April 19, 2005
Class review — 4/19
Hey Class,
Recall that Onur, Andrea, Man-Hong, and Xu Wang are giving talks on Thursday. I will provide snacks :) Also remember to put in a vote (via email) to me if you want an extra, bonus class on atomic collisions (and quantum mechanical scattering theory).
Today we talked about:
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Resonance fluorescence!
If we drive an atom with a strong laser field close to resonance, we see funky things:
The Mollow triplet - sidebands appear in the fluorescence spectrum.
Funny features in the correlation function g(2). Not only do we observe anti-bunching, but wiggles in g(2).
The dressed-state picture is useful for understanding the spectrum of the emitted light. In that picture, we have new states that are combination of the ground and excited atomic states and photon states with different occupation numbers. Spontaneous decay between the dressed states at different frequencies are possible because all of the dressed states have excited state and ground state character.
A radiative cascade picture is useful for understanding what's going on with g(2). In that picture, decay occurs between the bare states with the same photon number in the field (technically, this only makes sense for an atom in a cavity, where decay represents light that leaves the cavity). After a photon is emitted, you have to wait for Rabi oscillations to re-excite the atom. What you see in g(2) depends on when you look relative to the Rabi oscillation pi-time, and how fast the Rabi rate is compared with the spontaneous decay rate.
Cheers,
Brian
Posted by Brian at 06:54 PM | Comments (0)
April 15, 2005
Class Revew — 4/14
Class,
Make sure that you read this entire entry -- there is information on the talks you will be giving!
For your talks, plan on 13 minutes plus 5 minutes for questions. Do not plan on using the board -- instead, use Powerpoint or transparencies. I can provide you with transparencies of various kinds. I can also supply a laptop for use during your talk. Let me know in advance if you will be using Powerpoint.
I need to clean up the lecture notes on Sysiphus cooling before I post them. There will be no pre-flight for Monday, but I will post reading material on resonance fluorescence for those of you who are interested.
Yesterday in class we talked about:
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The Doppler limit, which is the limiting temperature (about 150 microK) for simple Doppler cooling. It comes from the competition between Doppler cooling and recoil heating.
So-called Sysiphus cooling (or polarization gradient cooling, or sub-Doppler cooling, or...). This is a tricky cooling process which relies on optical pumping among hyperfine ground states as an atom moves through the optical potentials produced by overlapping laser beams. Sysiphus cooling can produce a very large force for low-velocity atoms, which lets us cool to lower temperatures (few-10 microK)
Magneto-optic traps (MOTs). MOTs are the workhorses of atomic physics -- they let us trap and cool billions of atoms from a room temperature vapor. A MOT is an optical molasses with an added quadrupole magnetic field. The magnetic field adds a position dependence to the optical force using the Zeeman effect. The overall trap is characterized by a spring-constant and a damping rate.
Cheers,
Brian
Posted by Brian at 10:20 AM | Comments (0)
April 12, 2005
Class review — 4/12
Hey Class,
Don't forget that there is a pre-flight for Thursday. Don't do it at the last minute!
Today we talked about:
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Some issues brought up in the last class.
The spectrum of scattered light is equal to the driving light
The spectrum of spontaneous emission (what comes out of an atom if we start in an an excited state) is a Lorentzian with a width set by Gamma.
The fact that I dislike my old picture for why saturation occurs. I think it's just an effect related to damping in a two-level system.
A puzzle (based on the micromaser) related to quantum measurement.
Laser cooling. We saw how Doppler cooling works by creating velocity-dependent, or friction-like, force.
We also calculated the cooling rate for Doppler cooling, and we could see how it cannot work well for atoms with large initial velocities (something above a few-10 m/sec).
We talked about recoil heating, which occurs for any atom that is scattering light. The so-called "recoil limit" is around a few micro-Kelvin.
More next time!
Brian
Posted by Brian at 07:08 PM | Comments (0)
April 08, 2005
Class review — 4/7
Hey Class,
Remember that problem 2 is due on Tuesday.
There will be reading material for next class and pre-flights up by tomorrow afternoon.
A few questions came up in class; the answers follow. First: the scattered light for a single, two-level atom in free-space is always exactly at the frequency of the driving field (the spectrum is a delta-function). Second: one should not employ an intermediate state when arguing this (like I did, when I said that the atom was in the excited state). Hmmm... it seems as if there were more questions. Comment on this blog entry if you can think of them! I still haven't figured out what's going on with the entropy of the universe.
