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

• petergreat
In summary, the Schrodinger wavefunction for the hydrogen atom scales as r^l for small r, where l is the orbital angular momentum. This does not change dramatically for the Dirac equation wavefunction, as the upper component of the Dirac spinor gives similar results to the Schrodinger case. However, there is limited information on the lower components of the Dirac spinor. The full solution for the Dirac hydrogen atom can be found in "Relativistic Wave Mechanics" by Corinaldesi and Strocchi. This information is relevant for understanding the effects of an electron electric dipole moment on the atomic spectrum.
petergreat
The Schrodinger 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.

Last edited:
petergreat said:
The Schrodinger 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 Schrodinger's hydrogen atom. (Of course, the angular momentums are the same, too.)
(So the "radial" boundary conditions are from zero to infinity in both Schrodinger 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.

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

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

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

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 Schrodinger 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.

## 1. What is the Dirac equation for the hydrogen atom?

The Dirac equation is a relativistic wave equation that describes the behavior of a free spin-half particle, such as an electron, in a quantum field. For the hydrogen atom, the Dirac equation can be solved to determine the energy levels and wave functions of the electron.

## 2. How does the Dirac equation differ from the Schrödinger equation for the hydrogen atom?

The Dirac equation takes into account the electron's spin and relativistic effects, while the Schrödinger equation does not. This makes the Dirac equation more accurate for describing the behavior of electrons in high energy states.

## 3. What is meant by "small r behavior" in the context of the Dirac equation for the hydrogen atom?

Small r behavior refers to the behavior of the electron at very small distances from the nucleus, where the electron-nucleus interaction is strongest. This behavior is important for understanding the stability and structure of the atom.

## 4. How does the Dirac equation predict the small r behavior of the hydrogen atom?

The Dirac equation predicts that at very small distances, the electron will have a non-zero probability of being found inside the nucleus. This is in contrast to the Schrödinger equation, which predicts that the electron's probability of being found inside the nucleus is zero.

## 5. What implications does the small r behavior of the hydrogen atom have for our understanding of the atom?

The small r behavior of the hydrogen atom has important implications for our understanding of the atom's stability and structure. It also allows us to make more accurate predictions about the behavior of electrons in high energy states, which has applications in fields such as nuclear physics and quantum computing.

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