# Can you calculate the energy needed to ionize an electron in hydrogen? [closed]

If I have an electron in the 2s orbital,it would have different energy with the 2p orbital.Can we calculate the difference in the energy ?

Because the single electron in the hydrogen atom feels a purely symmetric Coulomb potential, the states with different orbital angular momentum $\ell$ are degenerate in non-relativistic quantum mechanics and the binding energy of the electron is given by

$$E_{n,\ell}=-\frac{\mathcal{R}_\text{H}}{n^2},$$ where $\mathcal{R}_\text{H}$ is the mass-corrected Rydberg constant for the hydrogen atom and $n$ is the principal quantum number.

When Dirac introduced his relativistic theory for one-electron systems it was shown that the spin-orbit coupling in the hydrogen atom results in a splitting between the 2P$_{3/2}$ level on one hand and the 2S$_{1/2}$ and 2P$_{1/2}$ levels on the other hand (the latter two are degenerate). The subscript number indicates the total angular momentum $J=S+L$.

Later, Willis Lamb was able to measure a shift between the 2S$_{1/2}$ and 2P$_{1/2}$ levels that was not predicted by Dirac's theory and this splitting was explained by Hans Bethe (traveling back by train from the conference where he learned about Lamb's results) and marked the birth of quantum electrodynamics (QED).

Effect of Dirac theory and QED on the non-relativistic energy of 2S and 2P levels of atomic hydrogen (taken from H. C. Wolf H. Haken, The physics of atoms and quanta, 7 ed., Springer, 2005.).

To come back to your question, yes, it is possible to calculate the splitting between the electronic states of atomic hydrogen, however, it requires some advanced maths. If you are really interested in calculating it yourself, I recommend Quantum Mechanics of One- and Two-Electron Atoms by Bethe and Salperer.

• Thanks for the explaantion sir ! It would be out of my e Sep 25 '17 at 12:46