Yesterday we spoke about:
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We learned about Mollow's transformation, which tells us that an EM field in a coherent state interacting with an atom is equivalent to a classical field plus the quantum vacuum interacting with the atom.
We talked about "scattering" and the Optical Bloch Equations (OBEs). Thi is just a density matrix description of light scattering (laser incident on an atom and a detector measuring scattered light). We define the fluorescence rate as the decay rate times the excited state population. The fluorescence rate has a Lorentzian dependence on detuning, and saturates at high driving intensities (there are many ways of understanding that).
We saw spontaneous Raman scattering in the context of quantum jumps.
We ended with a summary of what we know about light and atoms now, and the different pictures that we have.
Cheers,
Brian
Posted by Brian at 10:27 AM | Comments (0)
April 01, 2005
Class Review — 3/31
Hey Class,
Remember: Homework 2, problem 1 is due 4/7 and problem 2 is due 4/11.
And: Class is cancelled on 4/5.
Lecture notes will be up on the website today or tomorrow. There will be no pre-flight for next class.
Yesterday in class we learned about:
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A quick answer to what happens inside of a simple laser (single-mode, gain medium has three levels). The number of photons in the laser mode increases dramatically at a "threshold" pumping rate. Well above that threshold, the number of photons in the mode is described by a Poisson distribution that has a larger variance than a coherent state.
The Jaynes-Cummings Hamiltonian. This Hamiltonian describes the interaction of an atom with a single mode of the quantum EM field. The J-C Hamiltonian has two terms, emission + |e> -> |g> and absorption + |g> -> |e>, which results in a Rabi rate that depends on the number of photons in the mode. We saw how micromaser experiments are a realization of the J-C Hamiltonian, and how one can observe "vacuum Rabi oscillations." Note that the J-C Hamiltonian does not describe atoms in free space.
Spontaneous emission, as described by the Wigner-Weisskopf theory (which is not pertubation theory). The W-W solution is a solution to Schrodinger's equation (quantum EM field + quantum atom), in the limit that our quantization volume gets very large. An atom in the excited state decays exponentially to the ground state with a rate that depends on the dipole matrix element squared.
Cheers,
Brian
Posted by Brian at 12:23 PM | Comments (0)
March 29, 2005
March 29 — class review;
Hi Class,
Casimir's wife's name was Josina Jonker; they were married in 1933.
Remember! No class on April 5, and the homework is due on April 7.
There is a pre-flight for Thursday.
We got embroiled in a discussion of g(2) at the end of class. Correlation functions are confusing, and I'll bring a concrete example to next class.
Today we talked about:
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Quantum States of light. We talked about Fock, or number, states. For these states, the expectation value of the electric field is 0, but the variance depends on n.
Coherent states. These states are produced by a laser (maybe) and classical currents (an antenna). Coherent states are a specific superposition of number states, where the probability distribution is Poisson-ian in the number state basis. Coherent states are also characterized by a single parameter that is a complex number. The expectation value of the field is finite, and shows classical-like behavior. That means that the electric field strength oscillates in time. What is not classical about coherent states is the extra quantum "fuzz" on the electric field, although the size of the fuzz as a fraction of the electric field strength shrinks as the expectation value of n grows.
We talked about the correlation function g(2) as something that distinguishes Fock from coherent states. More on this next time!
Cheers,
Brian
Posted by Brian at 04:16 PM | Comments (0)
March 18, 2005
Class Review — 3/17
Hey Class,
Lecture notes on stimulated Raman transitions / lambda-systems are up on the website now. Pre-flights for next class will go up some time next week.
Have a great Spring Break!
Yesterday we talked about:
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Quantization of the EM field. The procedure to turn the EM field into something qauntum mechanical is to first quantize the field and then second quantize the field. We use the Coulomb, or transverse, gauge for working with quantum EM fields (and in QED).
First quantization is purely classical, and just means writing down the fields as expansions in an othornormal basis set of functions. In class, we did this by considering the fields in free space and choosing a box as our quantization volume. We're going to care about the vector potential A when we deal with light interacting with an atom, so we write down the wave equation for A and solve it; E and B are just derivatives of A.
Then we write down the total energy in the field, and that's our classical Hamiltonian. Cleverly, we can invent real valued coordinates Q and P and show that this Hamiltonian is just a simple harmonic oscillator Hamiltonian in Q and P. The fields are made quantum mechanical (this is second quantization) by the usual procedure — the coordinates are made operators and we introduce raising and lowering operators. The Hamiltonian is then diagonal in the mode occupation basis, or photon number basis.
