Skip to main content

The Lamb Shift and Science Communications

Detail View of the Lamb Shift Apparatus from [5]
Note to the reader  The following was intended to be a summary of the Lamb shift experiment and the associated QED theory.  Instead, I was so impressed by Lamb, Retherford, and Bethe's communication abilities that while I point out the very abstract highlights of both the experiment and the theory, I get a little carried away with lauding the accomplishments of Lamb, Retherford, and Bethe.

In quantum mechanics II today we studied the Lamb shift. The Dirac equation predicts that the S and P electron states of an atom such as hydrogen should have an accidental degeneracy, (an identical predicted energy state of the atom's electron for two different state of the atom-electron system). While several researchers tried to measure the energy of the two levels to determine if there were in fact degeneracies using optical techniques, they were unsuccessful.  In 1947 Willis Lamb and his graduate student Robert Retherford performed an experiment[3][4][5][6] that showed the theoretically predicted degeneracy did not in fact exist.

The experiment, (apparatus shown above in picture 1), made use of microwave technology developed during World War II to get around the need for optical spectroscopy.  It turns out that there is a metastable excited state of the hydrogen atom for n equal to 2 and the orbital angular momentum S, state, (denoted 2S), that takes around one seventh of a second to decay back to the ground state for the atom. It also turns out that hydrogen atoms in this state will induce a current in a metal plate while atoms in other states, say the 2P state will not.  Finally, 2S state atoms can be forced to a 2P state where they will quickly decay to the ground state by exposing them to a combination of microwave energy photons, and a magnetic field of a specific strength.

Lamb and Retherford built their experiment around these phenomena.  They sent a beam of 2S hydrogen atoms towards a metal plate detector.  A sensitive galvanometer registered an electric current caused by the 2S atoms striking the plate.  Between the source of the hydrogen beam and detector, they exposed the beam to microwave energy and a magnetic field.  At specific frequencies and field strengths  the microwave/magnet combination caused 2S to 2P transitions in the hydrogen atoms and the galvanometer current was reduced, (quenched). These combinations of microwave frequency and magnetic field strength could be calculated given an assumed energy of the hydrogen atom in the 2S state.  Since Dirac had predicted a degeneracy between the 2S and 2P states, they calculated the field/frequency combinations using the known energy of the 2P state.  The calculated values are shown as solid lines in the graph below, (picture 2), from [3].  By recording the field and frequency values where the beam was actually quenched, Lamb and Retherford were able to plot the open circles shown in the graph, and construct a set of experimental lines that corresponded to the theoretical ones, but with the energy of the 2S state shifted down by 1040 MHz.  In doing this, they showed experimentally that the 2S and 2P energy levels were in fact not degenerate.

What do you do with this information once you have it?  Lamb traveled to the Shelter Island Conference on Quantum Mechanics, (a short train ride from Columbia on the LIRR), and presented the results.  We lived near Shelter Island for a few years while we were working at Brookhaven National Laboratory.  The island is right across the aptly named Shelter Island Sound from the Blue Duck Bakery, (picture 3), famous for, (what else?), their blue duck cookies.

View Larger Map

Present at the meeting were such quantum mechanics dignitaries as Schwinger, Bethe, and Feynman.  Reading Bethe's article[1] on his initial Lamb shift equations, it seems like they had been sitting on a new quantum electrodynamics method for awhile without any experimental data to try it out on when Lamb turned up at the conference.  On the train ride home from the conference, (having lived there, one imagines Bethe riding the LIRR back towards New York), he ran a quick calculation to see if he got the same results measured by Lamb and Retherford.

Bethe compliments Lamb and Retherford on having a beautiful experiment, but his article in turn is a masterpiece of physics communication.  In three pages, Bethe introduces the reader to QED giving them only the initial formula that was needed for his calculation while making the semi-quantum mechanics savvy feel very comfortable without delving too far into the details.  It's difficult to emphasize what kind of accomplishment this is.  It seems that Bethe had chosen the entire physics community as his audience, not just the guys  on Shelter Island that were already familiar with all the methods.  It also seems that he was intent on making his audience feel comfortable with his presentation as well as the fact that he was ushering them into a new type of physics that had just matched with experimental results quite nicely.

