Principles of Quantum Mechanics · Problem 1.9.1

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · functions-of-operators

Problem

We know that the series

f(x) = \sum_{n=0}^{\infty} x^n

may be equated to the function

f(x) = (1-x)^{-1}

|x| < 1

. By going to the eigenbasis, examine when the

q

number power series

f(\Omega) = \sum_{n=0}^{\infty} \Omega^n

of a Hermitian operator

\Omega

may be identified with

(1 - \Omega)^{-1}