In Dirac solution for hydrogen atom, the energy levels are calculated as positive

\begin{equation} E=\frac{mc^{2}}{R(t)\sqrt{1+\frac{z^{2}\alpha^{2}}{\left(n+\sqrt{\left(j+\frac{1}{2}\right)^{2}-z^{2}\alpha^{2}}\right)^{2}}}} \end{equation}

, while in Bohr's model the energy levels are negative

\begin{equation} E=\frac{-Ze^{2}}{8\pi\epsilon_{0}r} \end{equation}

How are these two related to each other?

  • $\begingroup$ Why the downvote? The answer wasn't obvious to me at first glance so it seems a perfectly reasonable question. $\endgroup$ – John Rennie Sep 21 '15 at 11:44

The $E$ in your expression is the quantity calculated using the operator $i\hbar\frac{\partial}{\partial t}$, so it is the total energy. As $n \rightarrow \infty$ this energy $E$ goes to the rest energy of the electron $m_ec^2$ as we'd expect. For finite $n$ the energy is lower than $m_ec^2$ with the difference being the binding energy of the electron.

To compare the Dirac $E$ with the Bohr energies just subtract off $m_ec^2$.


I have found the answer myself here

Energy in Dirac model $E_d$ is related to energy in Bohr's model $E_b$ as

$E_b \approx E_d - m_ec^2$

where $m_e$ is mass of electron and $c$ is speed of light. The answer above is not useful.

  • $\begingroup$ Exactly in what paragraph in the link you find this relation? $\endgroup$ – anna v Sep 21 '15 at 8:17
  • $\begingroup$ Just look for sub heading Energy levels. Anyway John has also confirmed above the same thing. $\endgroup$ – amateurRebel Sep 21 '15 at 13:41

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