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:
If is hermitian, the exponential is unitary. If we try to calculate the exponential of two operators such as , the result isn’t as simple as we might hope if and don’t commute. To see the problem, we can write this out as a power series
The problem appears first in the fourth term in the series, since we can’t condense the sum into if . In fact, the expansion of can be written entirely in terms of the commutators of and with each other, nested to increasingly higher levels. This formula is known as the Baker-Campbell-Hausdorff formula. Up to the fourth order commutator, the BCH formula gives
There is no known closed form expression for this result. However, an important special case that occurs frequently in quantum theory is the case where , where is a complex scalar and is the usual identity matrix. Since commutes with all operators, all terms from the third order upwards are zero, and we have
We can prove this result as follows. Start with the operator function
where is a scalar parameter (not necessarily time!).
From its definition,
The inverse is
and the derivative is
Note that we have to keep the factor to the left of the factor because . Now we multiply:
We used Hadamard’s lemma in the penultimate line, which in this case reduces to
because so all higher order commutators are zero.
We end up with an expression in which has disappeared. This gives the differential equation for :
We try a solution of the form (this apparently appears from divine inspiration):
From which we get
Comparing this to 18, we have
Setting this equal to the original definition of in 7 and then taking we have
If we swap with and use the fact that , and also , we have
This is the restricted form of the BCH formula for the case where is a scalar.