Franz Schwabl Quantum Mechanics 4th Edit · Problem 2.5

One-Dimensional Problems

Physics — Quantum Mechanics · operator-identities-baker-hausdorff

Problem

(a) Show that

[A, B^n] = nB^{n-1}[A,B]

under the assumption

[[A,B],B]=0

. (b) Demonstrate the Baker–Hausdorff relation (2.44). Hint: Consider the Taylor expansion of the function

f(\lambda) = e^{\lambda A}Be^{-\lambda A}

. (c) Using (a), show that (2.47) holds under the conditions stated in the main text. Hint: Discuss the derivative of the function

e^{\lambda A}e^{\lambda B}

. (d) Assuming

\lambda \ll 1

, expand the operator expression

(A - \lambda B)^{-1}

in a power series in

\lambda