Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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

share|improve this question
1  
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
add comment

2 Answers

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


Hans

share|improve this answer
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.