Two hermitian commutator anticommut {A,B}=AB+BA=0

Thread starter dustu
Start date Oct 5, 2013
Tags

Commutator Hermitian

In summary, the commutator of two Hermitian operators, represented by {A,B}, measures how much two operators do not commute with each other. This has significant implications in quantum mechanics, as seen in the commutator anticommutator relation {A,B}=AB+BA=0. This relation affects the measurement of physical quantities, as it restricts the precision with which certain pairs of observables can be measured. The commutator anticommutator relation can also be extended to more than two operators, and is closely related to the symmetry of a physical system.

Oct 5, 2013

dustu

Two hermitian commutator anticommute: {A,B}=AB+BA=0.Is it possible to have a simultaneous eigenket of A and B?illustrate...
Thank you in advance

Physics news on Phys.org

Oct 5, 2013

Jazzdude

Sounds like homework?

Cheers,

Jazz

EDIT: Anyway, just assume that |v> is an eigenvector to A and B, then apply the anticommutator to it and draw your conclusions.

Last edited: Oct 5, 2013

1. What does the commutator of two Hermitian operators mean?

The commutator of two Hermitian operators, represented by the notation {A,B}, is a mathematical operation that measures how much two operators do not commute with each other. In simpler terms, it quantifies the extent to which the order in which two operators act on a given state affects the end result.

2. What is the significance of the commutator anticommutator relation in quantum mechanics?

The commutator anticommutator relation, {A,B}=AB+BA=0, is a fundamental property of quantum mechanics that reflects the non-commutative nature of quantum operators. It is used to define important concepts such as uncertainty principles and quantum entanglement.

3. How does the commutator anticommutator relation affect the measurement of physical quantities?

The commutator anticommutator relation has a direct impact on the measurement of physical quantities in quantum mechanics. It implies that certain pairs of observables, such as position and momentum, cannot be measured simultaneously with arbitrary precision. This is known as the Heisenberg uncertainty principle.

4. Can the commutator anticommutator relation be generalized to more than two operators?

Yes, the commutator anticommutator relation can be extended to more than two operators. For example, the commutator of three operators A, B, and C can be written as {A,B,C}=ABC+BCA+CAB-ACB-BAC-CBA=0.

5. How is the commutator anticommutator relation related to the symmetry of a physical system?

The commutator anticommutator relation is closely linked to the symmetry of a physical system. In particular, if two operators commute with each other, it implies that they share a common set of eigenvectors. This leads to conserved quantities in a system, which can be interpreted as symmetries of the system.