How to Use Kets, the Hermitian Conjugate, and Bra-ket Notation
How to Use Operators for Quantities in Quantum Physics
How to Find the Hermitian Adjoint

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:

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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:

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Note in particular that any operator commutes with itself:

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

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

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And the Hermitian adjoint of a commutator works this way:

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

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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:

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The definition of commutators tells you the following:

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In accordance with the properties of adjoints,

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Therefore,

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But for Hermitian operators,

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But BA – AB is just

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so you have the following:

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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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