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

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