I am really struggling with something fundamental.

I keep coming across two versions of the hamiltonian for spin orbit coupling:

$H_{soc}=\frac{\mu_B}{2c^2}(v \times E) \cdot \sigma $

$\mu_B =$ bohr magnetron

$v =$ velocity

$E = $ Electric Field

$\sigma = $ pauli matrices


$H_{soc} = \alpha L\cdot S$

$\alpha =$ constant

$L = $ Orbital Angular Momentum

$S = $ Spin angular momentum

Are these equivalent? If not what situations are they referring to.


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