In this post, I will use the stationary(time-independent) first order perturbation theory, to find out the relativistic correction to the Energy of the nth state of an Hydrogen Atom.

In order to find out the relativistic correction to the Energy, we would need to consider and use relativistic relations.

The relativistic Kinetic Energy is given as:

The first term in the above equation is the total energy of a relativistic particle and the second term is the rest mass energy of a particle. So we get the Kinetic energy by subtracting those.

Now, using the binomial expansion on the Kinetic energy we can write it as:

Note Binomial Expansion:

If we ignore the rest of the higher order terms and take only the first three terms of the Binomial expansion, then

We can now see that the Kinetic Energy is actually modified and not just as in the classical case. Since the second term would be very small due to in the denominator, we can take it as a perturbation, and use the time-independent perturbation theory to find out the correction to the energy levels.

Let perturbation,

Then the first order energy correction to the nth level is given as:

From Schrodinger’s Equation:

Using the above relation,

From Virial Theroem for Hydrogen atom, we know that the expectation value of V:

So it all boils down to finding the expectation value of .

To do that we would need to use the following relations, from the Harmonic Oscillator:

So we need to find the expectation value of for H-atom.

We already did this in this post, using the Hellmann-Feynman Theorem and found that:

Substituting this back in equation

We know that,

and,

Therefore,

If you have any questions or doubts regarding the above proof, feel free to post a comment down below.

Ph.D. researcher at Friedrich-Schiller University Jena, Germany. I’m a physicist specializing in computational material science. I write efficient codes for simulating light-matter interactions at atomic scales. I like to develop Physics, DFT, and Machine Learning related apps and software from time to time. Can code in most of the popular languages. I like to share my knowledge in Physics and applications using this Blog and a YouTube channel.

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The correction seems to have a unit problem inside the parentheses as one term is dimensionless and the other (3En) is not — that problem happened in the 2nd to last step.

The final answer appear to have an error, because one of the terms in the parentheses is dimensionless and the other is not. The problems appears in the last few lines.

As Dor Ben-Amotz said, the final answer is wrong. For those who need the correct answer, it should have the E_n outside the brackets be squared and the -3E_n should be replaced by simply -3. (page 298 of Griffith’s introduction to Quantum Mechanics 3rd edition)