Measurement in quantum mechanics

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

Contents 1 Formalism of measurement 1.1 Measurable quantities ("observables") as operators 1.2 Eigenstates and projection 1.3 Wavefunction collapse 1.4 von Neumann measurement scheme 1.5 Example 2 Optimal quantum measurement 3 Philosophical problems of quantum measurements 3.1 What physical interaction constitutes a measurement? 3.2 Does measurement actually determine the state? 3.3 The quantum entanglement problem 4 See also 5 External links 6 Further reading

[edit] Formalism of measurement

[edit] Measurable quantities ("observables") as operators

An observable quantity is represented mathematically by an Hermitian or self adjoint operator. The set of the operator's eigenvalues represent the set of possible outcomes of the measurement. For each eigenvalue there is a corresponding eigenstate (or "eigenvector"), which will be the state of the system after the measurement. Some properties of this representation are

1. The eigenvalues of Hermitian matrices are real. The possible outcomes of a measurement are precisely the eigenvalues of the given observable.
2. A Hermitian matrix can be unitarily diagonalized (See Spectral theorem), generating an orthonormal basis of eigenvectors which spans the state space of the system. In general, the state of a system can be represented as a linear combination of eigenvectors of any Hermitian operator. Physically, this is to say that any state can be expressed as a superposition of the eigenstates of an observable.

Important examples are:

• The Hamiltonian operator, representing the total energy of the system; with the special case of the nonrelativistic Hamiltonian operator: .
• The momentum operator: (in the position basis).
• The position operator: , where (in the momentum basis).

Operators can be noncommuting. In the finite dimensional case, two Hermitian operators commute if they have the same set of {normalized} eigenvectors. Noncommuting observables are said to be incompatible and can not be measured simultaneously. This can be seen via the uncertainty principle.

[edit] Eigenstates and projection

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. Let be a measurement operator, an observable, with eigenstates WikiTeX: latex reported a failure, namely:

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for and corresponding eigenvalues . If the measurement outcome is , the system will then "collapse" to the state WikiTeX: latex reported a failure, namely:

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after measurement.

The case of a continuous spectrum is more involved, since, physically speaking, the basis has uncountably many eigenstates, but the general concept is the same. In the position representation, for instance, the eigenstates can be represented by the set of delta functions, indexed by all possible positions of the particle. In the experimental setting, 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 discrete spectrum).

[edit] Wavefunction collapse

Given any quantum state which is a superposition of eigenstates at time t

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

(this result is chosen randomly according to probability given by

),

then the system's quantum state after the measurement is

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so any repeated measurement of energy will yield E₂.

Figure 1. The process of wavefunction collapse illustrated.

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 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.

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

1. 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).
2. 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.

[edit] 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

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are eigenstates of the operator that needs to be measured. In order to make the measurement, the measured system described by WikiTeX: latex reported a failure, namely:

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needs to interact with the measuring apparatus described by the quantum state WikiTeX: latex reported a failure, namely:

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(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./1903ce74af8b58825d8dbafa3c01f.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*} |\phi\rang

                                \end{equation*}

[1] (./1903ce74af8b58825d8dbafa3c01f.aux) )
(see the transcript file for additional information)
Output written on 1903ce74af8b58825d8dbafa3c01f.dvi (1 page, 260 bytes).

Transcript written on 1903ce74af8b58825d8dbafa3c01f.log.

, so that the total wave function before the interaction is WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./3445d6fcb5bfe7804a3ea4882e804
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
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(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./3445d6fcb5bfe7804a3ea4882e804.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*} |\psi\rang

                                |\phi\rang \end{equation*}

! Undefined control sequence.
l.5 \begin{equation*} |\psi\rang |\phi\rang

                                           \end{equation*}

[1] (./3445d6fcb5bfe7804a3ea4882e804.aux) )
(see the transcript file for additional information)
Output written on 3445d6fcb5bfe7804a3ea4882e804.dvi (1 page, 268 bytes).

Transcript written on 3445d6fcb5bfe7804a3ea4882e804.log.

. After the interaction, the total wave function exhibits the unitary evolution WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./e461db6fba50ba332e7206d9859d2
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./e461db6fba50ba332e7206d9859d2.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*} |\psi\rang
                                 |\phi\rang \rightarrow \sum_n c_n |\psi_n\r...

! Undefined control sequence.
l.5 \begin{equation*} |\psi\rang |\phi\rang
                                            \rightarrow \sum_n c_n |\psi_n\r...