At this point, the vacuum energy shows up. This is real, and hard to understand. The vacuum energy has physical consequences, such as in the Casimir and Casimir-Polder effects.
Cheers,
Brian
Posted by Brian at 09:23 AM | Comments (0)
March 15, 2005
3/15 — Class Review
Hey Class,
Pre-flights for next class are up!
Today we talked about:
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Stimulated Raman transitions. We can drive Rabi oscillations between two ground states in a three-state system by applying light at two different frequencies. Adiabatic elimination is an approximation that lets us ignore the excited states. More or less, the beat frequency from the two beams drives the transition between ground states.
Recoil. When atoms undergo stimulated Raman transitions, they recoil. This can be worked out from the semi-classical theory, but a photon picture becomes more useful. Recoil makes stimulated Raman transitions really useful for making atom interferometers.
Cheers,
Brian
Posted by Brian at 03:12 PM | Comments (0)
March 11, 2005
Class review — 3/10
Hey Class,
There is no pre-flight for Tuesday — we are playing pre-flight catchup. Lecture notes will be posted sometime today or tomorrow.
Remember! You need to tell me something about your project by March 15!
In the summary below (check the blog), there is a comment on Matt White's comment on the so-called "dipole approximation".
Yesterday in class we talked about:
The AMO secret handshake — what AMO physicists really mean by polarization. We talk about light having three polarizations: right-circular, left-circular, and pi.
To calculate how light really couples ground and excited electronic states, we have to calculate matrix elements of r. There is a big long formula for working out the angular-momentum part of that matrix element, but most AMO people use handy charts. The reduced matrix element for r must be measured, and is on the order of ea0.
So-called "lin perp lin" lattices work because when light is far detuned from a resonance (but not too far compared with the fine structure), the potential experienced by atoms is polarization-dependent (it's also state-dependent).
We can see clean, direct Rabi oscillations between ground and excited states (without spontaneous emission) if we use a "forbidden" transition. We saw an example in 40Ca+.
Matt White made a comment about electric quadrupole transitions being possible in our semi-classical picture if we expand the operator:
to higher order. I think that he is basically right. But, expanding that operator to higher order (or just not expanding it at all, which gives you recoil) is basically admitting to photons. We'll see that this operator is intimately connected to the idea of a photon.
Cheers,
Brian
Posted by Brian at 10:14 AM | Comments (0)
March 08, 2005
Class review — 3/8
Hey Class,
The pre-flight for Thursday is ready. Remember that you need to tell me something about your final project soon!
Today we talked about some new things:
Dipole traps. A focused laser beam can trap atoms! The trap depth and trapping frequencies are functions of the light intensity and detuning. A far-detuned trap can confine all Zeeman levels in a hyperfine ground state manifold, unlike a magnetic trap. Traps can be red-detuned (focused beam) or blue-detuned (focused beam with a dark spot in the middle).
Optical lattices. One type of optical lattice is formed from a standing wave of light. Lattices can be 1, 2, or 3 dimensional.
Cheers,
Brian
Posted by Brian at 06:23 PM | Comments (0)
March 03, 2005
Class review — 3/3
Hey Class,
Boy am I tired. Expect many updates to the website tomorrow, like the last couple set of lecture notes and the pre-flights for Tuesday.
Today an issue came up about the energies of the "dressed states" of an atom interacting with light. I worked out the details very carefully after class. I was right about which state is the ground state. Remember — we're working in the interaction picture, and to get the total energy we have to add the energy from H0. This type of solution is fundamentally different that what we did with the two-level problem before. I will explain the solution very carefully on Tuesday. You can check out the lecture notes on this topic once they're up tomorrow.
Today we learned about:
T1 processes are not the same as decay. They go along with an interaction with a bath. T1 processes cannot occur without T2 processes at the same time.
A model: classical atom interacting with classical light. When the electron gets moving, it radiates. This is equivalent to a damping force, so the whole problem looks like a damped, forced harmonic oscillator. The emitted radiation has a Lorentzian lineshape. Our ability to drive the atom has the same Lorentzian lineshape.
The physics of a quantum atom interacting with classical light. This just looks like a two-level system! We looked at the physics in the limit of far detuning, where the ground state shift is
For "red-detuned" light, the atom is attracted to regions of high intensity.
