How to Derive Pauli's Equation from Dirac's Equation in a Weak Field?

[maverick280857]

Hi,

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

$$\psi = \left(\begin{array}{cc}\chi \\ \varphi \end{array}\right)$$

into the Dirac equation, I get two coupled equations

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

where $\pi = \boldsymbol{p} - e\boldsymbol{A}$.

Substituting $\chi = e^{-imt}X$ and $\varphi = e^{-imt}\Phi$, we get

$$i\frac{\partial X}{\partial t} = (\sigma\cdot\pi)\Phi + eA^{0}X$$ ------- (*)
$$i\frac{\partial \Phi}{\partial t} = (\sigma\cdot\pi)X - (2m - eA^{0})\Phi$$

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

$$i\frac{\partial \Phi}{\partial t} = (\sigma\cdot\pi)X - 2m\Phi$$ -------- (**)

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

$$i\frac{\partial^{2}\Phi}{\partial t^2} = (\sigma\cdot\pi)^2\Phi - 2im\frac{\partial \Phi}{\partial t}$$

How does one get Pauli's equation from this?

Do I also have to make an explicit nonrelativistic approximation:

$$E = \sqrt{p^2 + m^2} \approx m$$

so that $exp(-imt) = exp(-iEt)$

?

Thanks in advance.

Cheers
Vivek.

 

[maverick280857]

Anyone?

 

[Maxim Zh]

In the equation (**) you should also neglect the first derivative term:

$$0 = (\sigma\cdot\pi)X - 2m\Phi$$ ------- (**')

See also:
L. D. Landau and E. M. Lifgarbagez, Course of Theoretical Physics,
Vol. 4 Quantum Electrodynamics, section 33

 

[maverick280857]

In the equation (**) you should also neglect the first derivative term:

$$0 = (\sigma\cdot\pi)X - 2m\Phi$$ ------- (**')

See also:
L. D. Landau and E. M. Lifgarbagez, Course of Theoretical Physics,
Vol. 4 Quantum Electrodynamics, section 33

Thanks!