! Undefined control sequence.
l.5 ...hi\rang \rightarrow \sum_n c_n |\psi_n\rang
                                                   |\phi_n\rang \end{equation*}

! Undefined control sequence.
l.5 ...tarrow \sum_n c_n |\psi_n\rang |\phi_n\rang
                                                   \end{equation*}
[1] (./e461db6fba50ba332e7206d9859d2.aux) )
(see the transcript file for additional information)
Output written on e461db6fba50ba332e7206d9859d2.dvi (1 page, 432 bytes).
Transcript written on e461db6fba50ba332e7206d9859d2.log.

, where WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./7aa91eb457be68dc79f496e4ddd32
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./7aa91eb457be68dc79f496e4ddd32.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*} |\phi_n\rang
                                  \end{equation*}
[1] (./7aa91eb457be68dc79f496e4ddd32.aux) )
(see the transcript file for additional information)
Output written on 7aa91eb457be68dc79f496e4ddd32.dvi (1 page, 312 bytes).
Transcript written on 7aa91eb457be68dc79f496e4ddd32.log.

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 WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./8caaad7d59f4cdc1fbca4f3453ed2
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./8caaad7d59f4cdc1fbca4f3453ed2.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*} |e\rang

                             \end{equation*}

[1] (./8caaad7d59f4cdc1fbca4f3453ed2.aux) )
(see the transcript file for additional information)
Output written on 8caaad7d59f4cdc1fbca4f3453ed2.dvi (1 page, 260 bytes).

Transcript written on 8caaad7d59f4cdc1fbca4f3453ed2.log.

, so that, after the interaction, the total wave function takes a form WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./35d0e560c5321f81dfd2ab5a1960b
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./35d0e560c5321f81dfd2ab5a1960b.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*} \sum_n c_n |\psi_n\rang
                                              |\phi_n\rang |e_n \rang\end{eq...

! Undefined control sequence.
l.5 ...tion*} \sum_n c_n |\psi_n\rang |\phi_n\rang
                                                   |e_n \rang\end{equation*}
! Undefined control sequence.
l.5 ..._n c_n |\psi_n\rang |\phi_n\rang |e_n \rang
                                                  \end{equation*}
[1] (./35d0e560c5321f81dfd2ab5a1960b.aux) )
(see the transcript file for additional information)
Output written on 35d0e560c5321f81dfd2ab5a1960b.dvi (1 page, 428 bytes).
Transcript written on 35d0e560c5321f81dfd2ab5a1960b.log.

, 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 WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./a361a8ac0dcf493ac9dce22545919
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./a361a8ac0dcf493ac9dce22545919.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 ...ion*} |\psi\rangle \rightarrow |\psi_n\rang
                                                   \end{equation*}
[1] (./a361a8ac0dcf493ac9dce22545919.aux) )
(see the transcript file for additional information)
Output written on a361a8ac0dcf493ac9dce22545919.dvi (1 page, 324 bytes).
Transcript written on a361a8ac0dcf493ac9dce22545919.log.

on the level of the measured system, but can also be understood as a process WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./6b65334076382efe917da38e51755
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./6b65334076382efe917da38e51755.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 ...ion*} |\phi\rangle \rightarrow |\phi_n\rang

                                                  \end{equation*}

[1] (./6b65334076382efe917da38e51755.aux) )
(see the transcript file for additional information)
Output written on 6b65334076382efe917da38e51755.dvi (1 page, 324 bytes).

Transcript written on 6b65334076382efe917da38e51755.log.

on the level of the measuring apparatus, or as a process WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./91716e253a1555dfdc8490a683708
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./91716e253a1555dfdc8490a683708.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 ...{equation*} |e\rangle \rightarrow |e_n\rang

                                                  \end{equation*}

[1] (./91716e253a1555dfdc8490a683708.aux) )
(see the transcript file for additional information)
Output written on 91716e253a1555dfdc8490a683708.dvi (1 page, 324 bytes).

Transcript written on 91716e253a1555dfdc8490a683708.log.

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 WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./20a2f91dc1305504025eb17c8ec7e
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./20a2f91dc1305504025eb17c8ec7e.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*} \{ |\psi_n\rang

                                    \} \end{equation*}

[1] (./20a2f91dc1305504025eb17c8ec7e.aux) )
(see the transcript file for additional information)
Output written on 20a2f91dc1305504025eb17c8ec7e.dvi (1 page, 320 bytes).

Transcript written on 20a2f91dc1305504025eb17c8ec7e.log.

, WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./60b5eec13c64f79486e467c75aa45
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
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h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
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(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
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(./60b5eec13c64f79486e467c75aa45.aux)
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(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*} \{ |\phi_n\rang

                                    \} \end{equation*}

[1] (./60b5eec13c64f79486e467c75aa45.aux) )
(see the transcript file for additional information)
Output written on 60b5eec13c64f79486e467c75aa45.dvi (1 page, 320 bytes).

Transcript written on 60b5eec13c64f79486e467c75aa45.log.

, or WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./8fd02e506ae7042fb45302e9f98cc
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
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h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
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(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
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(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
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(./8fd02e506ae7042fb45302e9f98cc.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
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(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*} \{ |e_n\rang

                                 \} \end{equation*}

[1] (./8fd02e506ae7042fb45302e9f98cc.aux) )
(see the transcript file for additional information)
Output written on 8fd02e506ae7042fb45302e9f98cc.dvi (1 page, 320 bytes).

Transcript written on 8fd02e506ae7042fb45302e9f98cc.log.

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 Everett interpretation which both deny the reality of wavefunction collapse; they both, though, predict 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. 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.

[edit] Example

Suppose that we have a particle in a box. If the energy of the particle is measured to be then the corresponding state of the system is WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./dc2edb0dafef78a96a42a63a7a6d9
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
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h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
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(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./dc2edb0dafef78a96a42a63a7a6d9.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*}|\psi_N\rang
                                  = \int | x\rang \lang x|\psi_N\rang dx\end...

! Undefined control sequence.
l.5 \begin{equation*}|\psi_N\rang = \int | x\rang
                                                  \lang x|\psi_N\rang dx\end...

! Undefined control sequence.
l.5 ...quation*}|\psi_N\rang = \int | x\rang \lang
                                                   x|\psi_N\rang dx\end{equa...

! Undefined control sequence.
l.5 ..._N\rang = \int | x\rang \lang x|\psi_N\rang
                                                   dx\end{equation*}
[1] (./dc2edb0dafef78a96a42a63a7a6d9.aux) )
(see the transcript file for additional information)
Output written on dc2edb0dafef78a96a42a63a7a6d9.dvi (1 page, 444 bytes).
Transcript written on dc2edb0dafef78a96a42a63a7a6d9.log.

where WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./0f9d3232df6772f3d9f5b7d3331ce
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./0f9d3232df6772f3d9f5b7d3331ce.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
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(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*}\lang

                          x|\psi_N\rang = \lang x|N\rang = \sqrt{ \frac{2}{...

! Undefined control sequence.
l.5 \begin{equation*}\lang x|\psi_N\rang

                                        = \lang x|N\rang = \sqrt{ \frac{2}{...

! Undefined control sequence.
l.5 \begin{equation*}\lang x|\psi_N\rang = \lang

                                                x|N\rang = \sqrt{ \frac{2}{...

! Undefined control sequence.
l.5 ...ation*}\lang x|\psi_N\rang = \lang x|N\rang

                                                  = \sqrt{ \frac{2}{L} }~{\...

[1] (./0f9d3232df6772f3d9f5b7d3331ce.aux) )
(see the transcript file for additional information)
Output written on 0f9d3232df6772f3d9f5b7d3331ce.dvi (1 page, 588 bytes).

Transcript written on 0f9d3232df6772f3d9f5b7d3331ce.log.

, which is determined by solving the Time-Independent Schrödinger equation for the given potential. Alternatively, if instead of knowing the energy of the particle the particle's position is determined to be a distance from the left wall of the box, the corresponding system state is WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./899ff93cb75dd22f3882bb254d6d1
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
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(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
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(./899ff93cb75dd22f3882bb254d6d1.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
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(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*}|\psi_S\rang
                                  = \int | x\rang \lang x|\psi_S\rang dx\end...

! Undefined control sequence.
l.5 \begin{equation*}|\psi_S\rang = \int | x\rang
                                                  \lang x|\psi_S\rang dx\end...

! Undefined control sequence.
l.5 ...quation*}|\psi_S\rang = \int | x\rang \lang
                                                   x|\psi_S\rang dx\end{equa...

