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.
Spin-Orbit CouplingToday'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 CorrectionThis 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 AtomTaking 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.
I ran over with a bunch of people to help him, and we found that his head was OK (he hit the hoses, not the pump itself), but that he had excruciating pain in his thigh. My immediate thought was that he had broken his hip. He wanted help to stand up, but couldn't, even with help. I told him that it was best if he just tried not to move, and to call the paramedics. His hip might not be broken, I told him, but that if it was, standing up -- even with our help -- could make it much worse.
He refused to let us call the paramedics, and insisted that we help him get back into his car. We obliged, as gently as we could. It was a very difficult; he was taller than me, with a sturdy build, and seemed to be made entirely out of bone and gristly muscle. He started sweating from the pain, and winced whenever he moved his legs.
I brought his card to the attendant, and filled up his tank for him. This took a few minutes, and distracted him enough to get him to calm down. He seemed to be in perfect command of his faculties, but was deeply embarrassed and frustrated. The pain seemed to subside a little, but every time he tried to move his legs back into his car, he would howl with agony. He wanted to drive home, so I asked him, "Even if you can control the car, what will you do once you get there? How will you get out of the car? Do you have stairs? Is there anyone at home who can carry you, if you fell again?"
He said he would drive to the hospital. So, I told him that if he's going to to the hospital, he may as well have the paramedics take him. He again refused to let us call an ambulance.
Two guys from a nearby auto body shop came over, and offered to drive him to he hospital. He agreed. We then lifted him out of the driver's seat, and placed him in the back seat where he could lie down. It took six people to move him, but we were able to support him by his shoulders and back, and moved his legs as little as possible. I was very worried about this, but he said that we didn't cause any more of the sharp pains he felt when he tried to get up the first time.
I warned the guys from that body shop that UCLA charges insane parking fees, and there was a good chance that even if he is discharged today, his family might not be able to arrange to retrieve his car from the hospital for some time. Some years ago, a good friend of mine had to pay a huge parking fee for his father's car after an extended hospital stay. I think that's a rotten thing for hospitals to do to people. Fortunately, the man lives nearby, so the guys from the body shop said that if his family will come and pick him up from the hospital, they would drive his car back to his house. This seemed to be a great relief to the guy, and he busied himself with his seatbelt.
I'm not sure if this was the right thing to do. As heroic as those fellows are, I still feel that we should have called the paramedics immediately. However, as long as he seemed to be in control of his faculties, I couldn't justify overruling his wishes. Should I have?
Couldn't Bush have at least spared a few empty platitudes for New Orleans in his State of the Union Address?
Then, we looked at the Zeeman effect (neglecting spin effects) using perturbation theory. I think perturbation theory is becoming less perturbing to me.
In other news, someone crashed into Abers' car while it was parked. Evidently, the person who hit his car also hit a couple of other parked cars, and then abandoned their vehicle. I saw something like this happen once a few years ago. The police thought the guy was drunk, but he came up negative on the breath test, and he looked more surprised than inebriated. It turned out that the driver had undiagnosed diabetes, and had passed out at the wheel.
Today Abers covered sections 7.1.1, 7.1.2 and 7.2.1 in his book. They are as follows.
Introduction to bound-state perturbation theoryGenerally, one uses bound-state perturbation theory to solve problems that don't have exact solutions, but are very close to problems that do have exact solutions. If you can represent something as a problem with an exact solution plus some small wiggle, you can use perturbation theory. For example, the infinite square well potential has an exact solution. The infinite square well with a lump at the bottom probably doesn't, but is probably amenable to perturbation theory.
The process goes like this :
- Write the Hamiltonian as the sum of the unperturbed Hamiltonian and some H', which is small (we hope).
- Note that the presence of the H' will change the energy eigenvalue, hopefully by some small amount. Call this delta.
- We want to solve for delta. It is usually not possible to do this directly. However, we can get a zeroth-order approximation of delta by taking the expectation value of the perturbed Hamiltonian H' with the unperturbed eigenstate. The error will be one order higher.
- To get the higher order terms, we need to expand the state vector (the state vector, not the unperturbed state vector) in powers of H' (or its matrix elements, if that makes any sense). Equation 7.8 in Abers does the job.
- Successive iterations of the process yields successfully higher order terms.
- Care must be given to the normalization of the perturbed eigenstates; different books (e.g. Griffiths) use slightly different conventions.
- There must be a one-to-one correspondence between eigenstates of the unperturbed Hamiltonian and the eigenstates of the full Hamiltonian.
- There is a second restriction, explained incoherently in the book and mysteriously in today's lecture, that has something to do with how one handles degeneracies.
Perturbation of the first excited state of HydrogenWe went through the derivation of the energy splitting due to the presence of a weak electric field (the Stark effect). There was much confusion resulting from the change in bases required to avoid dividing by zero when evaluating the formula for obtaining the first order terms. Abers noted that many textbooks treat "degenerate" perturbation theory separately, and that this was silly because all you had to do was switch into a basis that diagonalizes H', find the higher order terms, and then switch back into the original basis.
However, by the time we had returned from our basis-changing digression, it was rather unclear what had happened. Fortunately, it seemed to work, and we calculated something for the energy shift in terms of the energy eigenstates.
To solve for the shift directly, we integrated over all space in spherical coordinates. As usual, the spherical harmonics are separable, and the result for the perturbed radial wave function is -3*sqrt(3)a, where a is the Bohr radius. The energy shift is -3*e*E*a.
I will be back in the Spring.
But the minimum wage should be the same everywhere: $0. Labor is a commodity; governments make messes when they decree commodities' prices. Washington, which has its hands full delivering the mail and defending the shores, should let the market do well what Washington does poorly. But that is a good idea whose time will never come again.Really? I mean, Really? Why does Mr. Will pine for the good old days of the 19th century? There is really only one coherent coherent criticism of the Federal minimum wage that has survived scrutiny; that it is a floor without a ceiling, and therefor helps to institutionalize inflation. However, the minimum wage is so low that the aggregate purchasing power of all people at or near the minimum plays and insignificant role in inflation -- and even raising it to $7.25 won't change this. Although fewer in number, the wealthy command much greater aggregate purchasing power, and thus play a larger role in inflation. If we really treated labor as a commodity, then we should also have a Federal maximum wage.
But of course, this is America. Every economic policy needs to be attached to some sort of appeal to the emotions. This is exactly what George Will is doing; his is encouraging the reader to join him in waxing nostalgic for the simpler days of buggy whips and gas lamps, before everything got so damn political.
Evidently, poor Mr. Will was denied a proper education. The decent and proper thing to do would be to take up a collection so he can attend a few introductory history and economics classes at his local community college.
I've never liked fast food; it doesn't taste good, it's unhealthy, the vegetarian selection sucks, and the places are loud, tacky and uncomfortable. The food and sauces have so much salt that instead of feeling full, I end up feeling like a beached whale bloating in the sun, and still hungry.
Now I have one more reason to hate them. They buy produce farmed and prepared so inexpertly that they can't keep keep the shit off of it. I've worked on a farm myself, and I find it astonishing that in 2007, we have problems like this. Running a farm is difficult, complicated work, but making sure the food is safe to eat is one of the easiest parts of the job. It involves very simple and inexpensive decisions like making sure not to pour liquid pig shit on the ground directly uphill from the where you grow the onions.
Aren't you glad that George Bush has gutted the USDA? Don't worry. Industry can inspect itself.