterministic]]. (However, there is considerable dispute over this issue; in some other [[interpretations of quantum mechanics|interpretations]], the result merely ''appears'' random and indeterministic, as elaborated below.)
Another important aspect of measurement is [[wavefunction collapse]]. After one has measured some aspect of the quantum state, the quantum state immediately ''changes'' in such a way that if the measurement is repeated without re-preparing the state, one finds the same result as the first measurement. (The detailed nature of this collapse is another source of disagreement among interpretations of quantum mechanics.)
===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]].
====Measurable quantities ("observables") as operators====
{{main|Observable}}
It is a postulate of quantum mechanics that all measurements have an associated [[linear operator|operator]] (called an '''observable operator''', or just an '''observable'''), with the following properties:
#The observable is a [[Hermitian operator|Hermitian]] ([[self-adjoint operator|self-adjoint]]) [[linear operator|operator]] mapping a [[Hilbert space]] (namely, the [[state space (physics)|state space]], which consists of all possible [[quantum state]]s) into itself.
#The observable's [[eigenvalue]]s are [[real number|real]]. The possible outcomes of the measurement are precisely the eigenvalues of the given observable.
#For each eigenvalue there are one or more corresponding [[eigenvector]]s (which in this context are called [[eigenstate]]s), which will make up the state of the system after the measurement.
#The observable has a set of eigenvectors which [[linear span|span]] the state space. It follows that each observable generates an [[orthonormal]] [[basis (linear algebra)|basis]] of [[eigenvector]]s (called an [[eigenbasis]]). Physically, this is the statement that any quantum state can always be represented as a [[quantum superposition|superposition]] of the eigenstates of an observable.
Important examples of observables are:
* The [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator, representing the total [[energy]] of the system; with the special case of the [[Hamiltonian (quantum mechanics)|nonrelativistic Hamiltonian]] operator: <math> {\hat H} = {\hat p^2 \over 2m} + V( \hat x ) </math>.
* The [[momentum]] operator: <math> {\hat p} = -i\hbar {\partial \over \partial x} </math> (in the position basis).
* The [[position operator]]: <math> {\hat x} </math>, where <math> {\hat x} = i\hbar {\partial \over \partial p} </math> (in the momentum basis).
Operators can be [[commutator|noncommuting]]. 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]], 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 of an operator|spectrum]] (i.e., set of eigenvalues) of the observable.
===== Discrete, nondegenerate spectrum =====
Let <math> {\hat O} </math> be an observable, and suppose that it has discrete [[eigenstate]]s <math>|n> </math> (in [[bra-ket notation]]) for <math> n = 1, 2, 3, ... </math> and corresponding eigenvalues <math>O_1, O_2, O_3, ...</math>, no two of which are equal.
Assume the system is prepared in [[quantum state|state]] <math>|\psi\rang</math>. Since the eigenstates of an observable form a [[basis (linear algebra)|basis]] (the eigenbasis), it follows that <math>|\psi\rang</math> can be written in terms of the eigenstates as
:<math>|\psi\rang = c_1 | 1 \rang + c_2 | 2 \rang + c_3 | 3 \rang + \cdots</math>
(where <math>c_1,c_2,...</math> are complex numbers). Then measuring <math> {\hat O} </math> can yield any of the results <math>O_1, O_2, O_3, ...</math>, with corresponding probabilities given by
:<math> \Pr( O_n ) = \frac{ | c_n |^2 }{\sum_k | c_k |^2} = \frac{ |\lang n | \psi \rang|^2}{\lang \psi | \psi\rang} </math>
Usually <math>|\psi\rang</math> is assumed to be [[Normalisable wave function|normalized]], in which case this expression reduces to
:<math> \Pr( O_n ) = | c_n |^2 = |\lang n | \psi \rang|^2 </math>
If the result of the measurement is <math>O_n</math>, then the system's [[quantum state]] after the measurement is
:<math> | \psi' \rang = | n \rang </math>
so any repeated measurement of <math>{\hat O}</math> will yield the same result <math>O_n</math>. (This phenomenon is called [[wavefunction collapse]].)