! Undefined control sequence.
l.5 ..._S\rang = \int | x\rang \lang x|\psi_S\rang
                                                   dx\end{equation*}
[1] (./899ff93cb75dd22f3882bb254d6d1.aux) )
(see the transcript file for additional information)
Output written on 899ff93cb75dd22f3882bb254d6d1.dvi (1 page, 444 bytes).
Transcript written on 899ff93cb75dd22f3882bb254d6d1.log.

where WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./b31d85bb55993ecac8aea82bb580b
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
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h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
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(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./b31d85bb55993ecac8aea82bb580b.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*}\lang

                          x|\psi_S\rang = \lang x|S\rang = \delta( S - x ) ...

! Undefined control sequence.
l.5 \begin{equation*}\lang x|\psi_S\rang

                                        = \lang x|S\rang = \delta( S - x ) ...

! Undefined control sequence.
l.5 \begin{equation*}\lang x|\psi_S\rang = \lang

                                                x|S\rang = \delta( S - x ) ...

! Undefined control sequence.
l.5 ...ation*}\lang x|\psi_S\rang = \lang x|S\rang

                                                  = \delta( S - x ) \end{eq...

[1] (./b31d85bb55993ecac8aea82bb580b.aux) )
(see the transcript file for additional information)
Output written on b31d85bb55993ecac8aea82bb580b.dvi (1 page, 400 bytes).

Transcript written on b31d85bb55993ecac8aea82bb580b.log.

. These two state functions WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./f3c5b66bd18474c99d31f0b7ce224
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
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h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./f3c5b66bd18474c99d31f0b7ce224.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*}|\psi_N\rang
                                 \end{equation*}
[1] (./f3c5b66bd18474c99d31f0b7ce224.aux) )
(see the transcript file for additional information)
Output written on f3c5b66bd18474c99d31f0b7ce224.dvi (1 page, 312 bytes).
Transcript written on f3c5b66bd18474c99d31f0b7ce224.log.

and WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./aa7d12c0edf2e6a4bf65bc7504a38
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
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h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
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(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./aa7d12c0edf2e6a4bf65bc7504a38.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*}|\psi_S\rang

                                \end{equation*}

[1] (./aa7d12c0edf2e6a4bf65bc7504a38.aux) )
(see the transcript file for additional information)
Output written on aa7d12c0edf2e6a4bf65bc7504a38.dvi (1 page, 312 bytes).

Transcript written on aa7d12c0edf2e6a4bf65bc7504a38.log.

are distinct functions (of the position after we left multiply by the bra state WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./52df332f9b09b35a11e21d1c83605
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
ahasa, basque, bulgarian, catalan, croatian, czech, danish, dutch, esperanto, e
stonian, finnish, greek, icelandic, irish, italian, latin, magyar, norsk, polis
h, portuges, romanian, russian, serbian, slovak, slovene, spanish, swedish, tur
kish, ukrainian, nohyphenation, loaded.
(/usr/share/texmf/tex/latex/base/article.cls
Document Class: article 2004/02/16 v1.4f Standard LaTeX document class
(/usr/share/texmf/tex/latex/base/size10.clo))
(/usr/share/texmf/tex/latex/amsfonts/amssymb.sty
(/usr/share/texmf/tex/latex/amsfonts/amsfonts.sty))
(/usr/share/texmf/tex/latex/amsmath/amsmath.sty
For additional information on amsmath, use the `?' option.
(/usr/share/texmf/tex/latex/amsmath/amstext.sty
(/usr/share/texmf/tex/latex/amsmath/amsgen.sty))
(/usr/share/texmf/tex/latex/amsmath/amsbsy.sty)
(/usr/share/texmf/tex/latex/amsmath/amsopn.sty))
(/usr/share/texmf/tex/latex/amsmath/amscd.sty)
(/usr/share/texmf/tex/latex/concmath/concmath.sty)
(./52df332f9b09b35a11e21d1c83605.aux)
(/usr/share/texmf/tex/latex/concmath/ot1ccr.fd)
(/usr/share/texmf/tex/latex/concmath/omlccm.fd)
(/usr/share/texmf/tex/latex/concmath/omsccsy.fd)
(/usr/share/texmf/tex/latex/concmath/omxccex.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsa.fd)
(/usr/share/texmf/tex/latex/amsfonts/umsb.fd)
! Undefined control sequence.
l.5 \begin{equation*}\lang

                          x|\end{equation*}

[1] (./52df332f9b09b35a11e21d1c83605.aux) )
(see the transcript file for additional information)
Output written on 52df332f9b09b35a11e21d1c83605.dvi (1 page, 260 bytes).

Transcript written on 52df332f9b09b35a11e21d1c83605.log.

