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How to Find the Commutator of Operators

In quantum physics, the measure of how different it is to apply operator A and then B, versus B and then A, is called the operators’ commutator. Here’s how you define the commutator of operators A and B:

image0.png

Two operators commute with each other if their commutator is equal to zero. That is, it doesn’t make any difference in what order you apply them:

image1.png

Note in particular that any operator commutes with itself:

image2.png

And it’s easy to show that the commutator of A, B is the negative of the commutator of B, A:

image3.png

It’s also true that commutators are linear— that is,

image4.png

And the Hermitian adjoint of a commutator works this way:

image5.png

You can also find the anticommutator, {A, B}:

image6.png

Here’s another one: What can you say about the Hermitian adjoint of the commutator of two Hermitian operators? Here’s the answer. First, write the adjoint:

image7.png

The definition of commutators tells you the following:

image8.png

In accordance with the properties of adjoints,

image9.png

Therefore,

image10.png

But for Hermitian operators,

image11.png

But BA – AB is just

image12.png

so you have the following:

image13.png

A and B here are Hermitian operators. When you take the Hermitian adjoint of an expression and get the same thing back with a negative sign in front of it, the expression is called anti-Hermitian, so the commutator of two Hermitian operators is anti-Hermitian. (And by the way, the expectation value of an anti-Hermitian operator is guaranteed to be purely imaginary.)

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