Fermi · Concept Library · Courses · Books · Problems · Simulations · Math Wiki · High School Fermi › Problems Quantum Mechanics: Concepts and Applications · Problem 7.11
Identical Particles — Nouredine Zettili
Physics — Quantum Mechanics · pauli-matrices-rotations
Problem Using the commutation relations between the Pauli matrices, show that:
(a) e i α σ y σ x e − i α σ y = σ x cos ( 2 α ) + σ z sin ( 2 α ) e^{i\alpha\sigma_y}\sigma_x e^{-i\alpha\sigma_y} = \sigma_x\cos(2\alpha) + \sigma_z\sin(2\alpha) e i α σ y σ x e − i α σ y = σ x cos ( 2 α ) + σ z sin ( 2 α ) e i α σ z σ x e − i α σ z = σ x cos ( 2 α ) − σ y sin ( 2 α ) e^{i\alpha\sigma_z}\sigma_x e^{-i\alpha\sigma_z} = \sigma_x\cos(2\alpha) - \sigma_y\sin(2\alpha) e i α σ z σ x e − i α σ z = σ x cos ( 2 α ) − σ y sin ( 2 α ) e i α σ x σ y e − i α σ x = σ y cos ( 2 α ) − σ z sin ( 2 α ) e^{i\alpha\sigma_x}\sigma_y e^{-i\alpha\sigma_x} = \sigma_y\cos(2\alpha) - \sigma_z\sin(2\alpha) e i α σ x σ y e − i α σ x = σ y cos ( 2 α ) − σ z sin ( 2 α ) Browse the problem library