Principles of Quantum Mechanics · Problem 1.3.4

Mathematical Introduction — R. Shankar

Physics — Quantum Mechanics · triangle-inequality

Problem

Prove the triangle inequality starting with

|V+W|^2

. You must use

\text{Re}\langle V|W\rangle \leq |\langle V|W\rangle|

and the Schwarz inequality. Show that the final inequality becomes an equality only if

|V\rangle = a|W\rangle

where

a

is a real positive scalar.