References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 1.9.
Although the result in this post isn’t covered in Shankar’s book, it’s a result that is frequently used in quantum theory, so it’s worth including at this point.
We’ve seen how to define a function of an operator if that function can be expanded in a power series. A common operator function is the exponential:
Here we’ll look at a special function of two operators of the form
If , we can cancel the two exponentials and get the result . However, if the two exponentials must remain separated by the middle operator. To get a simpler form for this function, we’ll consider the auxiliary function
where is some parameter. We’ll need the first 3 derivatives at :
We can now write a Taylor expansion of 3 around :
Taking gives the required expansion
This is known as Hadamard’s lemma.
If we introduce the notation
and in general is the th order commutator of with , then we can write Hadamard’s lemma as