Finding wave-fuctions of a Dirac particle for given 4-momentum and spin 4-vector

Question

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

Answer 1

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

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

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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)

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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)

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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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Hans

Answer 2

By using the notation that p = {E,P} c=hbar = m = 1, m^2 = E^2-P^2 , so your initial condition is wrong. But nevermind that, use solve the equations by superposition of 4 independent solutions for the free particle at rest solution of the dirac equation. Than boosting with boost operator. You can also use the solution for particle not at rest.Insert $ \psi = \psi\left( \begin{array}{c} u_A(p) \\u_B (P) \end{array} \right) exp(ip \cdot \frac{x}\hbar-i\frac{Et}\hbar)$

to the Dirac equation, and you get a diff equation for each index of u which is a spin vector