), but they are in general not orthogonal to each other: WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
entering extended mode
(./3f0fd776fe0dedeedcebc375d7633
LaTeX2e <2003/12/01>
Babel  and hyphenation patterns for american, french, german, ngerman, b
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! Undefined control sequence.
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! Undefined control sequence.
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! Undefined control sequence.
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! Undefined control sequence.
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! Undefined control sequence.
l.5 ...\psi_N\rang = \lang S | N\rang = \int \lang
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! Undefined control sequence.
l.5 ...= \lang S | N\rang = \int \lang S | x \rang
                                                   \lang x | N\rang dx =\int...

! Undefined control sequence.
l.5 ...g S | N\rang = \int \lang S | x \rang \lang
                                                   x | N\rang dx =\int_0^L ~...

! Undefined control sequence.
l.5 ...g = \int \lang S | x \rang \lang x | N\rang
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Overfull \hbox (26.32954pt too wide) detected at line 5
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ML/ccm/m/it/10 S\OMS/ccsy/m/n/10 j\OML/ccm/m/it/10 N \OT1/ccr/m/n/10 = [] \OML/
ccm/m/it/10 S\OMS/ccsy/m/n/10 j\OML/ccm/m/it/10 xx\OMS/ccsy/m/n/10 j\OML/ccm/m/
it/10 Ndx \OT1/ccr/m/n/10 = [][]  []\OML/ccm/m/it/10 ^^N\OT1/ccr/m/n/10 (\OML/c
cm/m/it/10 S \OMS/ccsy/m/n/10 ^^@ \OML/ccm/m/it/10 x\OT1/ccr/m/n/10 ) [][] [][]
 [] \OML/ccm/m/it/10 dx \OT1/ccr/m/n/10 = [] [][] []
[1] (./3f0fd776fe0dedeedcebc375d7633.aux) )
(see the transcript file for additional information)
Output written on 3f0fd776fe0dedeedcebc375d7633.dvi (1 page, 912 bytes).
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.

The two systems are therefore distinct; a position measurement is instantaneous whereas a definite value of energy 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:

WikiTeX: latex reported a failure, namely:

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kish, ukrainian, nohyphenation, loaded.
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(./882237a4140f69cf18b87ab9ea46e.aux)
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! Undefined control sequence.
l.5 \begin{equation*} |S\rang
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Overfull \hbox (107.45667pt too wide) detected at line 5
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L/ccm/m/it/10 n\OMS/ccsy/m/n/10 i [] \OT1/ccr/m/n/10 = [] [] \OMS/ccsy/m/n/10 j
\OML/ccm/m/it/10 x\OMS/ccsy/m/n/10 ih\OML/ccm/m/it/10 x\OMS/ccsy/m/n/10 j\OML/c
cm/m/it/10 n\OMS/ccsy/m/n/10 i [] \OML/ccm/m/it/10 dx \OT1/ccr/m/n/10 = [] \OMS
/ccsy/m/n/10 j\OML/ccm/m/it/10 x\OMS/ccsy/m/n/10 i[] [] [] [] []  [][] [] \OML/
ccm/m/it/10 dx \OT1/ccr/m/n/10 = [] \OMS/ccsy/m/n/10 j\OML/ccm/m/it/10 x\OMS/cc
sy/m/n/10 i\OML/ccm/m/it/10 ^^N\OT1/ccr/m/n/10 (\OML/ccm/m/it/10 S \OMS/ccsy/m/
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[1] (./882237a4140f69cf18b87ab9ea46e.aux) )
(see the transcript file for additional information)
Output written on 882237a4140f69cf18b87ab9ea46e.dvi (1 page, 880 bytes).
Transcript written on 882237a4140f69cf18b87ab9ea46e.log.

, i.e. WikiTeX: latex reported a failure, namely:

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entering extended mode
(./2b92a56760189d8556af13c107a33
LaTeX2e <2003/12/01>
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! Undefined control sequence.
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[1] (./2b92a56760189d8556af13c107a33.aux) )
(see the transcript file for additional information)
Output written on 2b92a56760189d8556af13c107a33.dvi (1 page, 400 bytes).
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and

WikiTeX: latex reported a failure, namely:

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! Undefined control sequence.
l.5 \begin{equation*} |N\rang
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! Undefined control sequence.
l.5 \begin{equation*} |N\rang = \int ~|s\rang
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! Undefined control sequence.
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[1] (./4e14d04276cbb681fa3b376061a02.aux) )
(see the transcript file for additional information)
Output written on 4e14d04276cbb681fa3b376061a02.dvi (1 page, 600 bytes).
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.

