Commutator

From Wikipedia, the free encyclopedia

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

Group theory

The commutator of two elements, g and h, of a group G, is the element

[g, h] = g⁻¹h⁻¹gh.

It is equal to the group's identity if and only if g and h commute (i.e., if and only if gh = hg). The subgroup of G generated by all commutators is called the derived group or the commutator subgroup of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent and solvable groups.

The above definition of the commutator is used by some group theorists, as well as throughout this article. However, many other group theorists define the commutator as

[g, h] = ghg⁻¹h⁻¹.

Identities (group theory)

Commutator identities are an important tool in group theory. The expression a^x denotes the conjugate of a by x, defined as x⁻¹ax.

1. ${\displaystyle x^{y}=x[x,y].}$
2. ${\displaystyle [y,x]=[x,y]^{-1}.}$
3. ${\displaystyle [x,zy]=[x,y]\cdot [x,z]^{y}}$ and ${\displaystyle [xz,y]=[x,y]^{z}\cdot [z,y].}$
4. ${\displaystyle \left[x,y^{-1}\right]=[y,x]^{y^{-1}}}$ and ${\displaystyle \left[x^{-1},y\right]=[y,x]^{x^{-1}}.}$
5. ${\displaystyle \left[\left[x,y^{-1}\right],z\right]^{y}\cdot \left[\left[y,z^{-1}\right],x\right]^{z}\cdot \left[\left[z,x^{-1}\right],y\right]^{x}=1}$ and ${\displaystyle \left[\left[x,y\right],z^{x}\right]\cdot \left[[z,x],y^{z}\right]\cdot \left[[y,z],x^{y}\right]=1.}$

Identity (5) is also known as the Hall–Witt identity, after Philip Hall and Ernst Witt. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section).
N.B., the above definition of the conjugate of a by x is used by some group theorists. Many other group theorists define the conjugate of a by x as xax⁻¹. This is often written ${\displaystyle {}^{x}a}$. Similar identities hold for these conventions.

Many identities are used that are true modulo certain subgroups. These can be particularly useful in the study of solvable groups and nilpotent groups. For instance, in any group, second powers behave well:

${\displaystyle (xy)^{2}=x^{2}y^{2}[y,x][[y,x],y].}$

If the derived subgroup is central, then

${\displaystyle (xy)^{n}=x^{n}y^{n}[y,x]^{\binom {n}{2}}.}$

Ring theory

The commutator of two elements a and b of a ring or an associative algebra is defined by

${\displaystyle [a,b]=ab-ba.}$

It is zero if and only if a and b commute. In linear algebra, if two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra.

The anticommutator of two elements a and b of a ring or an associative algebra is defined by

${\displaystyle \{a,b\}=ab+ba.}$

Sometimes the brackets [ ]₊ are also used to denote anticommutators, while [ ]₋ is then used for commutators. The anticommutator is used less often than the commutator, but can be used for example to define Clifford algebras, Jordan algebras and is utilised to derive the Dirac equation in particle physics.

The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanics, since it quantifies how well the two observables described by these operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about such commutators, by virtue of the Robertson–Schrödinger relation. In phase space, equivalent commutators of function star-products are called Moyal brackets, and are completely isomorphic to the Hilbert-space commutator structures mentioned.

Identities (ring theory)

The commutator has the following properties:

Lie-algebra identities

1. ${\displaystyle [A+B,C]=[A,C]+[B,C]}$
2. ${\displaystyle [A,A]=0}$
3. ${\displaystyle [A,B]=-[B,A]}$
4. ${\displaystyle [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0}$

The third relation is called anticommutativity, while the fourth is the Jacobi identity.

Additional identities

1. ${\displaystyle [A,BC]=[A,B]C+B[A,C]}$
2. ${\displaystyle [A,BCD]=[A,B]CD+B[A,C]D+BC[A,D]}$
3. ${\displaystyle [A,BCDE]=[A,B]CDE+B[A,C]DE+BC[A,D]E+BCD[A,E]}$
4. ${\displaystyle [AB,C]=A[B,C]+[A,C]B}$
5. ${\displaystyle [ABC,D]=AB[C,D]+A[B,D]C+[A,D]BC}$
6. ${\displaystyle [ABCD,E]=ABC[D,E]+AB[C,E]D+A[B,E]CD+[A,E]BCD}$
7. ${\displaystyle [AB,CD]=A[B,CD]+[A,CD]B}$

