January 27, 2005
Class review — 1/27
Hi Class,
The pre-flight for Tuesday will be posted Saturday, and the first homework set should be posted by Monday.
The lecture notes from Thursday are posted now to the website; I believe that there are no math errors, but please let me know if you find a problem.
Here is a summary from last class:
Hydrogen wavefunctions are products of a radial wavefunction and spherical harmonics. The wavefunction is completely specified by quantum numbers n,l,m, and the energy only depends on n. Only s-states have non-zero amplitude at the origin.
note: It does look like those 3-d images of the hydrogen wavefunctions are OK. The phase variation outside of the f coordinate is binary, and just represents sign changes.The fine-structure in an atom is due to relativistic corrections to the electronic wavefunction. There are three parts to fine structure: kinetic energy corrections, spin-orbit interaction, and the mysterious Darwin term. It's difficult to make sense of these terms without using the Dirac equation. The fine-structure Hamiltonian is diagonal with respect to the total angular momentum J, and, with fine-structure present, the energy of the atomic states depends on the quantum numbers n and j.
The degeneracy between Hydrogenic states with the same n and j is lifted by the Lamb shift (known as a radiative correction), which is a pure QED effect. The first measurement of the Lamb shift prompted the development of QED; current measurements are getting to the level where the effect of the proton charge distribution can be resolved. A (mostly) incorrect model from Welton can be used to conceptually understand the Lamb shift and estimate the splitting between the Hydrogen 2s1/2 and 2p1/2 states to within an order of magnitude.
See you Tuesday! Definitely bring a calculator; we will be calculating fine and hyperfine structure for the alkali atoms.
Brian
Posted by Brian at 02:30 PM | Comments (0)
January 25, 2005
Class Review — 1/25
Class,
In class on 1/25, we talked about:
The spherical basis.
Irreducible tensors. These are operators that transform like spherical harmonics under rotations. They have a rank k, which is like l, and components q, which are like m. We're mostly going to see vector operators in this class, which are rank 1 tensors. We'll also see rank 0 tensors, or scalars, wich are kind of boring.
The Wigner-Eckart theorem (WET). I regret bringing this up before we're going to use it for something. Regardless, it is extremely important in many areas of physics. It tells you that the matrix element (between states with angular momentum quantum numbers j and j') of the q component of a rank k vector is proportional to:
the Clebsch-Gordan coefficient for adding j and k to get j'
the reduced matrix element, which is independent of m, m', and q
We talked about selection rules based on the Clebsch-Gordan coefficient in the WET for vector operators. There was some confusion (caused by me) over the so-called "parity rule." This is indeed, as Shizhong stated, really a rule that comes from direct evaulation of the matrix element of a vector operator with respect to real-space angular momentum states. I will clear this up at the beginning of next class.
The projection theorem. Ellen — what I wrote down in class was correct. This tells you that matrix elements of any vector operator (for states of well-defined angular momentum) are proportional to the matrix elements of the angular momentum operator. We will see how this is useful in a homework problem on the Zeeman effect in Hydrogen.
Lecture notes on angular momentum and the WET are on the website now. I have cleaned up my notation, and standardized to Sakurai's notation (which is just as confusing as anyone else's notation). There are some good tricks in the lecture notes for evaluating matrix elements of tensor products.
Learn to ignore notation! It's a red herring that will prevent you from learning physics. In real life, no one uses the same notation for anything. We'll see in class that people in my field can not even agree to within a factor of 2 on the definition of the Rabi rate for transitions in a two-level system. Also, once your brain starts to decay as you age past 30, you also won't be able to remember the details behind subjects like irreducible tensors. Unless you're Julian Schwinger, you'll just remember the important ideas. Then you look up details in a book when you want to calculate something.
See you and Hydrogen tomorrow!
Brian DeMarco
Posted by Brian at 02:33 PM | Comments (0)
January 20, 2005
Class review — 1/20
Hi Class,
In class today, we spoke about:
AMO physicists are not consistent when it comes to units. That's OK, as long as you have a conversion table and some idea of what's going on in terms of the physics.
Angular momentum in quantum mechanics is all about commutation relations and rotations. Any operator that satisfies the right commutation relations will have eigenkets that rotate like vectors.
You can rotate quantum mechanical states using rotation operators. You just need the right matrix elements to write the rotated state in terms of the unrotated basis states. You can find the non-trivial matrix elements of the rotation operator on the Clebsch-Gordan handout.
Addition of two angular momenta J1 and J2 produces new states that are eigenkets of J2, J12, J22 and Jz. These states |j,m> can be written in terms of the |j1,m1> and |j2,m2> states using Clebsch-Gordan coefficients. The quantum number j takes values that differ by integers between j1+j2 and |j1-j2|. It's always true that m=m1+m2. This means that |j,m> is only made up of states |j1,m1> and |j2,m2> where m=m1+m2. I didn't emphasize this point or the notion of "stretched states" enough in class. We'll briefly go over those ideas on Tuesday. 3-j symbols are Clebsch-Gordan coefficients in disguise.
Three angular momenta can be added in different ways. The different representations that you end up with can be connected by 6-j symbols, which we will see in other contexts.
I flubbed the right explanation of what's happening is real space for the problem that we did at the end of class (the decay problem). We'll go over the right picture on Tuesday, as well as talk about the meaning of parity in that problem.
On Tuesday we'll continue with spherical tensors and the Wigner-Eckart theorem. I think that we'll also be able to get to Hydrogen.
I'll send another email when pre-flight set 3 are ready. Lecture notes will be on the website shortly.
Cheers,
Brian
Posted by Brian at 06:44 PM | Comments (0)
January 18, 2005
Class Review — 1/18
Hi class,
If you are planning to withdraw from this class (please don’t — we'll have fun, I promise!), please do it soon, especially since the class is already over-enrolled. I noticed that there are some of you who are enrolled but did not show up today. Remember to read the syllabus in case I forgot to mention something important in class today (or you weren't there).
There is a list of probable topics which will be covered on the main page of the website now.
Here are some potentially useful things to take away from class today:
Modern AMO physics exists because of the invention of the tunable laser.
AMO physics is published in a lot of different journals! The best way to search journals is to use ISI's Web of Science. JSTOR is also very useful for finding old journal articles.
The American Institute of Physics (AIP) is the place to go for statistics on physics. You can even call them!
It's hard to figure out what type of research different funding agencies are supporting. The best way to research this is by looking at where different groups get their moolah. Check out the class notes from today for a couple of links to sites that keep track of different AMO groups.
The National Academy and National Research Council write reports that Congress reads about different areas of science. These reports are important to look at (and you'll have to for the pre-flights for next week).
Remember to do the pre-flights before class on Thursday. The lecture notes on units will be posted after class; read the e-reserve from Bransden and Joachain on units before class (see the class notes section of the website for 1/20). We'll spend Thursday talking about atomic units and angular momentum in quantum mechanics.
See you on Thursday!
Brian
Posted by Brian at 02:30 PM | Comments (0)