Required math: calculus
Required physics: Schrödinger equation
References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Chapter 3, Post 13.
The commutator of two operators is defined as
In general, a commutator is non-zero, since the order in which we apply operators can make a difference. In practice, to work out a commutator we need to apply it to a test function , so that we really need to work out and then remove the test function to see the result. This is because many operators, such as the momentum, involve taking the derivative.
We’ll now have a look at a few theorems involving commutators.
Proof: The LHS is:
The RHS is:
where is the momentum operator.
Proof: Using and letting the commutator operate on some arbitrary function :
Removing the function gives the result . QED.
Again, letting the commutator operate on a function :
Removing gives the result . QED.