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Pauli Eqn from Dirac Eqn?

  1. Jan 18, 2010 #1
    Hi,

    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 I also have 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.
     
    Last edited: Jan 18, 2010
  2. jcsd
  3. Jan 19, 2010 #2
  4. Jan 19, 2010 #3
    In the equation (**) you should also neglect the first derivative term:

    [tex]
    0 = (\sigma\cdot\pi)X - 2m\Phi
    [/tex] ------- (**')

    See also:
    L. D. Landau and E. M. Lifgarbagez, Course of Theoretical Physics,
    Vol. 4 Quantum Electrodynamics, section 33
     
  5. Jan 19, 2010 #4
    Thanks!
     
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