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](https://www.dummies.com/wp-content/uploads/397674.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](https://www.dummies.com/wp-content/uploads/397675.image1.png)
Note in particular that any operator commutes with itself:
![image2.png](https://www.dummies.com/wp-content/uploads/397676.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](https://www.dummies.com/wp-content/uploads/397677.image3.png)
It’s also true that commutators are linear— that is,
![image4.png](https://www.dummies.com/wp-content/uploads/397678.image4.png)
And the Hermitian adjoint of a commutator works this way:
![image5.png](https://www.dummies.com/wp-content/uploads/397679.image5.png)
You can also find the anticommutator, {A, B}:
![image6.png](https://www.dummies.com/wp-content/uploads/397680.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](https://www.dummies.com/wp-content/uploads/397681.image7.png)
The definition of commutators tells you the following:
![image8.png](https://www.dummies.com/wp-content/uploads/397682.image8.png)
In accordance with the properties of adjoints,
![image9.png](https://www.dummies.com/wp-content/uploads/397683.image9.png)
Therefore,
![image10.png](https://www.dummies.com/wp-content/uploads/397684.image10.png)
But for Hermitian operators,
![image11.png](https://www.dummies.com/wp-content/uploads/397685.image11.png)
But BA – AB is just
![image12.png](https://www.dummies.com/wp-content/uploads/397686.image12.png)
so you have the following:
![image13.png](https://www.dummies.com/wp-content/uploads/397687.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.)