# Exact energies of spherical harmonic oscillator in Dirac equation

The potential is given by: $$V(r) = {1\over 2} \omega^2 r^2$$ and we are solving the radial Dirac equation (in atomic units): $$c{d P(r)\over d r} + c {\kappa\over r} P(r) + Q(r) (V(r)-2mc^2) = E Q(r)$$ $$-c{d Q(r)\over d r} + c {\kappa\over r} Q(r) + P(r) V(r) = E P(r)$$ What is the analytic expression for the eigenvalues $E$ in atomic units?

It is ok to provide source code (any language) to obtain it if one needs to solve some simple analytic equation. Here are the (I think correct) energies from my numerical code that I would like to compare against analytic solution (for $c = 137.03599907$ and $\omega=1$):

  n  l  k kappa        E

1  0  0 -1       1.49999501
2  0  0 -1       3.49989517
2  1  0 -2       2.49997504
2  1  1  1       2.49993510
3  0  0 -1       5.49971547
3  1  0 -2       4.49983527
3  1  1  1       4.49979534
3  2  0 -3       3.49994176
3  2  1  2       3.49987520
4  0  0 -1       7.49945592
4  1  0 -2       6.49961564
4  1  1  1       6.49957571
4  2  0 -3       5.49976206


If one only solves the radial Schroedinger equation, then the analytic formula is $$E_{nl} = \omega (2n - l - {1\over 2})$$ I am looking for the relativistic version.

I found for example the paper: Qiang Wen-Chao: Bound states of the Klein-Gordon and Dirac equations for scalar and vector harmonic oscillator potentials. Vol 11, No 8, 2002, Chin. Phys. Soc., but it only shows a formula for nonzero scalar and vector potentials in Dirac equation (above we only have the scalar potential, the vector potential is zero).

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## 1 Answer

Have you tried to search "Dirac Oscillator"? there's Moshinsky and Szczepaniak article ...

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Yes. The article you linked does not provide a formula for the energies. –  Ondřej Čertík Aug 30 '12 at 6:11
My apologies --- they do provide an energy, the equation 14a, 14b. Would you be willing to edit your answer to include these? If it reproduces the correct energies, then I would like to give you the bounty. –  Ondřej Čertík Aug 30 '12 at 6:18
I've implemented their formula, see my script: gist.github.com/3523473, unfortunately, it doesn't agree with my numbers above (I am quite sure that my numerical numbers are correct). @nate, any ideas? –  Ondřej Čertík Aug 30 '12 at 6:50
Since nobody gave a better answer, the bounty is yours. But my question is not answered, until I find a formula that gives the correct energies against my numerical code. –  Ondřej Čertík Aug 30 '12 at 20:50
@Ondřej Čertík, I'm sorry for not replying earlier, I'm out of town and just flew for more than 20 hours to vacation... I'd check articles that cite the Moshinsky and Szczepaniak article... –  bla Aug 31 '12 at 14:08