===== Continuous, nondegenerate spectrum =====
Let <math> {\hat O} </math> be an observable, and suppose that it has a [[continuous spectrum]] of eigenvalues filling the [[interval (mathematics)|interval]] (a,b). Assume further that each eigenvalue ''x'' in this range is associated with a unique eigenstate <math>|x\rang</math>.
Assume the system is prepared in [[quantum state|state]] <math>|\psi\rang</math>, which can be written in terms of the eigenbasis as
:<math>|\psi\rang = \int_a^b c(x) | x \rang dx</math>
(where <math>c(x)</math> is a complex-valued function). Then measuring <math> {\hat O} </math> can yield a result anywhere in the interval (a,b), with [[probability density function]] <math>|c(x)|^2</math>; i.e., a result between ''y'' and ''z'' will occur with probability
:<math> \Pr( y<x<z ) = \frac{ \int_y^z | c(x) |^2 dx }{\int_a^b | c(x) |^2 dx} </math>
Again, <math>|\psi\rang</math> is often assumed to be [[Normalisable wave function|normalized]], in which case this expression reduces to
:<math> \Pr( y<x<z ) = \int_y^z | c(x) |^2 dx </math>
If the result of the measurement is ''x'', then the new wave function will be
:<math> |\psi'\rang = |x\rang.</math>
Alternatively, it is often possible and convenient to analyze a continuous-spectrum measurement by taking it to be the [[limit (mathematics)|limit]] of a different measurement with a discrete spectrum. For example, an analysis of [[scattering]] involves a continuous spectrum of energies, but by adding a [[particle in a box|"box" potential]] (which bounds the volume in which the particle can be found), the spectrum becomes [[discrete spectrum|discrete]]. 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 [[eigenspace|eigen''spaces'']]. The probability of measuring a particular eigenvalue is the squared component of the [[quantum state|state vector]] in the corresponding eigenspace, and the new state after measurement is the [[projection (linear algebra)|projection]] of the original state vector into the appropriate eigenspace.
===== Density matrix formulation =====
{{main|Density matrix}}
Instead of performing quantum-mechanics computations in terms of [[wavefunction]]s ([[bra-ket notation|kets]]), it is sometimes necessary to describe a quantum-mechanical system in terms of a [[density matrix]]. 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 <math> {\hat O} </math> be an observable, and suppose that it has discrete [[eigenvalue]]s <math>O_1,O_2,O_3,...</math>, associated with [[eigenspace]]s <math>V_1,V_2,...</math> respectively. Let <math>P_n</math> be the [[projection (linear algebra)|projection operator]] into the space <math>V_n</math>.
Assume the system is prepared in the [[quantum state|state]] described by the density matrix ''ρ''. Then measuring <math> {\hat O} </math> can yield any of the results <math>O_1, O_2, O_3, ...</math>, with corresponding probabilities given by
:<math> \Pr( O_n ) = \mathrm{Tr}(P_n \rho)</math>
where Tr denotes [[trace (linear algebra)|trace]]. If the result of the measurement is ''n'', then the new density matrix will be
:<math> \rho' = \frac{P_n \rho P_n}{\mathrm{Tr}(P_n \rho)}</math>
Alternatively, one can say that the measurement process results in the new density matrix
:<math> \rho'' = \sum_n P_n \rho P_n</math>
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 <math>\hat O</math>, on a state whose quantum state is <math>|\psi\rang</math>.
*The [[expected value|mean]] (average) value of the measurement is (see [[Expectation value (quantum mechanics)]])
:<math>\lang \psi | \hat O | \psi \rang </math>.
*The [[variance]] of the measurement is
:<math>\lang \psi | \hat O^2 | \psi \rang - (\lang \psi | \hat O | \psi \rang)^2</math>
*The [[standard deviation]] of the measurement is
:<math>\sqrt{\lang \psi | \hat O^2 | \psi \rang - (\lang \psi | \hat O | \psi \rang)^2}</math>
These are direct consequences of the above formulas for measurement probabilities.
==== Example ====
Suppose that we have a [[particle in a box|particle in a 1-dimensional box]], set up initially in the ground state <math>|\psi_1\rang</math>. As can be computed from the [[Schrödinger equation|time-independent Schrödinger equation]], the energy of this state is <math>E_1=\frac{\pi^2\hbar^2}{2mL^2}</math> (where ''m'' is the particle's mass and ''L'' is the box length), and the spatial wavefunction is <math>\lang x|\psi_1\rang = \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{\pi x}{L}\right)</math>. If the energy is now measured, the result will always be <math>E_1</math>, and this measurement will not affect the wavefunction.