An additional identity may be found for this last expression, in the form:

1. ${\displaystyle [AB,CD]=A[B,C]D+[A,C]BD+CA[B,D]+C[A,D]B}$
2. ${\displaystyle [[A,C],[B,D]]=[[[A,B],C],D]+[[[B,C],D],A]+[[[C,D],A],B]+[[[D,A],B],C]}$

If A is a fixed element of a ring R, the first additional identity can be interpreted as a Leibniz rule for the map ${\displaystyle \operatorname {ad} _{A}:R\rightarrow R}$ given by ${\displaystyle \operatorname {ad} _{A}(B)=[A,B]}$. In other words, the map ad_A defines a derivation on the ring R. The second and third identities represent Leibniz rules for more than two factors that are valid for any derivation. Identities 4–6 can also be interpreted as Leibniz rules for a certain derivation.

Hadamard's lemma, applied on nested commutators holds, and underlies the Baker–Campbell–Hausdorff expansion of log(exp(A) exp(B)):

• ${\displaystyle e^{A}Be^{-A}\equiv B+[A,B]+{\frac {1}{2!}}[A,[A,B]]+{\frac {1}{3!}}[A,[A,[A,B]]]+\cdots \equiv e^{\operatorname {ad} (A)}B.}$

This formula is valid in any ring or algebra where the exponential function can be meaningfully defined, for instance in a Banach algebra or in a ring of formal power series.

Use of the same expansion expresses the above Lie group commutator in terms of a series of nested Lie bracket (algebra) commutators,

• ${\displaystyle \ln \left(e^{A}e^{B}e^{-A}e^{-B}\right)=[A,B]+{\frac {1}{2!}}[(A+B),[A,B]]+{\frac {1}{3!}}\left({\frac {1}{2}}[A,[B,[B,A]]]+[(A+B),[(A+B),[A,B]]]\right)+\cdots .}$

These identities can be written more generally using the subscript convention to include the anticommutator:^[8] (defined above), for instance

1. ${\displaystyle [AB,C]_{-}=A[B,C]_{\mp }\pm [A,C]_{\mp }B}$
2. ${\displaystyle [AB,CD]_{-}=A[B,C]_{\mp }D\pm AC[B,D]_{\mp }+[A,C]_{\mp }DB\pm C[A,D]_{\mp }B}$
3. ${\displaystyle \left[A,[B,C]_{\pm }\right]+\left[B,[C,A]_{\pm }\right]+\left[C,[A,B]_{\pm }\right]=0}$

Graded rings and algebras

When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as

${\displaystyle [\omega ,\eta ]_{gr}:=\omega \eta -(-1)^{\deg \omega \deg \eta }\eta \omega .}$

Derivations

Especially if one deals with multiple commutators, another notation turns out to be useful, the adjoint representation:

${\displaystyle \operatorname {ad} (x)(y)=[x,y].}$

Then ad(x) is a linear derivation:

${\displaystyle \operatorname {ad} (x+y)=\operatorname {ad} (x)+\operatorname {ad} (y)}$ and ${\displaystyle \operatorname {ad} (\lambda x)=\lambda \operatorname {ad} (x)}$

and, crucially, it is a Lie algebra homomorphism:

${\displaystyle \operatorname {ad} ([x,y])=[\operatorname {ad} (x),\operatorname {ad} (y)]~.}$

By contrast, it is not always an algebra homomorphism; it does not hold in general:

${\displaystyle \operatorname {ad} (xy)\,{\stackrel {?}{=}}\,\operatorname {ad} (x)\operatorname {ad} (y)}$

Examples

${\displaystyle {\begin{aligned}\operatorname {ad} (x)\operatorname {ad} (x)(y)&=[x,[x,y]\,]\\\operatorname {ad} (x)\operatorname {ad} (a+b)(y)&=[x,[a+b,y]\,]\end{aligned}}}$

General Leibniz rule

The general Leibniz rule, expanding repeated derivatives of a product, can be written abstractly using the adjoint representation:

${\displaystyle x^{n}y=\sum _{k=0}^{n}{\binom {n}{k}}\left(\operatorname {ad} (x)\right)^{k}(y)\,x^{n-k}}$

Replacing x by the differentiation operator ${\displaystyle \partial }$, and y by the multiplication operator ${\displaystyle m_{f}:g\mapsto fg}$, we get ${\displaystyle \operatorname {ad} (\partial )(m_{f})=m_{\partial (f)}}$, and applying both sides to a function g, the identity becomes the general Leibniz rule for ${\displaystyle \partial ^{n}(fg)}$.

