Introduction to quantum field theory/Exercise ABJRC

Introduction to quantum field theory

Grading assignment

To be submitted latest by 4 PM on Tue, Dec 12

The Lande g-factor is defined as the proportionality constant in the relationship: [math]\displaystyle{ \langle \Psi_{n\ell j}^{m'} \vert\mathbf{L} + g_{s} \mathbf{S} \vert\Psi_{n\ell j}^{m} \rangle = g_{n j \ell} \langle \Psi_{n\ell j}^{m'} \vert \mathbf{J} \vert \Psi_{n\ell j}^{m} \rangle\,. }[/math] Using the fact that [math]\displaystyle{ \mathbf{J}\Psi_{n\ell j}^{m} }[/math] can be written as a linear combination of [math]\displaystyle{ \Psi_{n\ell j}^{m^{\prime\prime}} }[/math] involving all possible values of [math]\displaystyle{ m^{\prime\prime} }[/math] (and fixed [math]\displaystyle{ n, \ell,\text{ and } j }[/math]), show that

[math]\displaystyle{ g_{n j \ell} \equiv g_{j \ell} = 1 + (g_s - 1)\left[ \dfrac{j(j+1) - \ell(\ell+1) + 3/4}{2j(j+1)} \right]. }[/math]

Hence prove that, for a constant magnetic field [math]\displaystyle{ \mathbf{B} = B\hat{z}, }[/math] [math]\displaystyle{ \langle \Psi_{n\ell j}^{m'} \vert \delta H \vert \Psi_{n\ell j}^{m} \rangle = \left( \frac{e\hbar g_{j\ell}B}{2 m_e c} \right)m\delta_{mm^{\prime}}.\\ }[/math]