Measurement in quantum mechanics

From Academic Kids

Template:Quantum-theory The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications.

Contents

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 the system in mutual agreement between all members of this system, and therefore by a particular method which is reproducible by all members of the system, at least in principle.

Measurable quantities ("observables") as operators

Observable quantities are represented mathematically by a Hermitian operator, with its eigenvalues representing any definite result value which might be obtained as a result of the measurement, and the corresponding eigenstate being the state of the system during the trial. This representation is possible and appropriate because

  1. Its eigenvalues are real, and the possible result values of a measurement are correspondingly real numbers.
  2. 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.
  3. 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, this state is then 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> then 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>.

The case of a continuous spectrum is more problematic, since the basis has uncountably many eigenvectors. These can be represented by a set of Delta functions. Since the delta function is in fact not a function, and moreover, doesn't belong to the Hilbert space of square-integrable functions, this can causes difficulties such as singularities and infinite values. In all practical cases, 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 countable spectrum).

Wavefunction collapse


Given any quantum state which is a superposition of eigenstates
<math> | \psi \rang = c_1 e^{-i E_1 t} | 1 \rang + c_2 e^{-i E_2 t} | 2 \rang + c_3 e^{-i E_3 t} | 3 \rang + \cdots \ \ , <math>
if we measure, for example, the energy of the system and receive En
(this result is chosen randomly according to probability given by
<math> \Pr( E_n ) = \frac{ | A_n |^2 }{\sum_k | A_k |^2} <math> ),
than the system's quantum state instantly becomes
<math> | \psi \rang = e^{-i E_2 t} | 2 \rang <math>
so any further measurement of energy will always yield E2.

Figure 1. The process of wavefunction collapse illustrated mathematically.

The process in which a quantum state instantly becomes one of the eigenstates of the operator corresponding to the measured observable (precisely which eigenstate is random, though the probabilities are determined by the square of the amplitude with which that eigenstate contributes to the overall state) is called "collapse", or "wavefunction collapse". The collapse process has no trace or corresponding mathematical description in the mathematical formulation of quantum mechanics. Moreover, the Schrodinger equation, which determines the evolution of the system in time, does not predicts such a process, yet the process of collapse was demonstrated in many experiments (such as 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.

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.

Example

(should be reviewed and cleaned up)

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 Schrödinger 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 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:

<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 Schrödinger 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 problems of quantum measurements

What physical interaction constitutes a measurement?

A major conceptual problem of quantum mechanics and especially the Copenhagen interpretation is the lack of a distinctive criterion for a given physical interaction to qualify as "a measurement" and cause a wavefunction to collapse. This is best illustrated by the Schrödinger's cat paradox.

Major philosophical and metaphysical question surround this issue:

  • Does a measurement depends on the existence of a self-aware observer?
  • What interactions are strong enough to constitute a measurement?
  • 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?

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 answer is an unqualified "yes".

See also:

The quantum entanglement problem

See EPR paradox.

See also

External links

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