Riley hobson bence · Problem 19.9

Quantum operators

Mathematics — Methods · quantum-operators-BCH-commutator-expansion

Problem

Let

F(\lambda) = e^{\lambda A} B e^{-\lambda A}

where

A

and

B

are quantum operators. By showing that if

F^{(n)}

is the

n

th derivative of

F(\lambda)

then

F^{(n+1)} = \left[A, F^{(n)}\right]

, and using a Taylor series in

\lambda

to evaluate

F(1)

, show that the expression

e^{i\theta L_x} L_y e^{-i\theta L_x}

reduces to

\cos\theta\, L_y - \sin\theta\, L_z