Cheers,
Brian
Posted by Brian at 08:15 PM | Comments (0)
March 01, 2005
Class review — 3/1/2005
Hey Class,
Note that there is no pre-flight for Thursday. I want to get the pre-flights and the classes synchronized. Also, there was a problem with the example that I gave regarding Rabi oscillations in the presence of a T1-type process. I will clear that up next class. And, the lecture notes from today need some cleaning up; they will probably not be posted before Thursday.
Today we talked about:
We reviewed the equations of motion for the density matrix and Bloch vector from last class.
We considered a T2-type process in a Ramsey experiment. With a fluctuating magnetic field present during the free-evolution time, decoherence results from an ensemble measurement. Note that decoherence is showing up with only pure Hamiltonian evolution. With T2-type decoherence present, the coherences decay exponentially in time.
We started talking about light, and all of the ways that light can interact with an atom. Only two of these processes (plain Rabi oscillations and stimulted Raman transitions) can be explained semi-classically (quantum atom but classical field). You can't get spontaneous emission without quantizing the electro-magnetic field.
We laid out a cheesy (mmm...cheese) classical model where light is an oscillating electric field driving an electron that is bound harmonically to the nucleas. The electron will only be driven strongly when the electric field oscillates at the resonance frequency, and when the electron moves it will radiate!
Cheers,
Brian
Posted by Brian at 04:08 PM | Comments (0)
February 22, 2005
Class review — 2/22
Hey Class,
The VPython files (and a link to the VPython website) are on the website class notes for today. Read the VPython website carefully — you need to install Python first, and then the visual module.
The lecture notes posted for 2/17 cover today's class. Pre-flights for Thursday are ready!
Today we talked about:
Visual simulations of the Bloch vector and Rabi oscillations. Play with them - you might learn something!
If you look at Rabi oscillations over an ensemble that has atoms with different Larmor frequencies or pulse areas, then you need to average what you see over that ensemble. Examples are finite transit times in a beam experiment, or the effects of an inhomogeneous field.
Ramsey experiments, which are confusing to just about everyone. You can play with the VPython files to get a feeling for how the lineshape comes about. You should be able to understand the lineshape for small detunings when the free precession time is long by using the Bloch vector picture. The key ideas:
Use a single rotating frame, at the drive frequency.
The first pi/2 pulse rotates the Bloch vector almost into the x-y plane.
During the free precession time, the Bloch vector precesses with an angular frequency equal to the detuning.
The second pi/2 pulse rotates the Bloch vector somewhere, with a projection onto the z-axis not equal to 1. One fact: the azimuthal angle in the plane before the second pulse is roughly equal to delta*t. Second fact: that angle is fixed with respect to the effective torque vector. If the effective torque vector is almost in the x-y plane, then the angle between the plane and the Bloch vector after the second pi/2 pulse is just delta*t. Then do some geometry!
Cheers,
Brian
Posted by Brian at 05:02 PM | Comments (0)
February 17, 2005
Class review — 2/17
Hi Class,
A couple of notes:
Two questions came up regarding the homework. One is why I believe that that I can take a Hamiltonian like:
use the projection theorem:
and rewrite the Hamiltonian like:
Another question came up regarding the operator:
and why I claim that this operator behaves like a rank-1 irreducible tensor for any matrix element.
I have put together cogent explanations of these two points, and I will discuss them at the beginning of class on Tuesday.
The lineshape that I drew for an n-pi pulse is wrong (I went back and looked at some data!) — I mixed up the Rabi and Ramsey envelopes in class. I will clear this up on Tuesday. I have a hard time visualizing how that particular problem works for finite detuning using a Bloch vector picture. So, I'm working on a 3-D Bloch vector simulation that we can play with on Tuesday. I'm using VPython, which is a great tool that I'll show you.
Homework solutions, the pre-flight for Tuesday, and possible topics for the final project will all be posted sometime tomorrow.
Tuesday we'll cement our understanding of quantum two-levels systems and Bloch vectors, talk about Ramsey / clock experiments, and how density matrices and decoherence work in two-level systems.
Today we talked about:
A real two-level quantum system: Hypefine states in an atom. There are polarization selection rules that tell you which states can be connected by an oscillating magnetic field. State detection can be done using resonance fluorescence, which we'll talk about soon. For that type of state detection, one state looks bright and the other dark.