Next suppose that the particle's position is measured. The position ''S'' will be measured with probability density
:<math> \Pr(S<x<S+dS) = \frac{2}{L}~{\rm sin}^2\left(\frac{\pi S}{L}\right)dS.</math>
If the measurement result was ''S'', then the wavefunction after measurement will be the position eigenstate <math>|S\rang</math>. If the particle's position is immediately measured again, the same result will be obtained.
The new wavefunction <math>|S\rang</math> can, like any wavefunction, be written as a superposition of eigenstates of any observable. In particular, using energy eigenstates, we have
:<math>|S\rang = \sum_n | \psi_n \rangle \left\langle \psi_n | S \right\rangle = \sum_n | \psi_n \rangle \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{n \pi S}{L}\right)</math>
If we now leave this state alone, it will smoothly evolve in time according to the [[Schrödinger equation]]. But suppose instead that an energy measurement is immediately taken. Then the possible energy values <math>E_n</math> will be measured with probabilities:
:<math>\Pr(E_n) = |\lang \psi_n | S \rang|^2 = \frac{2}{L}~{\rm sin}^2\left(\frac{n \pi S}{L}\right)</math>
and moreover if the measurement result is <math>E_n</math>, then the new state will be the energy eigenstate <math>|\psi_n\rang</math>.
So in this example, due to the process of wavefunction collapse, a particle initially in the ground state can end up in any energy level after two subsequent measurements are made.
== Wavefunction collapse ==
{{main|Wave function collapse}}
The process in which a quantum state becomes one of the eigenstates of the operator corresponding to the measured [[observable]] is called "collapse", or "[[Wave function collapse|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 [[Principle of locality|locality]], as demonstrated in the [[EPR paradox]] and later in [[Greenberger-Horne-Zeilinger state|GHZ entanglement]]. (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]], 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
<math> |\psi\rang = \sum_n c_n |\psi_n\rang </math>,
where <math> |\psi_n\rang </math> are [[eigenstates]] of the operator that needs to be measured. In order to make the measurement, the measured system described by
<math> |\psi\rang </math> needs to interact with the measuring apparatus described by the quantum state <math> |\phi\rang </math>, so that the total wave function before the interaction is <math> |\psi\rang |\phi\rang </math>. After the interaction, the total wave function exhibits the [[unitary operator|unitary]] evolution
:<math> |\psi\rang |\phi\rang \rightarrow \sum_n c_n |\psi_n\rang |\phi_n\rang </math>,
where <math> |\phi_n\rang</math> 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 <math> |e\rang </math>, so that, after the interaction, the total wave function takes a form
:<math> \sum_n c_n |\psi_n\rang |\phi_n\rang |e_n \rang</math>,
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 [[Wave function collapse|wavefunction collapse]] does not need to be understood as a process <math> |\psi\rangle \rightarrow |\psi_n\rang </math> on the level of the measured system, but can also be understood as a process
<math> |\phi\rangle \rightarrow |\phi_n\rang </math> on the level of the measuring apparatus, or as a process <math> |e\rangle \rightarrow |e_n\rang </math> 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
;<math> \{ |\psi_n\rang\} </math>, <math> \{ |\phi_n\rang\} </math>, or <math> \{ |e_n\rang\} </math>
represents a set of states that do not overlap in space, the appearance of collapse can be generated by either the [[Bohm interpretation]] or the [[many worlds interpretation|Everett interpretation]] which both deny the reality of [[Wave function collapse|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]]. 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]].
