Measurement in quantum mechanics

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The framework of Quantum Mechanics admits a careful definition of Measurement, and a thorough discussion of its practical and philosophical implications.

The mathematical formalism of measurement

The goal of a particular measurement of a particular system, in any experimental trial, is to obtain a characterization of this system in mutual agreement between all members of this system, and therefore by a particular method which is reproducible by all members of this system, at least in principle.

Measurable Quantities ("Observables") as Operators

The corresponding method of measurement of a particular physical quantity, which is to be applied to the given observational data of the particular trial, is therefore represented mathematically by a particular Hermitian operator, with its eigenvalues representing any definite result value which might be obtained as result of the measurement; and the state of the system during the trial as its corresponding eigenstate. This representation is possible and appropriate because

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

Important examples are:

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

Many operators are pairwise noncommuting; that is, for a given set of observational data, from a particular trial, one may obtain a definite real result value for one quantity, but not for the other, or even for neither. Even if the state of the system in one particular trial corresponds to one particular eigenstate of one operator, it is then to be represented as a nontrivial linear combination of eigenstates of the other operator.

Eigenstates and projection

In Quantum mechanics, when you take a measurement of a system with state vector (wave function) <math>|\psi\rang</math> where the corresponding measurement operator <math> {\hat O} </math> has eigenstates <math>|n\rang </math> for <math> n = 1, 2, 3, ... </math>, and if you found one definite result value <math> O_N</math> and the state which the system had in this trial is consequently represented as <math> |N\rang </math>, the system may be said having been forced or "collapsed" into the state <math>|N\rang </math>.

Suppose we knew that a particle had been confined throughout in a box potential (see, for example, the particle in a box problem) and we had found its energy value to be <math> E_N </math>; with the corresponding system state <math>|\psi_N\rang = |N\rang = \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{N \pi x}{L}\right)</math> as solution of the [[Schr�dinger equation]], under assumption of a box potential. Suppose further, that in one particular trial over the course of obtaining the energy measurement, we had met the particle at a particular distance value <math> S </math> from one potential wall of the box; corresponding to system state <math>|\psi_S\rang = |S\rang = \delta( S - x ) </math>.

The state functions <math>|\psi_N\rang</math> and <math>|\psi_S\rang</math> are distinct functions (of distance <math> x </math>), but they are in general not orthogonal to each other:

<math> \lang \psi_S | \psi_N\rang = \lang S | N\rang = \int_0^L dx~\delta( S - x)~\sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{N \pi x}{L}\right) = \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{N \pi S}{L}\right) </math>.

The two trials from which observations were collected in order to obtain these measured values <math> S </math> and <math> E_N </math> were therefore distinct trials; a meeting between the particle and "us" (or someone who will be able to assert the distance value <math> S </math>) is instantaneous, while a definite value of energy <math> E_N </math> is established only in the limit of a long-lasting trial.

Completeness of eigenvectors of Hermitean 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:

<math> |S\rang = \sum_n | n \rang \left\langle n | S \right\rangle = \frac{2}{L}~\sum_n {\rm sin}\left(\frac{n \pi x}{L}\right)~{\rm sin}\left(\frac{n \pi S}{L}\right) = \delta( S - x )</math>, and

<math> |N\rang = \int ds~|s\rang \left\langle s | N \right\rangle = \int ds~\delta( s - x )~\sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{N \pi s}{L}\right) = \sqrt{ \frac{2}{L} }~{\rm sin}\left(\frac{N \pi x}{L}\right) </math>.

The evolution of states is described by the [[Schr�dinger equation]], and in the given example with energy eigenvalues <math>E_n</math> it follows that

<math>|\psi( t )\rang = \sum_n |n\rang \lang n|\psi_S\rang ~e^{-i t E_n/\hbar} </math>,

where <math> t </math> represents the duration since the meeting had been observed, based on which the distance value <math> S </math> was measured. Consequently

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

at least for several distinct energy eigenstates <math>|n\rang </math>, for all values <math> t </math>, and for all <math> 0 < S < L </math>.

The particle state <math> |\psi_S \rang</math> therefore can not have evolved (in the above technical sense) into state <math> |\psi_N \rang </math> (which is orthogonal to all energy eigenstates, except itself), for any duration <math> t </math>. 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 <math> |\psi_N\rang </math>, it is perhaps worth emphasizing that any definite value of energy <math> E_N </math> can be established only in the limit of a long-lasting trial, i.�e. not for any one particular value of <math> t </math>.

Philosophical problem of Quantum measurements

Does measurement actually determine the state?

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

According to the Copenhagen interpretation, the unswer is an unqualified "yes".

According to actualist philosophers (such as Immanuel Kant) the answer is positive - measurement does determine the state of the system, but it is a logical determination. This answer is derived from Kant's actual principle that the world does not exist seperatly from the mind who watches it. This principle is based on the insight that the measurement is always involving the observer's mind that interprets the events (measurement reading) according to its apriori conventions. In other words, radical actualists hold that nothing exists until it is grasped by your mind. It is very hard to cope with this principle, which may sound like absurd with corollaries such as "The moon exists only when you look at it".

So according to actualism, taking a measurement is no more a passive action of observing the state of a system, but rather an active action which determines the state of the system. We should note that the determination of the state is done randomly, according to probability which is given by the wave function of the system.

The Quantum entanglement problem

Particular attention has been given to measurements based on mutual observations by system constituents that were separate from each other; such as measurements of

distance between two detectors,
momentum of one detector relative to an extended set of suitable detectors and
whether or not such an entire system had been closed, in terms of its momentum, throughout some particular sequence of trials;
angle of two detector pairs to each other,
angular momentum of one detector pair relative to a separate set of suitable detectors and
whether or not such an entire system had been closed, in terms of its angular momentum, throughout some particular sequence of trials.

Paradoxial conclusions may result if such relations and quantities are merely presumed or extrapolated from results of past trials, instead of being measured explicitly. (See EPR paradox.)

See Talk:Measurement in quantum mechanics.