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Introduction to quantum field theory/Exercise ABJRC

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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]\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] \mathbf{J}\Psi_{n\ell j}^{m} [/math] can be written as a linear combination of [math]\Psi_{n\ell j}^{m^{\prime\prime}}[/math] involving all possible values of [math]m^{\prime\prime}[/math] (and fixed [math]n, \ell,\text{ and } j[/math]), show that

[math]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]\mathbf{B} = B\hat{z},[/math] [math]\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]


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