== 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 [[Wave function collapse|collapse]]. This is best illustrated by the [[Schrödinger's cat]] paradox. Certain aspects of this question are now well understood in the framework of quantum decoherence theory, such as an understanding of [[weak measurement]]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 [[Wave function collapse|wavefunction collapse]].) For example, in most versions of the [[Copenhagen interpretation]], the measurement determines the state, and after measurement the state is definitely what was measured. But according to the [[Many-worlds interpretation]], 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 process|random]] or [[determinism|deterministic]]? ===
{{main|Hidden variable theory}}
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<ref>[http://arxiv.org/abs/quant-ph/0609163 Quantum mechanics: Myths and facts]</ref>
whether this is true, fundamental randomness, or merely "emergent" randomness resulting from underlying ''hidden variables'' which deterministically cause measurement results to happen a certain way each time. This continues to be an area of active research.<ref>S. Gröblacher ''et al.'', An experimental test of non-local realism, Nature '''446''', 871 (2007).</ref>
=== Does the measurement process [[Nonlocality|violate locality]]? ===
{{main|Nonlocality|Principle of locality}}
In physics, the '''Principle of locality''' is the concept that information cannot travel faster than the [[speed of light]] (also see [[special relativity]]). It is known experimentally (see [[Bell's theorem]], which is related to the [[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.<ref>[http://arxiv.org/abs/quant-ph/0609163 Quantum mechanics: Myths and facts]</ref>
== See also ==
* Measurement related problems and [[paradox]]es
** [[Measurement problem]]
** [[Wave function collapse|Wavefunction collapse]]
** [[EPR paradox]]
** [[Renninger negative-result experiment]]
** [[Elitzur-Vaidman bomb-testing problem]]
** [[Schrödinger's cat]]
** [[Popper's experiment]]
* [[Interpretation of quantum mechanics|Interpretations of quantum mechanics]]
** [[Transactional interpretation]]
** [[Copenhagen interpretation]]
** [[Many-worlds interpretation]]
** [[The Pondicherry interpretation of quantum mechanics]]
* Quantum mechanics formalism
** [[Quantum mechanics]]
** [[Mathematical formulation of quantum mechanics]]
** [[Schrödinger equation]]
** [[Bra-ket notation]]
** [[POVM|Generalized measurement]] (POVM, Positive operator valued measure)
== External links ==
* [http://www.analogsf.com/0410/altview2.shtml Analog: A Farewell to Copenhagen?]
* "[http://physicsweb.org/article/world/15/9/1 The Double Slit Experiment]". (physicsweb.org)
* Shahriar S. Afshar, "''[http://my.harvard.edu/cgi-bin/webevent/webevent.cgi?cmd=showevent&ncmd=calmonth&cal=9719&y=2004&m=3&d=23&id=10416384&token=G6409379:1&sb=0&cf=cal&lc=calmonth&swe=1&set=0&sa=0&sort=e,m,t&ws=0&sib=0&de=0&tf=0 Waving Copenhagen Good-bye: Were the founders of Quantum Mechanics wrong?]''"
"''[http://www.sciencefriday.com/images/shows/2004/073004/AfsharExperimentSmall.jpg Variation on the similar two-pin-hole "which-way" experiment]''". (reported in [[New Scientist]]; July 24), [http://www.irims.org/quant-ph/030503/ Reprint at irims.org]
* "[http://plato.stanford.edu/entries/qt-measurement/ Measurement in Quantum Mechanics]" Henry Krips in the Stanford Encyclopedia of Philosophy
* [http://arxiv.org/abs/quant-ph/0312059 Decoherence, the measurement problem, and interpretations of quantum mechanics]
* [http://arxiv.org/abs/quant-ph/0505070 Measurements and Decoherence]
* [http://thisquantumworld.com/ht/index.php This Quantum World] What is quantum mechanics trying to tell us about the nature of Nature?
* Yonina C. Eldar, Alexandre Megretski, and George C. Verghese. Designing optimal quantum detectors via semidefinite programming. <cite>IEEE Transactions on Information Theory</cite>, Vol. '''49''', No. 4, 1007--1012, 2003.
==Further reading==
* [[John A. Wheeler]] and [[Wojciech Hubert Zurek]] (eds), ''Quantum Theory and Measurement'', [[Princeton University Press]], (1983), ISBN 0-691-08316-9
* [[Vladimir B. Braginsky]] and [[Farid Ya. Khalili]], ''Quantum Measurement'', [[Camebridge University Press]], (1992), ISBN 0-521-41928-X
* [[Greenstein, G.]] and [[Zajonc, A.G.]], ''The Quantum Challenge'', [[Jones and Bartlett Publishers]], (2006), ISBN 0-7367-2470-X
==References==
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