Observable

An observable ( Latin observabilis , observable) is in physics , especially quantum physics , the formal name for a measurand and the operator assigned to it , which act in the state space , a Hilbert space . Examples are the energy , the position coordinates, the coordinates of the momentum and the components of the spin of a particle as well as the Pauli matrices .

Von-Neumann theory

In the traditional von Neumann mathematical formalism of quantum mechanics , observables are represented by self-adjoint , densely defined linear operators on a Hilbert space . This theory generalizes Born's probability interpretation . ${\displaystyle A}$ ${\displaystyle {\mathcal {H}}}$

The result of a measurement of the observables of a quantum mechanical system, the state of which is described by a normalized vector ( wave function in Bra-Ket notation ), is random . The probability with which a certain measured value can occur is given by the probability distribution ${\displaystyle A}$ ${\displaystyle |\Psi \rangle \in {\mathcal {H}}}$ ${\displaystyle B}$

${\displaystyle P[B]=\langle \Psi |\lambda _{A}(B)|\Psi \rangle }$

wherein the spectral measure of after the spectral theorem , respectively. ${\displaystyle \lambda _{A}}$ ${\displaystyle A}$

If the quantum mechanical state of the system is described more generally by a density operator , the probability distribution of the measurement result is given by ${\displaystyle \rho }$

${\displaystyle P[B]=\operatorname {Spur} (\lambda _{A}(B)\,\rho )}$

with the lane class operator . ${\displaystyle \operatorname {Spur} }$

The expected value of the measurement result, i.e. the expected value of the probability distribution , is given by or by . ${\displaystyle P}$ ${\displaystyle \langle \Psi |A|\Psi \rangle }$ ${\displaystyle \operatorname {Spur} (A\,\rho )}$

In the special case that the spectrum of is discrete and simple, the possible measurement results are the eigenvalues of . The probability of finding the eigenvalue as a measurement result is then or , where denotes a normalized eigenvector for the eigenvalue . ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle |\langle \phi _{a}|\Psi \rangle |^{2}}$ ${\displaystyle \langle \phi _{a}|\rho \phi _{a}\rangle }$ ${\displaystyle \phi _{a}}$ ${\displaystyle a}$

Examples:

• The observable “place” of a particle in one dimension corresponds (in position representation) to the multiplication operator with over the Lebesgue space , the position operator . ${\displaystyle x}$ ${\displaystyle L^{2}(\mathbb {R} )}$
• The observable “momentum” of a particle in one dimension corresponds (in position representation) to the differential operator over ; more precisely its self-adjoint continuation , the momentum operator . Here the reduced Planck quantum of action denotes . ${\displaystyle -\mathrm {i} \hbar {\tfrac {\mathrm {d} }{\mathrm {d} x}}}$ ${\displaystyle L^{2}(\mathbb {R} )}$ ${\displaystyle \hbar }$
• The observable “energy” corresponds to the Hamilton operator .

Description by POVM

→ Main article : Positive Operator Valued Probability Measure (POVM)

The description of time measurements does not fit into the traditional von Neumann formalism, e.g. B. the arrival time of a particle in a detector . A more precise, realistic formal modeling of real experiments shows that most real measurements on quantum systems are not exactly described by von Neumann observables. The more general description of quantum mechanical observables by POVM eliminates these defects.

See also

• Complementary observables
• Complete set of commuting observables