Transactional interpretation

From Wikipedia, the free encyclopedia

The transactional interpretation of quantum mechanics (TIQM) takes the psi and psi* wave functions of the standard quantum formalism to be retarded (forward in time) and advanced (backward in time) waves that form a quantum interaction as a Wheeler–Feynman handshake or transaction. It was first proposed in 1986 by John G. Cramer, who argues that it helps in developing intuition for quantum processes. He also suggests that it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and also resolves various quantum paradoxes. TIQM formed a minor plot point in his science fiction novel Einstein's Bridge.

More recently, he has also argued TIQM to be consistent with the Afshar experiment, while claiming that the Copenhagen interpretation and the many-worlds interpretation are not. The existence of both advanced and retarded waves as admissible solutions to Maxwell's equations was explored in the Wheeler–Feynman absorber theory. Cramer revived their idea of two waves for his transactional interpretation of quantum theory. While the ordinary Schrödinger equation does not admit advanced solutions, its relativistic version does, and these advanced solutions are the ones used by TIQM.

In TIQM, the source emits a usual (retarded) wave forward in time, but it also emits an advanced wave backward in time; furthermore, the receiver, who is later in time, also emits an advanced wave backward in time and a retarded wave forward in time. A quantum event occurs when a "handshake" exchange of advanced and retarded waves triggers the formation of a transaction in which energy, momentum, angular momentum, etc. are transferred. The quantum mechanism behind transaction formation has been demonstrated explicitly for the case of a photon transfer between atoms in Sect. 5.4 of Carver Mead's book Collective Electrodynamics. In this interpretation, the collapse of the wavefunction does not happen at any specific point in time, but is "atemporal" and occurs along the whole transaction, and the emission/absorption process is time-symmetric. The waves are seen as physically real, rather than a mere mathematical device to record the observer's knowledge as in some other interpretations of quantum mechanics. Philosopher and writer Ruth Kastner argues that the waves exist as possibilities outside of physical spacetime and that therefore it is necessary to accept such possibilities as part of reality.

Cramer has used TIQM in teaching quantum mechanics at the University of Washington in Seattle.

Advances over previous interpretations

TIQM is explicitly non-local and, as a consequence, logically consistent with counterfactual definiteness (CFD), the minimum realist assumption. As such it incorporates the non-locality demonstrated by the Bell test experiments and eliminates the observer dependent reality that plagues the Copenhagen Interpretation. Greenberger–Horne–Zeilinger state the key advance over Everett's Relative State Interpretation^[6] is to regard the conjugate state vector of the Dirac formalism as ontologically real, incorporating a part of the formalism that, prior to TIQM, had been interpretationally neglected. Having interpreted the conjugate state vector as an advanced wave, it is shown that the origins of the Born rule follow naturally from the description of a transaction.

The transactional interpretation is superficially similar to the two-state vector formalism (TSVF) which has its origin in work by Yakir Aharonov, Peter Bergmann and Joel Lebowitz of 1964.^[8][9] However, it has important differences—the TSVF is lacking the confirmation and therefore cannot provide a physical referent for the Born Rule (as TI does). Kastner has criticized some other time-symmetric interpretations, including TSVF, as making ontologically inconsistent claims.

Kastner has developed a new Relativistic Transactional Interpretation (RTI) also called Possibilist Transactional Interpretation (PTI) in which space-time itself emerges by a way of transactions. It has been argued that this relativistic transactional interpretation can provide the quantum dynamics for the causal sets program.

Debate

In 1996, Tim Maudlin proposed a thought experiment involving Wheeler's delayed choice experiment that is generally taken as a refutation of TIQM. However Kastner showed Maudlin's argument is not fatal for TIQM.

In his book, The Quantum Handshake, Cramer has added a hierarchy to the description of pseudo-time to deal with Maudlin's objection and has pointed out that some of Maudlin's arguments are based on the inappropriate application of Heisenberg's knowledge interpretation to the transactional description.

Transactional Interpretation faces criticisms. The following is partial list and some replies:

1. “TI does not generate new predictions / is not testable / has not been tested.”

