Principles of Quantum Mechanics · Problem 20.8

The Dirac Equation — R. Shankar

Physics — Quantum Mechanics · dirac-equation-energy-levels

Problem

The energy levels of the hydrogen atom according to the Dirac equation are given by Enj=mc2[1+(αn−(j+12)+(j+12)2−α2)2]−1/2E_{nj} = mc^2\left[1 + \left(\frac{\alpha}{n - (j+\frac{1}{2}) + \sqrt{(j+\frac{1}{2})^2 - \alpha^2}}\right)^2\right]^{-1/2} where α=e2/(4πϵ0ℏc)\alpha = e^2/(4\pi\epsilon_0 \hbar c) is the fine-structure constant. Expand this expression in powers of α\alpha and show that the leading correction to the rest mass energy gives the Bohr energy levels, and that the next correction gives the fine-structure splitting.

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