# Why does the Dirac equation work for the hydrogen atom?

The Dirac equation works well for predicting the spectrum of the hydrogen atom, famously incorporating relativistic effects like fine structure. Yet, there seems to be a sense in which this is accidental. Specifically, let me quote the following passage from Sidney Coleman's recently published lecture notes, where he argues against the usage of the Dirac equation as a single-particle wave-equation (as opposed to a quantum field theoretic description):

"As a general conclusion, the corrections of relativistic kinematics and corrections from multi-particle intermediate states are comparable; relativity forces you to consider many-body problems. There are however very special cases, due to the specific dynamics involved, where the kinematic effects of relativity are considerably larger than the effects of pair states. One of these is the hydrogen atom. That's why Dirac's theory gives excellent results to order $$(v/c)^2$$ for the hydrogen atom, even without considering pair production and multi-particle intermediate states. This is a fluke." - Quantum Field Theory, Lectures of Sidney Coleman (Page 1).

Of course the hydrogen atom is very special, being integrable, and being one of the few exactly solvable systems in quantum mechanics.

But physically, what exactly is it about the hydrogen atom which makes the single-particle Dirac equation work, where we expect it fails for generic relativistic systems? What is the fluke here?

More generally, are there generic physical conditions under which we can expect relativistic results to be faithfully represented by simplified single-body techniques?

• Perhaps you should provide a link to these lecture notes. – StephenG May 31 at 10:47
• @StephenG They're not freely available, but here is the publisher's website. You can also hear him talking about this in his lectures posted on YouTube here around the 10 minute mark (beware the video quality isn't great). He really doesn't go into anymore detail than what I quote here. – EuYu May 31 at 10:54
• A "fluke" is not a physical qualification. The argument needs to be more detailed. Also what is true for hydrogen is true for all atoms, molecules, materials. That makes it a very common "fluke". – my2cts May 31 at 11:07
• Thanks for the link. Even if you don't have a free source for the material, it's better to give a link (or at least a reference) as it may prove useful to others who find your question. – StephenG May 31 at 12:18
• Related: physics.stackexchange.com/q/65359. That question also asks about the "fluke" mentioned in Sidney Coleman's lectures on Quantum Field Theory. – Chiral Anomaly May 31 at 23:01

• @euyu . "I am not aware of any work on non-hydrogenic atoms which can be obtained without including quantum field theoretic effects" There is the large research field of relativistic quantum chemistry based on the many electron Dirac equation. Only small effects such a Lamb shifts require radiative and virtual pair formation corrections. The no pair approximation gives an error of order $(Z\alpha) ^3$. – my2cts May 31 at 13:53