TI is an exact interpretation of QM and so its predictions must be the same as QM. Like the many-worlds interpretation (MWI), TI is a "pure" interpretation in that it does not add anything ad hoc but provides a physical referent for a part of the formalism that has lacked one (the advanced states implicitly appearing in the Born rule). Thus the demand often placed on TI for new predictions or testability is a mistaken one that misconstrues the project of interpretation as one of theory modification.

2. “It is not made clear where in spacetime a transaction occurs.”

One clear account is given in Cramer (1986), which pictures a transaction as a four-vector standing wave whose endpoints are the emission and absorption events.

3. “Maudlin (1996, 2002) has demonstrated that TI is inconsistent.”

Maudlin's probability criticism confused the transactional interpretation with Heisenberg's knowledge interpretation. However, he raised a valid point concerning causally connected possible outcomes, which led Cramer to add hierarchy to the pseudo-time description of transaction formation. Kastner has extended TI to the relativistic domain, and in light of this expansion of the interpretation, it can be shown that the Maudlin Challenge cannot even be mounted, and is therefore nullified; there is no need for the 'hierarchy' proposal of Cramer. Maudlin has also claimed that all the dynamics of TI is deterministic and therefore there can be no 'collapse.' But this appears to disregard the response of absorbers, which is the whole innovation of the model. Specifically, the linearity of the Schrödinger evolution is broken by the response of absorbers; this directly sets up the non-unitary measurement transition, without any need for ad hoc modifications to the theory. The non-unitarity is discussed, for example in Chapter 3 of Kastner's book The Transactional Interpretation of Quantum Mechanics: The Reality of Possibility (CUP, 2012).

4. "It is not clear how the transactional interpretation handles the quantum mechanics of more than one particle."

This issue is addressed in Cramer's 1986 paper, in which he gives many examples of the application of TIQM to multi-particle quantum systems. However, if the question is about the existence of multi-particle wave functions in normal 3D space, Cramer's 2015 book goes into some detail in justifying multi-particle wave functions in 3D space. A criticism of Cramer's 2015 account of dealing with multi-particle quantum systems is found in Kastner 2016, "An Overview of the Transactional Interpretation and its Evolution into the 21st Century, Philosophy Compass (2016) . It observes in particular that the account in Cramer 2015 is necessarily anti-realist about the multi-particle states: if they are only part of a 'map,' then they are not real, and in this form TI becomes an instrumentalist interpretation, contrary to its original spirit. Thus the so-called "retreat" to Hilbert space (criticized also below in the lengthy discussion of note) can instead be seen as a needed expansion of the ontology, rather than a retreat to anti-realism/instrumentalism about the multi-particle states. The vague statement (under that "Offer waves are somewhat ephemeral three-dimensional space objects" indicates the lack of clear definition of the ontology when one attempts to keep everything in 3+1 spacetime.

Measurement in quantum mechanics

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The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus. The question of how the operational process measurement affects the ontological state of the observed system is unresolved, and called the measurement problem.

Measurement from a practical point of view

Measurement plays an important role in quantum mechanics, and it is viewed in different ways among various interpretations of quantum mechanics. In spite of considerable philosophical differences, different views of measurement almost universally agree on the practical question of what results from a routine quantum-physics laboratory measurement. To understand this, the Copenhagen interpretation, which has been commonly used, is employed in this article.

Qualitative overview

In classical mechanics, a simple system consisting of only one single particle is fully described by the position ${\displaystyle {\vec {x}}(t)}$ and momentum ${\displaystyle {\vec {p}}(t)}$ of the particle. As an analogue, in quantum mechanics a system is described by its quantum state, which contains the probabilities of possible positions and momenta. In mathematical language, all possible pure states of a system form an abstract vector space called Hilbert space, which is typically infinite-dimensional. A pure state is represented by a state vector in the Hilbert space.

Once a quantum system has been prepared in laboratory, some measurable quantity such as position or energy is measured. For pedagogic reasons, the measurement is usually assumed to be ideally accurate. The state of a system after measurement is assumed to "collapse" into an eigenstate of the operator corresponding to the measurement. Repeating the same measurement without any evolution of the quantum state will lead to the same result. If the preparation is repeated, subsequent measurements will likely lead to different results.

