Roger's Equations

Last time in Real and Virtual Particles - Part I, I discussed how the energy time uncertainty relation allows nature to circumvent conservation of energy for very short periods of time. I also discussed how particles that exist for only short periods of time have too large an energy uncertainty to be considered particles in the traditional sense. Today I'd like to discuss a measurable consequence of virtual particles, the Lamb Shift.

The Hydrogen Atom

What the hydrogen atom lacks in complexity it makes up for with its precision. Since the hydrogen atom is a relatively simple system, we can solve it with quantum mechanics to a precision we can't achieve with larger systems. This precision allows us to discern more subtle effects that might otherwise be lost in the "noise" of approximation. As we all know, the hydrogen atom consists of a proton and an electron.

Of course in quantum mechanics, you can't know the exact location of the electron. Actually a more accurate diagram of the hydrogen atom displays the probability of the electron about the proton nucleus as can be seen in the two diagrams below.

Diagram 1-Angular solutions for the hydrogen atom. Theses solutions are Spherical Harmonics.

Diagram 2-Radial Solutions for the Hydrogen Atom (Probability of Finding the electron at particular distances from the nucleus, 1s maximum is located at bohr radius)

What you see above are pictures of the angular and radial electron wavefunctions derived from the Hamiltonian of the hydrogen atom which is a separatable differential equation that yeilds spherical harmonics as the solution to the angular components and Laguerre polynomials as the solution to the radial part. Don't worry if you don't know what that means, all I'm trying to say is that quantum mechanics allows you to solve for the wavefunctions of the electron in the hydrogen atom exactly. Since the modulus squared of the wavefunction is the probability of finding the electron at that point, essentially the diagrams above shows the probability distribution of the electron in the Hydrogen atoms at different excitations. 1s being the ground state of the electron, the lowest energy state in the hydrogen atom. 2s and 2p are excited states, having the same energy as each other which is higher than the 1s groundstate, but with the electron configured in different ways (When the energies of two different wavefunctions are equal, they are called degenerate).

To feel more comfortable that this abstract concept of probabilty distributions actually represents nature accurately, take a look at diagram 2. In it you'll see a curve labelled 1s that represents the probability distribution of finding the electron a certain radial distance from the nucleus. You'll notice that the curve has a sharp peak, this peak corresponds to the bohr radius. In other words, the highest probable location of an electron is the ground state (1s) is at the bohr radius.

The Lamb Shift - 2s and 2p

In our discussion of the lamb shift, lets not worry about the spherical harmonics and concentrate on the Languerre polynomials which detail how each probability distribution varies as we move outward from the nucleus. In the calculations done to get these wavefunctions, the energy for each is calculated and we find that 1s has the lowest energy and 2s and 2p share the same energy and are higher in energy than the 1s state.

However, it can be shown experimentally that the 2p state is actually ever so slightly lower in energy (4 μeV) than the 2s state which seems to contradict the results above. To understand why, we first need to look at that the radial probability distribution of the 2s orbital as opposed to the 2p orbital. What we find is that the 2s orbital has a small hump in its probability distribution close to the nucleus whereas the 2p does not. That means that an electron in 2s gets closer to the nucleus than an electron in 2p. But why would that make its higher energy? If anything, when a proton and electron are closer together their attraction should lower the energy of the system, not increase it. The answer, essentially, is that the electron "feels" the electromagnetic force less effectively so close to the nucleus. I really like how the website Hyperphysics explains it:

"The "self-interaction" of the electron when it is near the proton causes the effective "smearing" of the electron charge so that its attraction to the proton is slightly weaker than it otherwise would have been. This means it has encountered an interaction which makes it slightly less tightly bound than a 2p electron, hence higher in energy."

So what is meant by "self interaction" and why does it occur more frequently closer to the proton (hydrogen nucleus)?

When an Electron isn't an Electron - Virtual Particles

Thanks to the heisenberg uncertainty principle, the conservation of energy can be violated for short periods of time. This allows for an electron to self interact. This occurs by the electron emitting and absorbing virtual photons. Sometimes an electron will emit a virtual photon and then absorb it. Sometimes the electron with emit a virtual photon that becomes a positron-electron pair, which annihilate and become a virtual photon which is absorbed back by the electron. The complexity has no limit (though generally the greater the complexity, the lower the probability of it occuring).

I while back I did a blog entry on The Principle of Least Action - Feynman's Many Paths. In it I discuss how the uncertainty relation causes an infinite number of paths from one momment in time to the next for a particle or system. Each possible path must be included and weighted (by its likelyhood) to calculate the overall observed effect. That is exactly what is occuring here. The electron, thanks to natures sneaky energy-time uncertainty relation, can be transformed into different types of particles in many different ways. The trick is it all has to happen quickly enough so that the energy-time uncertainty relation can work around conservation of energy. What this means is that when we think of an electron, we shouldn't just think of a plain old particle but instead say that an electron is actually the sum of all the possible self-interactions weighted appropriately by their likelyhood of occuring.

What's more, the self interaction of the electon can change slightly in different environments, like when it's close to a proton. This change is measurable and is the reason why two hydrogen levels that should be equal, the 2s and 2p, are not. It results in a shift in a small shift in the hydrogen spectrum that corresponds to this energy difference between those to levels. That shift in the spectrum is called the Lamb shift, named after Willis Lamb who recieved the Nobel Prize in Physics in 1955 for measuring it.

When that shift is measured, essentially what is really being measured is how the electron has changed by being in proximity of the proton nucleus. This change, which is very small and precise has been predicted successfully by quantum electrodynamics using virtual particles. The lamb shift is compelling support for the existence of virtual particles.

Living in a World of Virtual Particles

It's scary at first to picture a universe in constant flux filled with seething virtual particles willfully violating conservation of energy and making the world unnecessarily complicated. But maybe that's the point. The classical interpretation of physics in comforting and intuitive, as long as you are willing to not look to close.

But once you do decide to take a closer look, the universe turns out to be much more complicated than we could ever have imagined.

That's all for now. Thanks for reading.

Useful links:

http://www.sciam.com/askexpert_question.cfm?articleID=0004D0F8-772A-1526-B72A83414B7F0000
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/lamb.html
http://en.wikipedia.org/wiki/Lamb_shift