Lecture Summary 01/31/2007
Hyperfine SplittingHyperfine splitting is the effect on the electron energy levels due to the proton's (small) dipole moment. This dipole moment establishes a weak magnetic field, which interacts with the magnetic field due to the electron's spin. This effect is different from spin-orbit coupling; spin-orbit effects are due to the interaction of the electron dipole moment with the magnetic field arising from the changing electric field seen in the electron's rest frame. Hyperfine splitting is caused by the direct interaction of the proton and electron magnetic moments.
We are particularly interested in the effect on the ground state. Abers uses a somewhat impenetrable tensor approach to this problem. Griffiths uses a much more intuitive approach (though I'm sure it sacrifices something that the business with the tensors does not -- probably some generality). I couldn't follow what Abers was doing in lecture, so I read chapter 6.5 in Griffiths.
Using first order perturbation theory, one obtains a formula for the correction to the energy that requires the computation of two expectation values (Griffiths, equation 6.87). However, in the ground state, or any state with l=0, the wave function is spherically symmetric. The first expectation value vanishes, and the second one can be computed with a simple triple integral in spherical coordinates.
From this, we obtain an equation for the first order correction to the energy that is proportional to the expectation value of the scalar product of the proton spin and the electron spin. Griffiths dubs this spin-spin coupling to contrast it with spin orbit coupling, which has a first order energy correction proportional to the expectation value of the electron spin and the electron orbital angular momentum.
It is important to examine the properties of this Hamiltonian. By inspection, one can see that that the orbital angular momentum, the proton spin and the electron spin all fail to commute with the Hamiltonian. However, an operator constructed from the sum of all three does commute. It is not difficult to see why; in the presence of spin-orbit coupling and spin-spin coupling, a rotation performed on the proton, the electron, or the electron orbit will each change the energy of the system.
In the ground state the orbit is spherically symmetric, so rotations of the orbital angular momentum will not change anything. So, in our consideration of the effect on the ground state, we can discard the terms relating to the orbital angular momentum. The operator that commutes with the Hamiltonian is S=Se+Sp. This operator implies two configurations for the proton and electron spins: In the "triplet" state, the spins are parallel, the total spin is 1, and the eigenvalue of S2 is two h-bar squared (Oh, how I wish I could use LaTeX in blog posts). In the "singlet" state, the total spin is 0, and the eigenvalue is 0.
Plugging these expectation values back into the first order correction for the energy, we can calculate the hyperfine splitting of the ground state energy level. The triplet state elevates the energy by +1/4 (and some constants), and the singlet state depresses the energy by -3/4 (and some constants).
All electrons and all protons have an intrinsic spin, and thus a magnetic moment. Thus, all hydrogen atoms exhibit this feature. The exact degenerate ground state never, in fact, occurs. All ground-state hydrogen atoms are either in the singlet state or the triplet state. Because the triplet state has an elevated energy, there is a nonzero probability that it will decay into the singlet state. Taking the energy released by this transition and (by dividing by Plank's constant) calculating the frequency of a photon with this energy, you will find that the frequency is 1420 MHz, or a wavelength of 21 cm. This is the "21-centimeter line," the most abundant form of electromagnetic radiation in the universe. This band of microwave emission is one of the principle diagnostic tools of radio astronomy.
Multi-electron SystemsWe then set aside our beloved hydrogen atom and began examining multi-electron systems. The zero-order approach is to ignore both magnetic effects and the electrostatic interaction among the electrons. One simply adds the Hamiltonians for each electron together. We used this method to calculate the (very approximate) ground state energies of helium and a few helium-like ions. This rough approach seems to come withing about 25% of the observed energies.
To do better, we're going to need a new approximation technique. We'll save that for Monday, though.