1
$\begingroup$

The conserved 4-current is defined as

$j^\mu=\bar{\Psi} \gamma^\mu \Psi$

where $\Psi$ is the 4-component wave function. To get the wavefunctions, if we look at the Dirac orbital spinor solution for central potential

$ \psi=\left(\begin{array}{l} \chi \\ \phi \end{array}\right)=\left(\begin{array}{c} f(r) \mathscr{Y}_{j l m}(\theta, \phi) \\ i g(r) \mathscr{Y}_{j l^{\prime} m}(\theta, \phi) \end{array}\right)$ where $f(r)$ and $g(r)$ correspond to the large and small component, $\mathscr{Y}_{j l m}(\theta, \phi)$ is the spinor spherical harmonics.

I would like to know:

a. how to calculate the transition 4-current $j^\mu_{trans}=\bar{\Psi}_0 \gamma^\mu \Psi_n$, where $\Psi_0$ and $\Psi_n$ are the intial and final states from the above spinor solution.

b. how this could reduce to the non-relativisitic case $j^\mu_{trans}=(\rho,j)$, where $j=\frac{\hbar}{2mi}(\psi^*\nabla \psi -\psi\nabla \psi^*)$

$\endgroup$
0

1 Answer 1

0
$\begingroup$

I unearth this topic since it interests me. The idea of a transition current generating the photon electromagnetic field (both electric and magnetic field) instead of creation/annihilation operator (second quantization) has been proposed by Roger Boudet. He used this kind of transition current to describe the current source of the electromagnetic wave. This is far more stronger and simpler both in the principle and the mathematical framework than those of Fock space. That being said the notation used is a bit hard. More of that, using these transitions current, he was able to compute spontaneous emission, photoelectric effect and the 2 Lamb shifts. Note that it considered the second quantization as "artifice" to compute fields, as many physicists even Dirac himself...

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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