Unitary exponential sandwich
Posted by peeterjoot on September 27, 2010
One of the chapter II exercises in  involves a commutator exponential sandwich of the form
where is Hermitian. Asking about commutators on physicsforums I was told that such sandwiches (my term) preserve expectation values, and also have a Taylor series like expansion involving the repeated commutators. Let’s derive the commutator relationship.
Let’s expand a sandwich of this form in series, and shuffle the summation order so that we sum over all the index plane diagonals . That is
Assuming that these interior sums can be written as commutators, we’ll shortly have an induction exercise. Let’s write these out for a couple values of to get a feel for things.
This compares exactly to the double commutator:
And this compares exactly to the triple commutator
The induction pattern is clear. Let’s write the fold commutator as
and calculate this for the case to verify the induction hypothesis. We have
We now have to sum those binomial coefficients. I like the search and replace technique for this, picking two visibly distinct numbers for , and that are easy to manipulate without abstract confusion. How about , and . Using those we have
Straight text replacement of and with and respectively now gives the harder to follow, but more general identity
For our commutator we now have
That completes the inductive proof and allows us to write
Or, in explicit form
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.