Dirac Equation for H atom - what's the small r behaviour?

[petergreat]

The Schrödinger wavefunction for the hydrogen atom scales as r^l for small r, where l is the orbital angular momentum. Is this changed in any dramatic way for the Dirac equation wavefuction? Does the small component of the Dirac spinor have the same small-r asymptotic behaviour as the large component? Can someone tell me the small r behaviour for the 1S j=1/2 state and 2P j=1/2,3/2 states? Thanks in advance.

 

[ytuab]

The Schrödinger wavefunction for the hydrogen atom scales as r^l for small r, where l is the orbital angular momentum. Is this changed in any dramatic way for the Dirac equation wavefuction? Does the small component of the Dirac spinor have the same small-r asymptotic behaviour as the large component? Can someone tell me the small r behaviour for the j=1/2, n=1 state and j=1/2,3/2, n=2 states? Thanks in advance.

According to this book (Advanced Quantum mechanics by J.J. Sakurai), the 2 upper component of Dirac spinor (4 x 1) for hydrogen atom gives very similar results to those of Schrödinger's hydrogen atom. (Of course, the angular momentums are the same, too.)
(So the "radial" boundary conditions are from zero to infinity in both Schrödinger and Dirac's hydrogens. )
This means in both cases, the divergence scales or signs of "tangential" and "radial" kinetic energies in small r are almost same only in the 2 upper components .

But unfortunately, there are no more comments about the 2 lower components of the Dirac spinor for hydrogen atom in this book.
(As far as I know, this book explains this interpretation (of Dirac hydrogen) in more detail than other books.)

Of course, the lower components of Dirac spinor also have angular momentums, and are incorporated into the upper kinetic energy term. So these are indispensable.

 

[petergreat]

I want to find a table that enumerates the explicit solutions of the lowest energy states, but didn't find any... unlike the Schrödinger case which I can find almost anywhere.

 

[dextercioby]

Have you tried to look into Bethe & Salpeter's book ? I heard it's very good. It might contain what you're looking for.

 

[Bill_K]

The full solution for the Dirac hydrogen atom can be found in "Relativistic Wave Mechanics" by Corinaldesi and Strocchi, pp202-206.

 

[petergreat]

Thanks! I'll check the book.
By the way, I asked this question because I was reading about how an electron electric dipole moment, if it exists and is sufficiently large, would affect the atomic spectrum in an external electric field. It turns out that the Schrödinger equation predicts no change in energy level to first order, and one has to use the Dirac equation to take relativistic effects into account to compute the energy splitting.