Physical meaning of a commutator?

In summary, a commutator is a mathematical formula used to check if two operators commute or not. If the result is zero, then the observables can be measured simultaneously with a definite value. However, if they do not commute, then the observables cannot be measured with infinite precision and must obey the uncertainty principle. Additionally, if an operator commutes with the Hamiltonian, its expectation value will remain constant.
  • #1
Fellowroot
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I know how to use a commutator as a mathematical formula but I really don't understand what it means. Can anyone explain it to me.

Is a commutator nothing more than a check to see if it commutes or not since I know that if you use a commutator with a wave function and the result equals zero then you can have a definite value for the observables (operators) simultaneously?



Thanks.
 
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  • #2
https://www.physicsforums.com/showthread.php?t=131975

If two operators do not commute, then they cannot be measured simultaneously with infinite precision - the pair of observables has to obey Heisenberg's uncertainty principle.

If an operator commutes with the Hamiltonian, then its expectation value is constant.
 

1. What is a commutator in physics?

A commutator in physics is a mathematical operation used to describe the relationship between two physical quantities. It is represented by the symbol [A, B], where A and B are operators representing physical quantities.

2. What is the physical meaning of a commutator?

The physical meaning of a commutator is that it represents the degree to which two physical quantities "commute" or can be measured simultaneously. The value of the commutator indicates whether the two quantities are compatible or if there is uncertainty in their measurement.

3. How is a commutator used in quantum mechanics?

In quantum mechanics, the commutator is used to calculate the uncertainty in the measurement of two observables. If the commutator is zero, the observables can be measured simultaneously with no uncertainty. If the commutator is non-zero, there is inherent uncertainty in the measurement of the observables.

4. What is the difference between a commutator and an anti-commutator?

A commutator and an anti-commutator are both mathematical operations used in physics, but they have different properties. A commutator is defined as [A, B] = AB - BA, while an anti-commutator is defined as {A, B} = AB + BA. In quantum mechanics, the commutator is used to calculate uncertainty, while the anti-commutator is used to calculate expectation values.

5. How does the commutator relate to Heisenberg's uncertainty principle?

The commutator is directly related to Heisenberg's uncertainty principle, which states that there is a fundamental limit to the precision with which certain pairs of physical quantities can be known simultaneously. The commutator quantifies this uncertainty and shows that the more non-zero the commutator is, the greater the uncertainty in the measurement of the corresponding physical quantities.

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