Commensurability (quantum mechanics)

In quantum mechanics , two observables are called commensurable if they can be measured simultaneously with any precision. Observables that cannot be measured with any precision at the same time are called incommensurable . Two observables are commensurable if and only if the commutator of the associated operator vanishes.

Incommensurable observables whose commutator takes on the value are called complementary observables . ${\displaystyle \mathrm {i} \hbar }$

proof

According to the (generalized) Heisenberg uncertainty principle, the following applies to two operators and a state for their measurement uncertainties or in the state : ${\displaystyle A,B}$ ${\displaystyle |\psi \rangle }$ ${\displaystyle \sigma _{A}}$ ${\displaystyle \sigma _{B}}$ ${\displaystyle |\psi \rangle }$

${\displaystyle \sigma _{A}\sigma _{B}\geq {\frac {1}{2}}{\Big |}\langle \psi |[A,B]|\psi \rangle {\Big |}}$

It follows from for any state . ${\displaystyle \sigma _{A}=\sigma _{B}=0}$ ${\displaystyle [A,B]=0}$

On the other hand, it follows that there is a set of common eigenstates for the operators and . By measuring one of the two variables , the state collapses to the corresponding eigenstate and is already in an eigenstate of the second operator, so that a measurement of the other variable does not change the system again. ${\displaystyle [A,B]=0}$ ${\displaystyle A}$ ${\displaystyle B}$

Examples

• The position and momentum of a particle in the same spatial direction are incommensurable and complementary, because the following applies: ${\displaystyle [x_{i},p_{j}]=\mathrm {i} \hbar \delta _{ij}}$
• Various components of angular momentum are incommensurable, but not complementary, because the following applies: ${\displaystyle [L_{i},L_{j}]=\mathrm {i} \hbar \varepsilon _{ijk}L_{k}}$
• Energy and momentum are commensurate, because the following applies: ${\displaystyle [H,p_{i}]=0}$

literature

• Torsten Fließbach: Quantum Mechanics . 4th edition. Spektrum, Munich 2005, ISBN 3-8274-1589-6 , pp. 115 .