Advanced Quantum Mechanics Schwabl · Problem 8.1

The Foldy–Wouthuysen Transformation and Relativistic Corrections

Physics — Quantum Mechanics · dirac-equation-coulomb-pauli-spinors

Problem

Demonstrate the validity of the relation (σ⋅p)(σ⋅x^)f(r)φjmjℓ=−ir[r∂∂r+1±(j+12)]f(r)φjmjℓ.(\boldsymbol{\sigma} \cdot \mathbf{p})(\boldsymbol{\sigma} \cdot \hat{\mathbf{x}})f(r)\varphi^\ell_{jm_j} = -\frac{\mathrm{i}}{r}\left[r\frac{\partial}{\partial r} + 1 \pm \left(j + \frac{1}{2}\right)\right] f(r)\varphi^\ell_{jm_j}.

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