The predicted values of the measurement are described by a probability distribution, or an "average" (or "expectation") of the measurement operator based on the quantum state of the prepared system. The probability distribution is either continuous (such as position and momentum) or discrete (such as spin), depending on the quantity being measured.

The measurement process is often considered as random and indeterministic. Nonetheless, there is considerable dispute over this issue. In some interpretations of quantum mechanics, the result merely appears random and indeterministic, whereas in other interpretations the indeterminism is core and irreducible. A significant element in this disagreement is the issue of "collapse of the wave function" associated with the change in state following measurement. There are many philosophical issues and stances (and some mathematical variations) taken—and near universal agreement that we do not yet fully understand quantum reality. In any case, our descriptions of dynamics involve probabilities, not certainties.

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

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

1. The observable is a self-adjoint or Hermitian operator mapping a Hilbert space (namely, the state space, which consists of all possible quantum states) into itself.
   
2. Thus, the observable's eigenvectors (called an eigenbasis) form an orthonormal basis that span the state space in which that observable exists. Any quantum state can be represented as a superposition of the eigenstates of an observable.
   
3. Hermitian operators' eigenvalues are real. The possible outcomes of a measurement are precisely the eigenvalues of the given observable.
   
4. For each eigenvalue there are one or more corresponding eigenvectors (eigenstates). A measurement results in the system being in the eigenstate corresponding to the eigenvalue result of the measurement. If the eigenvalue determined from the measurement corresponds to more than one eigenstate ("degeneracy"), instead of being in a definite state, the system is in a sub-space of the measurement operator corresponding to all the states having that eigenvalue.

Important examples of observables are:

• The Hamiltonian operator ${\displaystyle {\hat {H}}}$, which represents the total energy of the system. In nonrelativistic quantum mechanics the nonrelativistic Hamiltonian operator is given by ${\displaystyle {\hat {H}}={\hat {T}}+{\hat {V}}={{\hat {p}}^{2} \over 2m}+V({\hat {x}})}$.
• The momentum operator ${\displaystyle {\hat {p}}}$ is given by ${\displaystyle {\hat {p}}=-i\hbar {\partial \over \partial x}}$ (in the position basis), or ${\displaystyle {\hat {p}}=p}$ (in the momentum basis).
• The position operator ${\displaystyle {\hat {x}}}$ is given by ${\displaystyle {\hat {x}}=x}$ (in the position basis), or ${\displaystyle {\hat {x}}=i\hbar {\partial \over \partial p}}$ (in the momentum basis).

Operators can be noncommuting. Two Hermitian operators commute if (and only if) there is at least one basis of vectors such that 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 discovered by Werner Heisenberg.

Measurement probabilities and wave function collapse

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

Discrete, nondegenerate spectrum

Let ${\displaystyle {\hat {O}}}$ be an observable. By assumption, ${\displaystyle {\hat {O}}}$ has discrete eigenstates ${\displaystyle |1\rangle ,|2\rangle ,|3\rangle ,...}$ with corresponding distinct eigenvalues ${\displaystyle O_{1},O_{2},O_{3},...}$. That is, the states are nondegenerate.
Consider a system prepared in state ${\displaystyle |\psi \rangle }$. Since the eigenstates of the observable ${\displaystyle {\hat {O}}}$ form a complete basis called eigenbasis, the state vector ${\displaystyle |\psi \rangle }$ can be written in terms of the eigenstates as

${\displaystyle |\psi \rangle =c_{1}|1\rangle +c_{2}|2\rangle +c_{3}|3\rangle +\cdots =\sum _{n}c_{n}|n\rangle }$,

where ${\displaystyle c_{1},c_{2},\ldots }$ are complex numbers in general. The eigenvalues ${\displaystyle O_{1},O_{2},O_{3},...}$ are all possible values of the measurement. The corresponding probabilities are given by

${\displaystyle \Pr(O_{n})={\frac {|\langle n|\psi \rangle |^{2}}{|\langle \psi |\psi \rangle |}}={\frac {|c_{n}|^{2}}{\sum _{k}|c_{k}|^{2}}}}$

Usually ${\displaystyle |\psi \rangle }$ is assumed to be normalized, i.e. ${\displaystyle \langle \psi |\psi \rangle =1}$. Therefore, the expression above is reduced to