The quantum two-level problem in the presence of an oscillating drive can be solved using a rotating frame, which amounts to defining new coefficients for wavefunctions. See the lecture notes for the gory details, and for an alternative, more modern solution. Using the rotating frame solution, we find new eigenstates which have an energy that depends on the Rabi frequency and detuning.
Rabi's formula for Rabi oscillations can be derived by writing one of the bare eigenstates (the eigenstates in the absence of the coupling) in terms of the new eigenstates. Then, you just time evolve the state using exponential factors, and calculate the probability to be found in the other bare state. Rabi oscillations occur at a rate that depends on the Rabi frequency and detuning. The lineshape associated with Rabi oscillations is surprisingly complicated.
Cheers,
Brian
Posted by Brian at 05:32 PM | Comments (0)
February 15, 2005
Class Review — 2/15
Hi Class,
Remember — homework #1 is due Thursday. Check the blog for corrections and hints! I picked these problems because I think that you will learn a lot from doing them, so investing some time is probably worth your while.
There's no pre-flight for Thursday. We're going to work on the quantum two-level problem, and fun stuff like Ramsey's method of separated oscillatory fields. I'll also introduce density matrices and decoherence. You can review relevant material from Cohen-Tannoudji's and Eberly's book if you like.
Today we talked about:
All two-level systems are equivalent to a spin-1/2 particle in a magnetic field.
Classical magnetic resonance. This gets expectation values for the quantum system correct, but cannot get the fluctuations right. Using the rotating frame, one can see that the effect of an oscillating transverse field is to create a static effective torque. The direction of the effective torque depends on the detuning and the magnitude of the oscillating field. The spin vector precesses around the effective torque (call it a magnetic field, if you will) in the rotating frame.
The Bloch, or pseudo-spin vector, can be rotated to point in any direction by using a combination of pulses with different phases and durations.
The pulse-area theorem. The rotation angle associated with the action of a time-dependent oscillating field only depends on the pulse area (time-integral of the magnitude of the oscillating field for a resonant drive).
Adiabatic rapid passage (ARP). It can be tough to flip a spin using a pi-pulse if you don't have good control or knowledge over the Larmor frequency and transverse field magnitude. ARP is another way to flip the spin — you just start with the drive far detuned, and sweep across resonance. In the rotating frame, the pseudo-spin adiabatically follows the effective magnetic field.
Cheers,
Brian
Posted by Brian at 10:08 PM | Comments (0)
February 10, 2005
Class review — 2/10
Hey Class,
Remember to check the blog for comments on the homework.
Pre-flights for Tuesday and the lecture notes from today will be ready by tomorrow (Friday) night.
Here's what we talked about today:
RF Paul traps. Classical motion of ions in a Paul trap is described by the Matheiu equations, which have exact solutions. An approximate solution can be put together using the "pseudopotential" approximation, where the slow, secular motion is separated from the fast micromotion at the RF frequency. Paul traps have regions of stable motion determined by the charge-to-mass ratio and the DC and RF voltages.
We started on 2-state systems! We'll use spin-1/2 language, where we always imagine a static magnetic field present along the z-direction.
Larmor precession. If a spin-1/2 particle is not prepared in an eigenstate, then the spin precesses around the magnetic field. Spin precession just means that áSxñ and áSyñ oscillate out of phase.
Mixing of states when we apply a static, transverse field. The new eigenstates rotate as the ratio of the transverse (B^) to z-field (B0) is varied. The energy of the two states also varies in a hyperbolic fashion as the ratio B^/B0changes.
Landau-Zener transitions. Note that the formula I showed today is only valid when B0 is changed with B^ fixed (that's the case that Zener considers in his paper). The beautiful two-state picture (with a transverse field) contains all of the Landau-Zener transition and Majorana physics in it.
See you next week!
Brian
Posted by Brian at 08:06 PM | Comments (0)
February 08, 2005
Class review — 2/8
Hi Class,
The pre-flights and reading material for Thursday are ready. I'll probably finish up magnetic and electric traps, and then move on to two-level systems.
Check the Homework #1 entry on the blog — new comments have already appeared!
Today we talked about:
Earnshaw's and Wing's theorem. The upshot is that we cannot trap charged particles using static electric fields, and we can only trap atoms with negative magnetic moments in magnetic traps. Wing's theorem is also a cool example of what you can do with vector calculus.