First, he explains that an energy shift of this type had been expected as a result of the electron interacting with electromagnetic fields present in the vacuum, in other words, the electron in the hydrogen atom is interacting with zero point energy, the same fields we calculate in my Casimir research.  However, he emphasizes that the associated calculations that determined how the electron would be effected by vacuum fields had an infinite divergence associated with the self-energy of the electron and as a result had been mostly ignored.  Bethe gently and concisely walks the reader through the issue and his proposed solution.  He points out that the linear diverging term in the calculations exists for both bound electrons, (think hydrogen atoms), and free electrons, (electrons propagating through space).  Judging this to mean that the divergence was simply a property of the electron, he drops the term altogether.  This leaves a term that diverges logarithmically.  He then simply points out that the leading term for the electron in the hole theory, (Dirac's theory), is logarithmically divergent, and shows how it can also be ignored leaving a convergent equation.  The remaining terms show that the electron should only be effected by vacuum electromagnetic fluctuations up to a frequency that corresponds to the rest energy of the electron itself.  All frequencies above that one can be ignored.  A few short and easy approximations later, Bethe arrives at a level of shift of 1000 MHz, differing by a mere 40 out of a 1000 MHz compared to the experimental result!

In summary, if you want to see some of the best, most complete, most easy to read physics articles around, check out references [1][5], and [6] from Bethe and Lamb and Retherford.  What makes the articles great is that they embrace and include the reader who obviously was supposed to be anyone in the physics community circa 1950.  Bethe's theory article presents rather deep QED issues as easy to understand background, and then postulating the correctness of a single formula he runs through a handful of easy calculations and arrives at a result that is within 4% of experimental results.  The tone of the article might be compared to a good old boy from Texas figuring out how many feet of barbed wire he was going to need to rebuild the fence.  The Lamb and Retherford articles present each technical issue of the experiment along with an associated experimental design, and measurements of the validity of the design, as well as any additional issues they ran into while implementing the design and how they addressed them.  All three articles are masterpieces of simple language directed at what they considered to be a reasonably intelligent and very broad audience.


1.  Bethe paper with rough theory of Lamb shift
Bethe H. (1947). The Electromagnetic Shift of Energy Levels, Physical Review, 72 (4) 339-341. DOI: 

2.  Relativistic corrections to the rough theory
Baranger M., Bethe H. & Feynman R. (1953). Relativistic Correction to the Lamb Shift, Physical Review, 92 (2) 482-501. DOI: 

3.  The Lamb experiment (free from the +American Physical Society )
Lamb W. & Retherford R. (1947). Fine Structure of the Hydrogen Atom by a Microwave Method, Physical Review, 72 (3) 241-243. DOI: 

4.  Lamb paper that describes the experiment in detail including the detector
Lamb W. & Retherford R. (1950). Fine Structure of the Hydrogen Atom. Part I, Physical Review, 79 (4) 549-572. DOI: 

5.  Fine Strucutre of the Hydrogen Atom Part I
Lamb W. & Retherford R. (1950). Fine Structure of the Hydrogen Atom. Part I, Physical Review, 79 (4) 549-572. DOI:

6.  Fine Structure of the Hydrogen Atom Part II
Lamb W. & Retherford R. (1951). Fine Structure of the Hydrogen Atom. Part II, Physical Review, 81 (2) 222-232. DOI:


Popular posts from this blog

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

Now available as a Kindle ebook for 99 cents! Get a spiffy ebook, and fund more physics
The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differe…

Division: Distributing the Work

Our unschooling math comes in bits and pieces.  The oldest kid here, seven year-old No. 1 loves math problems, so math moves along pretty fast for her.  Here’s how she arrived at the distributive property recently.  Tldr; it came about only because she needed it.
“Give me a math problem!” No. 1 asked Mom-person.

“OK, what’s 18 divided by 2?  But, you’re going to have to do it as you walk.  You and Dad need to head out.”

And so, No. 1 and I found ourselves headed out on our mini-adventure with a new math problem to discuss.

One looked at the ceiling of the library lost in thought as we walked.  She glanced down at her fingers for a moment.  “Is it six?”

“I don’t know, let’s see,” I hedged.  “What’s two times six?  Is it eighteen?”

One looked at me hopefully heading back into her mental math.

I needed to visit the restroom before we left, so I hurried her calculation along.  “What’s two times five?”

I got a grin, and another look indicating she was thinking about that one.

I flashed eac…

The Javascript Google URL Shortener Client API

I was working with the Google API Javascript Client this week to shorten the URLs of Google static maps generated by my ham radio QSL mapper. The client interface provided by Google is very useful. It took me a while to work through some of the less clear documentation, so I thought I'd add a few notes that would have helped me here. First, you only need to authenticate your application to the url shortener application if you want to track statistics on your shortened urls. If you just want the shortened URL, you don't need to worry about this. The worst part for me was that the smaple code only showed how to get a long url from an already shortened rul. If you follow the doucmentaiotn on the insert method, (the method for getting a shortened url from a long one), there is a reference to a rather nebulous Url resource required argument. It's not at all clear how to create one of these in Javascript. The following example code shows how:
var request = gapi.clie…