Observable

An observable (Latin observabilis = observable) is in physics, especially quantum physics, the formal name for a measure or for a special class of operators which act in the state space. Examples include the energy, the location coordinates, the coordinates of the pulse, and the components of the spin of the particle and the Pauli matrices.

From Neumann'sche - theory

In the traditional von Neumann's mathematical formalism of quantum mechanics, observables are represented by self-adjoint, densely defined linear operators on a Hilbert space. The term " observable " is often used synonymously for the measured variable, as well as for the associated operator. This theory generalizes the Born'sche probability interpretation. The result of a measurement of the observables of a quantum mechanical system whose state is given by a normalized vector is randomly. The probability that a particular measurement can occur is given by the probability distribution:

Where the spectral measure of the spectral theorem after referred.

Is generally of the quantum-mechanical state of the system described by a density operator, then the probability distribution of the measurement result is

Given.

The expected value of the measurement result, ie the expected value of the probability distribution is given by and by.

In the special case that the spectrum of discrete and simple, the possible results of measurement are the eigenvalues ​​of. The probability of finding the intrinsic value as a measurement result then reads, or, where a normalized eigenvector corresponding to the eigenvalue respectively.

Examples:

  • The observables "place" of a particle in one dimension corresponds ( in position representation ) of the multiplication operator on the Lebesgue space, the position operator.
  • The observables " momentum " of a particle in one dimension corresponds to about ( in position representation ) of the differential operator; more precisely its self-adjoint extension, the momentum operator. Herein, the reduced Planck constant.
  • The observables " energy " corresponds to the Hamiltonian.

Contemporary description by POVM

→ Main article: Positive operator valued probability measure ( POVM )

Not in the traditional von- Neumann's formalism fits the description of time measurements, for example, the arrival time of a particle in a detector. A more realistic formal modeling real experiments shows that even the most real quantum measurements are not described accurately by by - Neumann'sche observables. These defects fixes the general description of quantum mechanical observables by POVM.

  • Quantum mechanics
  • Quantum field theory
  • Theoretical Chemistry
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