${\displaystyle \Pr(O_{n})=|\langle n|\psi \rangle |^{2}=|c_{n}|^{2}}$

If the result of the measurement is ${\displaystyle O_{n}}$, then the system (after measurement) is in pure state ${\displaystyle |n\rangle }$. That is,

${\displaystyle |\psi '\rangle =|n\rangle }$

so any repeated measurement of ${\displaystyle {\hat {O}}}$ will yield the same result ${\displaystyle O_{n}}$. When there is a discontinuous change in state due to a measurement that involves discrete eigenvalues, that is called wave function collapse. For some, this is simply a description of a reasonably accurate discontinuous change in a mathematical representation of physical reality; for others, depending on philosophical orientation, this is a fundamentally serious problem with quantum theory; others see this as statistically-justified approximation resulting from the fact that the entity performing this measurement has been excluded from the state-representation. In particular, multiple measurements of certain physically extended systems demonstrate predicted statistical correlations which would not be possible under classical assumptions.

Continuous, nondegenerate spectrum

Let ${\displaystyle {\hat {O}}}$ be an observable. By assumption, ${\displaystyle {\hat {O}}}$ has continuous eigenstate ${\displaystyle |x\rangle }$, with corresponding distinct eigenvalue ${\displaystyle x}$. The eigenvalue forms a continuous spectrum filling the interval (a,b).

Consider a system prepared in state ${\displaystyle |\psi \rangle }$. Since the eigenstates of the observable ${\displaystyle {\hat {O}}}$ form a complete basis called eigenbasis, the state vector ${\displaystyle |\psi \rangle }$ can be written in terms of the eigenstates as

${\displaystyle |\psi \rangle =\int _{a}^{b}c(x)|x\rangle \,dx}$,

where ${\displaystyle c(x)}$ is a complex-valued function. The eigenvalue that fills up the interval ${\displaystyle (a,b)}$ is the possible value of measurement. The corresponding probability is described by a probability function given by

${\displaystyle \Pr(d$

where ${\displaystyle (d,e)\subseteq (a,b)}$. Usually ${\displaystyle |\psi \rangle }$ is assumed to be normalized, i.e. ${\displaystyle |\langle \psi |\psi \rangle \|=1}$. Therefore, the expression above is reduced to

${\displaystyle \Pr(d$

If the result of the measurement is ${\displaystyle x}$, then the system (after measurement) is in pure state ${\displaystyle |x\rangle }$. That is,

${\displaystyle |\psi '\rangle =|x\rangle .}$

Alternatively, it is often possible and convenient to analyze a continuous-spectrum measurement by taking it to be the limit of a different measurement with a discrete spectrum. For example, an analysis of scattering involves a continuous spectrum of energies, but by adding a "box" potential (which bounds the volume in which the particle can be found), the spectrum becomes 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 multiple eigenspaces. The probability of measuring a particular eigenvalue is the squared component of the state vector in the corresponding eigenspace, and the new state after measurement is the projection of the original state vector into the appropriate eigenspace.

Density matrix formulation

Instead of performing quantum-mechanics computations in terms of wave functions (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 wave function formulation above. The result for the discrete, degenerate case, for example, is as follows:

Let ${\displaystyle {\hat {O}}}$ be an observable, and suppose that it has discrete eigenvalues ${\displaystyle O_{1},O_{2},O_{3},\ldots }$, associated with eigenspaces ${\displaystyle V_{1},V_{2},\ldots }$ respectively. Let ${\displaystyle P_{n}}$ be the projection operator into the space ${\displaystyle V_{n}}$.

Assume the system is prepared in the state described by the density matrix ρ. Then measuring ${\displaystyle {\hat {O}}}$ can yield any of the results ${\displaystyle O_{1},O_{2},O_{3},\ldots }$, with corresponding probabilities given by

${\displaystyle \Pr(O_{n})=\mathrm {Tr} (P_{n}\rho )}$

where ${\displaystyle \mathrm {Tr} }$ denotes trace. If the result of the measurement is n, then the new density matrix will be

${\displaystyle \rho '={\frac {P_{n}\rho P_{n}}{\mathrm {Tr} (P_{n}\rho )}}}$

Alternatively, one can say that the measurement process results in the new density matrix

${\displaystyle \rho ''=\sum _{n}P_{n}\rho P_{n}}$

where the difference is that ${\displaystyle \rho ''}$ is the density matrix describing the entire ensemble, whereas ${\displaystyle \rho '}$ is the density matrix describing the sub-ensemble whose measurement result was ${\displaystyle n}$.