Quadrupole traps. You can trap atoms using quadrupole traps, but the hole of death will limit the density. The hole of death comes from non-adiabatic Marjorana transitions.
The adiabatic theorem. This is deep thing, man. It tells us that any transformation of a Hamiltonian is adiabatic if it is slow enough compared with "internal" timescales. There is also a specific statement of the adiabatic theorem for quantum mechanical systems, which relates the fractional rate of change of the Hamiltonian to the characteristic energy gap in the system.
Majorana transitions. These happen for a magnetic dipole in a field which changes direction, and cause trouble near the middle of a quadrupole trap. Near the center of a quadrupole trap, the Larmor frequency is small, and non-adiabatic spin-flip transitions become more likely.
TOP traps. This is a quadrupole trap with an additional rotating bias field. The hole death gets moved away from the atoms in this kind of trap.
Ioffe-Pritchard traps. These are a complicated mess of coils, which confine atoms using magnetic forces. They use static fields, and nominally have a bias field with a fixed direction.
Whew! See you Thursday!
Brian
Posted by Brian at 03:46 PM | Comments (0)
February 03, 2005
Class Review — 2/3
The pre-flights for Tuesday should be up tomorrow; I'll send around another email. The readings are already set, so see the class notes on the website for 2/8. Slogging through the material by Ghosh is not for the faint-of-heart. Brian King's thesis is a better place to start to learn about modern rf traps.
Review from today:
Today we learned all about the fun Zeeman effect, which is just how anything with a magnetic moment interacts with a static magnetic field.
The real story that we care about is for alkali atoms in magnetic fields. Some good things to keep in mind:
In the low field limit, the Zeeman effect is treated as a pertubation to the hyperfine Hamiltonian. The energy shift is m0gFmFB. Note that gF has a different sign for states with different F! That means that sometimes the m>0 states increase in energy with increasing magnetic fields, and sometimes the m<0 states increase in energy with increasing magnetic fields. This happens because the hyperfine interaction mixes together states with different magnetic moments (which is always dominated by the electron magnetic moment).
Always remember how the stretched states shift with field. These states have a fixed magnetic moment, which is, to a good approximation, the electron magnetic moment.
At high fields, the states group by their mJ quantum numbers. In this limit, the hyperfine interaction is treated as a pertubation to the Zeeman Hamiltonian. The hyperfine interaction breaks the degeneracy between states with the same mJ.
At intermediate fields, we have to solve the problem exactly. For the ground state this is not too hard — see the homework! The magnetic moment of the hyperfine states depends on field, which is because the makeup of the states in the |mI,mS> basis is changing!
We also started talking about traps today. We learned that nature foils our ability to trap both neutral and charged atoms. But, experimentalists are clever, and we trap all kinds of atoms anyway! We'll learn more next class.
Cheers,
Brian
Posted by Brian at 02:30 PM | Comments (0)
February 01, 2005
Class Review — 2/1
Hi Class,
I was talking trash at the end of class today. In fact, the hyperfine ground state splitting is small compared with room temperature, so the two states will be equally occupied. We see this any time we do spectroscopy in a room temperature vapor cell! Most AMO experiments begin at temperatures below 1 mK, and then only the lower ground state is occupied. The lifetime of the upper ground state is very long, and we will estimate it when we talk about light interacting with atoms.
Here is a review of class today:
Forget trying to calculate the electronic structure of real atoms. Screening is a serious effect in alkali atoms, for example. We use a semi-empirical formulation (quantum defect theory) for estimating quantities, and look up transition wavelengths in databases when we really care. The NIST database and the data from Dan Steck and Mike Gehm are valuable resources.
Hyperfine structure comes from the interaction of the electron with the nuclear electric and magnetic moments. The hyperfine interactions are diagonal in the F=I+J basis, where F is the total angular momentum. So, we construct states as |F,mF>, which are superpositions of states in the |mI,mS> basis. The hyperfine splittings (ground and excited states) can be calculated using some formulas and the A and B constants, which are measured.
Isotope shifts are important, since optical transitions can be different for different isotopes of the same atom. A big contributor to isotope shifts is the finite volume of the nucleus. I forgot to mention that there are other effects, such as effects due to the internal structure of the nucleus (non-uniform charge and current distributions).
These reviews will soon appear on a blog that I am starting for the class. You will be able to leave comments on the lectures and use it as a discussion board. More information to follow!
Brian
Posted by Brian at 02:30 PM | Comments (0)
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)