Information about Measurement In Quantum Mechanics

Quantum physics

Quantum mechanics

Introduction to... Mathematical formulation of...

Fundamental concepts

Decoherence Interference Uncertainty Exclusion Transformation theory Ehrenfest theorem Measurement

Experiments

Double-slit experiment Davisson-Germer experiment Stern–Gerlach experiment EPR paradox Popper's experiment Schrdinger's cat

Equations

Schrdinger equation Pauli equation Klein-Gordon equation Dirac equation

Advanced theories

Quantum field theory Wightman axioms Quantum electrodynamics Quantum chromodynamics Quantum gravity Feynman diagram

Interpretations

Copenhagen Ensemble Hidden variables Transactional Many-worlds Consistent histories Quantum logic Consciousness causes collapse

Scientists

Planck Schrdinger Heisenberg Bohr Pauli Dirac Bohm Born de Broglie von Neumann Einstein Feynman Everett Others

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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.

Measurement from a practical point of view

Measurement is viewed in different ways in the many interpretations of quantum mechanics; however, despite the considerable philosophical differences, they almost universally agree on the practical question of what results from a routine quantum-physics laboratory measurement. To describe this, a simple framework to use is the Copenhagen interpretation, and we will implicitly use this in this section; the utility of this approach has been verified countless times, and all other interpretations are necessarily constructed so as to give the same quantitative predictions as this in almost every case.

Qualitative overview

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

In fact, the result of the measurement is described not by a single number, but a probability distribution that specifies the likelihoods that the various possible results will be obtained. (This distribution can be either discrete or continuous, depending on what is being measured.) The measurement process is said to be random and indeterministic. (In some other interpretations, the result merely appears random and indeterministic, as elaborated below.)

Another important aspect of measurement is wavefunction collapse. After one has measured some aspect of the quantum state, the quantum state immediately changes in such a way that if you make a second measurement of the same thing immediately afterwards, one finds the same result as the first measurement. (The detailed nature of this collapse is another source of disagreement among interpretations of quantum mechanics.)

Quantitative details

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

Measurable quantities ("observables") as operators

Each observable quantity is represented mathematically by an Hermitian (self-adjoint) operator mapping a Hilbert space (namely, the state space, which consists of all possible quantum states) into itself. This operator corresponding to an observable quantity is called an observable operator, or just an observable. The operator's eigenvalues represent possible outcomes of the measurement. For each eigenvalue there are one or more corresponding eigenvectors (which in this context are called eigenstates), which will make up the state of the system after the measurement. Some relevant properties of this formalism are

1. The eigenvalues of Hermitian operators are real. The possible outcomes of a measurement are precisely the eigenvalues of the given observable.
2. Although not all Hermitian operators have a set of eigenvectors which span the state space, it is a definitional requirement that an observable Hermitian operator does. (In the special case where the state space is finite-dimensional, this is called being diagonalizable, and is true of all Hermitian operators; see Spectral theorem.) It follows that each observable generates an orthonormal basis of eigenvectors (called an eigenbasis. This mathematical statement, that the eigenvectors of an observable span the space, corresponds to the physical statement that any quantum state can always be represented as a superposition of the eigenstates of an observable.

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

Eigenstates and projection

Assume the system is prepared in state . Let be a measurement operator, an observable, with eigenstates for and corresponding eigenvalues . If the measurement outcome is , the system will then "collapse" to the state 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).

Wavefunction collapse

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

if we measure, for example, the energy of the system and receive E₂ this result is found with probability given by

,

then the system's quantum state after the measurement is

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.

von Neumann measurement scheme

The von Neumann measurement scheme, an ancestor of quantum decoherence theory, describes measurements by taking into account the measuring apparatus which is also treated as a quantum object. Let the quantum state be in the superposition , where are eigenstates of the operator that needs to be measured. In order to make the measurement, the measured system described by needs to interact with the measuring apparatus described by the quantum state , so that the total wave function before the interaction is . After the interaction, the total wave function exhibits the unitary evolution , where 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 , so that, after the interaction, the total wave function takes a form , which is related to the phenomenon of decoherence. The above is completely described by the Schrödinger equation and there are not any interpretational problems with this. Now the problematic wavefunction collapse does not need to be understood as a process on the level of the measured system, but can also be understood as a process on the level of the measuring apparatus, or as a process on the level of the environment. Studying these processes provides considerable insight into the measurement problem by avoiding the arbitrary boundary between the quantum and classical worlds, though it does not explain the presence of randomness in the choice of final eigenstate. If the set of states , , or represents a set of states that do not overlap in space, the appearance of collapse can be generated by either the Bohm interpretation 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.

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 where , 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 where .

These two state functions and are distinct functions (of the position after we left multiply by the bra state ), but they are in general not orthogonal to each other:

.

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:

, i.e.

and

.

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

,

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



at least for several distinct energy eigenstates , for all values , and for all .

The particle state therefore can not have evolved (in the above technical sense) into state (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 , 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.

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.

Philosophical problems of quantum measurements

What physical interaction constitutes a measurement?

Until the advent of quantum decoherence theory in the late 20th century, a major conceptual problem of quantum mechanics and especially the Copenhagen interpretation was the lack of a distinctive criterion for a given physical interaction to qualify as "a measurement" and cause a wavefunction to collapse. This best illustrated by the Schrödinger's cat paradox.

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?

The neutrality of this section is disputed. Please see the discussion on the talk page.

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.

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". According to the Many-worlds interpretation the answer is an unqualified "no".

See also:

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

The quantum entanglement problem

See EPR paradox.

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
• Popper's experiment
• 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)

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.

Further reading

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

References