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 ${\displaystyle {\hat {O}}}$, on a state whose quantum state is ${\displaystyle |\psi \rangle }$.

• The mean (average) value of the measurement is

${\displaystyle \langle \psi |{\hat {O}}|\psi \rangle }$ (see expectation value).

• The variance of the measurement is

${\displaystyle \langle \psi |{\hat {O}}^{2}|\psi \rangle -(\langle \psi |{\hat {O}}|\psi \rangle )^{2}}$

• The standard deviation of the measurement is

${\displaystyle {\sqrt {\langle \psi |{\hat {O}}^{2}|\psi \rangle -(\langle \psi |{\hat {O}}|\psi \rangle )^{2}}}}$

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

Example

Suppose that we have a particle in a 1-dimensional box, set up initially in the ground state ${\displaystyle |\psi _{1}\rangle }$. As can be computed from the time-independent Schrödinger equation, the energy of this state is ${\displaystyle E_{1}={\frac {\pi ^{2}\hbar ^{2}}{2mL^{2}}}}$ (where m is the particle's mass and L is the box length), and the spatial wave function is ${\displaystyle \langle x|\psi _{1}\rangle ={\sqrt {\frac {2}{L}}}~{\rm {sin}}\left({\frac {\pi x}{L}}\right)}$. If the energy is now measured, the result will always certainly be ${\displaystyle E_{1}}$, and this measurement will not affect the wave function.

Next suppose that the particle's position is measured. The position x will be measured with probability density

${\displaystyle \Pr(S$

If the measurement result was x=S, then the wave function after measurement will be the position eigenstate ${\displaystyle |x=S\rangle }$. If the particle's position is immediately measured again, the same position will be obtained.

The new wave function ${\displaystyle |x=S\rangle }$ can, like any wave function, be written as a superposition of eigenstates of any observable. In particular, using energy eigenstates, ${\displaystyle |\psi _{n}\rangle }$, we have

${\displaystyle |x=S\rangle =\sum _{n}|\psi _{n}\rangle \left\langle \psi _{n}|x=S\right\rangle =\sum _{n}|\psi _{n}\rangle {\sqrt {\frac {2}{L}}}~{\rm {sin}}\left({\frac {n\pi S}{L}}\right)}$

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 ${\displaystyle E_{n}}$ will be measured with relative probabilities:

${\displaystyle \Pr(E_{n})=|\langle \psi _{n}|S\rangle |^{2}={\frac {2}{L}}~{\rm {sin}}^{2}\left({\frac {n\pi S}{L}}\right)}$

and moreover if the measurement result is ${\displaystyle E_{n}}$, then the new state will be the energy eigenstate ${\displaystyle |\psi _{n}\rangle }$.
So in this example, due to the process of wave function collapse, a particle initially in the ground state can end up in any energy level, after just two subsequent non-commuting measurements are made.

Wave function collapse

According to the Copenhagen interpretation 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". 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 wave function collapse raises serious questions regarding "the measurement problem", as well as 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. Decoherence correctly predicts the form and probability distribution of the final eigenstates, and explains the apparent randomness of the choice of final state in terms of einselection.

von Neumann measurement scheme

The von Neumann measurement scheme, the ancestor of quantum decoherence theory, describes measurements by taking into account the measuring apparatus which is also treated as a quantum object.

"Measurement" of the first kind — premeasurement without detection

Let the quantum state be in the superposition ${\displaystyle \scriptstyle |\psi \rangle =\sum _{n}c_{n}|\psi _{n}\rangle }$, where ${\displaystyle \scriptstyle |\psi _{n}\rangle }$ are eigenstates of the operator for the so-called "measurement" prior to von Neumann's second apparatus. In order to make the "measurement", the system described by ${\displaystyle \scriptstyle |\psi \rangle }$ needs to interact with the measuring apparatus described by the quantum state ${\displaystyle \scriptstyle |\phi \rangle }$. The total wave function before the interaction with the second apparatus is then ${\displaystyle \scriptstyle |\psi \rangle |\phi \rangle }$. During the interaction of object and measuring instrument, the unitary evolution is supposed to realize the following transition from the initial to the final total wave function:

