# What is the relation between energy levels of hydrogen atom in Bohr's solution to that of Dirac solution

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?

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

## 2 Answers

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.

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