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I've been reading through various materials on relativistic quantum mechanics, but I find the lack of simple examples disturbing.

I'm acquainted with the general form the solutions to the Dirac equations have, but I'm having trouble just practically getting any specific example solution.

Since the motion of a free Dirac particle is entirely determined with four-momentum $p^\mu$ and the spin (polarization) four-vector $s^\mu$, how exactly does one find the wave-function corresponding to some given $p^\mu,s^\mu$?

E.g. $p^\mu=m\{\sqrt{2},0,0,1\}, s^\mu=\{1,0,0,\sqrt{2}\}$

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I suppose that you mean by the spin four vector the Pauli-Lubanski vector $s^{\mu} = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma}J_{\nu \rho} P_{\sigma}$, and this is the reason that you chose it to be orthogonal to the momentum vector. – David Bar Moshe May 15 '12 at 15:48
up vote 1 down vote accepted

Start with a plane-wave at rest with its spin pointing in the z-direction:

$\psi_L=\psi_R=\left(\begin{array}{c} 1\\ 0\end{array}\right)$

First rotate the spin to the required direction.

$\psi_L \rightarrow \exp\left(-\tfrac12 i\theta_i\,\sigma^i\right)\psi_L ~~~~~~~~~~~~ \psi_R \rightarrow \exp\left(-\tfrac12 i\theta_i\,\sigma^i\right)\psi_R$

(Find the Euler angles $\theta_i$ by transforming the spin-vector to the rest-frame)

Now boost the rotated spinors in the right direction.

$\psi_L \rightarrow \exp\left(-\tfrac12\vartheta_i\,\sigma^i\right)\psi_L ~~~~~~~~~~~~ \psi_R \rightarrow \exp\left(+\tfrac12\vartheta_i\,\sigma^i\right)\psi_R$

(Calculate the rapidity $\vartheta_i$ from the momentum)

Finally: Note that you can expand:

$\exp(i\phi_i\sigma^i) \longrightarrow \cos|\phi|+i \phi_i\sigma^i \sin|\phi|\,/\,|\phi|$

to get the matrices out of the exponent's argument


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