${\displaystyle |\psi \rangle |\phi \rangle \rightarrow \sum _{n}c_{n}|\psi _{n}\rangle |\phi _{n}\rangle \quad {\text{(measurement of the first kind),}}}$

where ${\displaystyle \scriptstyle |\phi _{n}\rangle }$ are orthonormal states of the measuring apparatus. The unitary evolution above is referred to as premeasurement. The relation with wave function collapse is established by calculating the final density operator of the object ${\displaystyle \scriptstyle \sum _{n}|c_{n}|^{2}|\psi _{n}\rangle \langle \psi _{n}|}$ from the final total wave function. This density operator is interpreted by von Neumann as describing an ensemble of objects being after the measurement with probability ${\displaystyle \scriptstyle |c_{n}|^{2}}$ in the state ${\displaystyle \scriptstyle |\psi _{n}\rangle .}$
The transition

${\displaystyle |\psi \rangle \rightarrow \sum _{n}|c_{n}|^{2}|\psi _{n}\rangle \langle \psi _{n}|\psi \rangle }$

is often referred to as weak von Neumann projection, the wave function collapse or strong von Neumann projection

${\displaystyle |\psi \rangle \rightarrow \sum _{n}|c_{n}|^{2}|\psi _{n}\rangle \langle \psi _{n}|\psi \rangle \rightarrow |\psi _{n}\rangle }$

being thought to correspond to an additional selection of a subensemble by means of observation.
In case the measured observable has a degenerate spectrum, weak von Neumann projection is generalized to Lüders projection

${\displaystyle |\psi \rangle \rightarrow \sum _{n}|c_{n}|^{2}P_{n},\;P_{n}=\sum _{i}|\psi _{ni}\rangle \langle \psi _{ni}|,}$

in which the vectors ${\displaystyle \scriptstyle |\psi _{ni}\rangle }$ for fixed n are the degenerate eigenvectors of the measured observable. For an arbitrary state described by a density operator ${\displaystyle \scriptstyle \rho }$ Lüders projection is given by

${\displaystyle \rho \rightarrow \sum _{n}P_{n}\rho P_{n}.}$

Measurement of the second kind — with irreversible detection

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

${\displaystyle |\psi \rangle |\phi \rangle \rightarrow \sum _{n}c_{n}|\chi _{n}\rangle |\phi _{n}\rangle ,}$

in which the states ${\displaystyle \scriptstyle |\chi _{n}\rangle }$ of the object are determined by specific properties of the interaction between object and measuring instrument. They are normalized but not necessarily mutually orthogonal. The relation with wave function collapse is analogous to that obtained for measurements of the first kind, the final state of the object now being ${\displaystyle \scriptstyle |\chi _{n}\rangle }$ with probability ${\displaystyle \scriptstyle |c_{n}|^{2}.}$ Note that many measurement procedures are measurements of the second kind, some even functioning correctly only as a consequence of being of the second kind. For instance, a photon counter, detecting a photon by absorbing and hence annihilating it, thus ideally leaving the electromagnetic field in the vacuum state rather than in the state corresponding to the number of detected photons; also the Stern–Gerlach experiment would not function at all if it really were a measurement of the first kind.

Decoherence in quantum measurement

One can also introduce the interaction with the environment ${\displaystyle \scriptstyle |e\rangle }$, so that, in a measurement of the first kind, after the interaction the total wave function takes a form

${\displaystyle \sum _{n}c_{n}|\psi _{n}\rangle |\phi _{n}\rangle |e_{n}\rangle ,}$

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 does not need to be understood as a process ${\displaystyle \scriptstyle |\psi \rangle \rightarrow |\psi _{n}\rangle }$ on the level of the measured system, but can also be understood as a process ${\displaystyle \scriptstyle |\phi \rangle \rightarrow |\phi _{n}\rangle }$ on the level of the measuring apparatus, or as a process ${\displaystyle \scriptstyle |e\rangle \rightarrow |e_{n}\rangle }$ 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

${\displaystyle \{|\psi _{n}\rangle \}}$, ${\displaystyle \{|\phi _{n}\rangle \}}$, or ${\displaystyle \{|e_{n}\rangle \}}$

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 wave function 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.

Quotes

A measurement always causes the system to jump into an eigenstate of the dynamical variable that is being measured, the eigenvalue this eigenstate belongs to being equal to the result of the measurement.

— P. A. M. Dirac (1958) in The Principles of Quantum Mechanics, p. 36