References: Mark Srednicki, Quantum Field Theory, (Cambridge University Press, 2007) – Chapter 1, Problem 1.3.
The number operator is defined as
Applied to a quantum state, it counts the number of particles in that state:
Another property of is that it commutes with any other operator that contains an equal number of creation and annihilation operators. To see this, look at the individual commutators as follows (where ).
Here we’ve used the commutation relations
Now suppose we have an operator which contains creation operators , and annihiliation operators , :
We can see that the commutator in the last line can be worked out recursively until we’ve processed all the creation operators up to , giving
The last commutator gives us
So if (the numbers of creation and annihiliation operators are equal), the operator commutes with . In particular, the hamiltonian we met last time satisfies this criterion, so and this hamiltonian conserves particle numbers.