The time dependence of the system states is determined by the Time Dependent Schrödinger equation. In the preceding example, with energy eigenvalues , it follows that the time dependent solution is

WikiTeX: latex reported a failure, namely:

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! Undefined control sequence.
l.5 \begin{equation*}|\psi( t )\rang
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! Undefined control sequence.
l.5 ...{equation*}|\psi( t )\rang = \sum_n |n\rang
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! Undefined control sequence.
l.5 ...ion*}|\psi( t )\rang = \sum_n |n\rang \lang
                                                   n|\psi_S\rang ~e^{-i t E_...

! Undefined control sequence.
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                                                   ~e^{-i t E_n/\hbar} \end{...

[1] (./104a3a6cef9cc8ddef720972a1df3.aux) )
(see the transcript file for additional information)
Output written on 104a3a6cef9cc8ddef720972a1df3.dvi (1 page, 616 bytes).
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,

where represents the time since the particle's location in space was measured. Consequently

WikiTeX: latex reported a failure, namely:

This is pdfeTeX, Version 3.141592-1.21a-2.2 (Web2C 7.5.4)
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LaTeX2e <2003/12/01>
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! Undefined control sequence.
l.5 \begin{equation*} \lang
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! Undefined control sequence.
l.5 \begin{equation*} \lang n|\psi( t )\rang
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! Undefined control sequence.
l.5 ...n{equation*} \lang n|\psi( t )\rang = \lang
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! Undefined control sequence.
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[1] (./8dadb970274dad4e73def68d93d15.aux) )
(see the transcript file for additional information)
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at least for several distinct energy eigenstates WikiTeX: latex reported a failure, namely:

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! Undefined control sequence.
l.5 \begin{equation*}|n\rang
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[1] (./a27292a0386d176d474b35678cc4c.aux) )
(see the transcript file for additional information)
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, for all values , and for all . The particle state WikiTeX: latex reported a failure, namely:

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LaTeX2e <2003/12/01>
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! Undefined control sequence.
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[1] (./6286cfb8f1cde9dd208083515d170.aux) )
(see the transcript file for additional information)
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therefore can not have evolved (in the above technical sense) into state WikiTeX: latex reported a failure, namely:

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(which is orthogonal to all energy eigenstates, except itself), for any duration . 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 WikiTeX: latex reported a failure, namely:

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, it is perhaps worth emphasizing that any definite value of energy can be established only in the limit of a long-lasting trial and never for any finite value of time.

[edit] Optimal quantum measurement

What is the optimal quantum measurement to distinguish mixed states from a given ensemble? This is a natural question of which the solution is well understood, and is given by a semidefinite programming.

More specifically, suppose a mixed state is drawn from the ensemble with probability , we wish to find a POVM measurement so that is maximized. This is clearly a semidefinit programming:

Interestingly, the dual problem has a nice description:

Let and be the solutions of the primal and the dual, we have

From this one can conclude that if all are pure states, then must also be of rank . Furthermore, if 's are in addition independent, then the optimal measure is a von Neumann measurement.

[edit] Philosophical problems of quantum measurements

[edit] 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.

Major philosophical and metaphysical questions surround this issue:

• 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:

• 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.

• 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.

[edit] Does measurement actually determine the state?

The question of whether a measurement actually determines the state, is deeply related to the Wavefunction collapse.

Most versions of the Copenhagen interpretation answer this question with an unqualified "yes".

See also:

• Philosophies: Copenhagen interpretation.
• People (actualist philosophers): Henri Poincaré, Niels Bohr.

[edit] The quantum entanglement problem

See EPR paradox.

[edit] See also

• Measurement related problems and paradoxes
  • Measurement problem
  • Wavefunction collapse
  • EPR paradox
  • Renninger negative-result experiment
  • Elitzur-Vaidman bomb-testing problem
  • Schrödinger's cat
• 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
  • Generalized measurement (POVM, Positive operator valued measure)

[edit] 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?"

"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
• 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. IEEE Transactions on Information Theory, Vol. 49, No. 4, 1007--1012, 2003.

[edit] Further reading

• John A. Wheeler and Wojciech Hubert Zurek (eds), Quantum Theory and Measurement, Princeton University Press, (1983), ISBN 0-691-08316-9

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This page uses content from the English-language version of Wikipedia. The original article was at Measurement in quantum mechanics. The list of authors can be seen in the page history. As with Psychology Wiki, the text of Wikipedia is available under the GNU Free Documentation License.