Complementary observables

In quantum mechanics, the complementarity of two measurable quantities ( observables ) is understood to mean the property that the operators belonging to the associated observables have a commutator that takes on the value . The reduced Planck quantum denotes . For two complementary operators and therefore applies: ${\displaystyle \pm \mathrm {i} \hbar }$ ${\displaystyle \hbar }$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle [A,B]=AB-BA=\pm \mathrm {i} \hbar }$

Due to the generalized Heisenberg uncertainty relation , it follows from this that both observables cannot be measured at the same time with arbitrary precision, but that for the variance of their measurement always

${\displaystyle \sigma _{A}\sigma _{B}\geq {\frac {\hbar }{2}}}$

applies. In particular, if the first variable is fully known , nothing at all can be said about the result of a quantum mechanical measurement of the second variable (all possible measurement results are equally likely).

A well-known pair of mutually complementary observables are the location and the pulse of an object. Since the classical trajectory is described by position and momentum, the complementarity of these two quantities means that the concept of classical orbital motion in quantum mechanics must be abandoned.

In this sense, the various components of the angular momentum are not complementary observables: they cannot be measured simultaneously either, but the commutator of the components of the angular momentum operator is not a number, but an operator itself. Quantities that cannot be measured with any precision at the same time, but that are not complementary, are called incommensurable .

Individual evidence

1. ↑ Torsten Fließbach: Quantum Mechanics . 4th edition. Elsevier, Munich 2005, p. 52 .