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Lecture Summary 01/29/2007

Posted by Russell on January 29, 2007 at 3:13 p.m.

Spin-Orbit Coupling

Today's lecture covered the spin-orbit coupling of the hydrogen atom. This phenomenon is due to the fact that because the electron has an intrinsic spin, it also has a finite magnetic moment. Because the electron is in motion, the magnetic moment couples with the proton-electron electric field.

One can think of the situation this way; most of the electron's motion is dominated by the fact that it has an electric charge -e, but some small part of the motion is due to the fact that there is a little magnet bolted onto the charge. A magnet at rest in a constant electric field suffers no force, but a magnet in motion in an electric field (or a stationary magnet in a changing electric field) does suffer a force.

If we pretend that the electron is moving slowly enough (i.e., that it is perfectly non-relativistic), you can do a classical frame-of-reference change such that the electron is at rest, and the proton is moving. If the electron was moving with velocity v, then in the electron frame, the proton is moving with velocity -v. We can then use the Biot-Savart law to find the magnetic field seen by the electron rest frame. Taking the dot product of this magnetic field with the electron's magnetic moment gives us the magnetic energy.

Sort of, anyway. The actual observed magnetic energy is half of what one obtains by this means. The reason for this factor-of-two error is that we relied on classical assumptions when we transformed into the accelerated frame of reference of the electron. Using the correct transformation yields the right coefficients. However, the classical result is correct to within a constant factor, which can be determined by observation anyway. So. Ahem. Moving on.

We can now take the magnetic energy and treat it as a perturbation on the Hamiltonian for the elementary hydrogen problem. The usual process applies. The result, to first order, can be found in section 7.3.2 of Abers' book.

The Relativistic Kinetic Energy Correction

This result is still wrong. We actually made two erroneous classical assumptions. The matter of the accelerated reference frame has been addressed (albeit in a completely unsatisfactory way). The first erroneous assumption was that the electron orbital velocity is slow. In fact, it is roughly 1% the speed of light, and thus relativistic effects are first order corrections.

This requires replacing the classical Hamiltonian of the electron with the correct relativistic Hamiltonian. We can expand in powers of the momentum, and keeping the first and second order result. Again, we treat this using perturbation theory.

The Fine Structure of the Hydrogen Atom

Taking the results of the spin-orbit coupling and the relativistic kinetic energy corrections together gives us the fine structure correction to the spectrum of the hydrogen atom. The fine structure correction depends on n (the energy level) and j (the electron spin orientation), but not l (the orbital angular momentum). This is where the fine structure constant alpha gets its name.
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