Quantum Mechanics: Concepts and Applications · Problem 7.5

Identical Particles — Nouredine Zettili

Physics — Quantum Mechanics · rotation-operator

Problem

The operator corresponding to a rotation of angle θ\theta about an axis n⃗\vec{n} is given by Un⃗(θ)=e−iθn⃗⋅J^/ℏU_{\vec{n}}(\theta) = e^{-i\theta\vec{n}\cdot\hat{J}/\hbar}. Show that the matrix elements of the position operator R^\hat{R} are rotated through an infinitesimal rotation like R⃗′=R⃗+θn⃗×R⃗\vec{R}' = \vec{R} + \theta\vec{n}\times\vec{R} (i.e., in the case where θ\theta is infinitesimal, show that Un+(θ)R^jUn(θ)=R^j+θ(n⃗×R^)jU_n^+(\theta)\hat{R}_j U_n(\theta) = \hat{R}_j + \theta(\vec{n}\times\hat{R})_j).

Related theory

Browse the problem library