Hi,(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Pauli Eqn from Dirac Eqn?

Loading...

Similar Threads - Pauli Dirac | Date |
---|---|

I How is Graphene's Hamiltonian rotationally invariant? | Mar 2, 2017 |

I From Pauli spinors to Dirac spinors | Jan 23, 2017 |

A Dirac Spin Exchange Operator | Aug 15, 2016 |

I A question on Bose enhancement & Pauli blocking | Aug 7, 2016 |

Dirac Equation and Pauli Matrices | Apr 8, 2015 |

**Physics Forums - The Fusion of Science and Community**