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I'm trying to get to Pauli's equation from Dirac's equation in the weak field regime. Specifically, if I substitute

[tex]\psi = \left(\begin{array}{cc}\chi \\ \varphi \end{array}\right)[/tex]

into the Dirac equation, I get two coupled equations

[tex]i\frac{\partial\chi}{\partial t} = (\sigma\cdot\pi)\varphi + (m + eA^{0})\chi[/tex]

[tex]i\frac{\partial\varphi}{\partial t} = (\sigma\cdot\pi)\chi + (m + eA^{0})\varphi[/tex]

where [itex]\pi = \boldsymbol{p} - e\boldsymbol{A}[/itex].

Substituting [itex]\chi = e^{-imt}X[/itex] and [itex]\varphi = e^{-imt}\Phi[/itex], we get

[tex]i\frac{\partial X}{\partial t} = (\sigma\cdot\pi)\Phi + eA^{0}X[/tex] ------- (*)

[tex]i\frac{\partial \Phi}{\partial t} = (\sigma\cdot\pi)X - (2m - eA^{0})\Phi[/tex]

In the weak field regime, [itex]2m >> eA^{0}[/itex], so the second of the last two equations becomes

[tex]i\frac{\partial \Phi}{\partial t} = (\sigma\cdot\pi)X - 2m\Phi[/tex] -------- (**)

Now, differentiating (**) wrt time to decouple (*) and (**) introduces a second derivative term in the 'almost Pauli' equation :tongue2:

[tex]i\frac{\partial^{2}\Phi}{\partial t^2} = (\sigma\cdot\pi)^2\Phi - 2im\frac{\partial \Phi}{\partial t}[/tex]

How does one get Pauli's equation from this?

Do Ialsohave to make an explicit nonrelativistic approximation:

[tex]E = \sqrt{p^2 + m^2} \approx m[/tex]

so that [itex]exp(-imt) = exp(-iEt)[/itex]

?

Thanks in advance.

Cheers

Vivek.

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# Pauli Eqn from Dirac Eqn?

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