Skip to main content

Posts

Showing posts from April, 2013

The Dirac Equation, Spin, and Open Access to the Royal Society Transactions and Proceedings

Hans Bethe on spin implied by angular momentum commutator
Here's a cool thing about spin and the Dirac equation I hadn't seen until I read Hans Bethe's book "Intermediate Quantum Mechanics".  Commuting the Hamiltonian of the Dirac equation with the orbital angular momentum of a particle indicates that total angular momentum isn't conserved[2].


If you're new to, or not in quantum mechanics, the commutator determines how two quantities behave in a multiplication when the order of the multiply is reversed.  In multiplication with real numbers, A times B is the same as B times A, (the commutative property).  Quantum mechanics uses matrices and in matrix multiplication, A times B is not always the same as B times A.  The commutator, [A,B] just subtracts B times A from A times B.  If the two quantities commute the result will be zero.

One last note for the non-QM inclined.  In quantum mechanics, if you take the commutator of an operator matrix with something ca…

Coherent States, Translation Operators and Sesame Street

I think I'm finally hitting my studying stride for finals.  I can tell because my thoughts on quantum mechanics are starting to merge with the text from Jr.'s Little Golden Books[1], (picture 1).  Sure, sure, to some this might mean that I'm studying too much or sleeping too little, but I see it as a sign of some sort of Zen integration of my personal and quantum mechanical lives :) hehehe  Hoo Boy!

I hadn't realized it on Friday, but looking back on the whiteboard where my professor kind of wowed me, his solution uses most of the basic tools we were taught this semester in an integrated form rather than as disparate facts which is how they'be been rattling around in my head.  So, in the tone of Jr.'s Sesame Street book on helping each other, here goes the solution of "Show that a time dependent force applied to a harmonic oscillator will produce a coherent state.  Here's the original white board in all it's glory (picture 2).  By the way, the con…

Science Locales Long Island, Brookhaven Laboratory, Tesla, and Radio History

In keeping with the new tradition of putting something pretty up here at least once a week, here are scenes located near the Shelter Island area on Long Island.  Looking back over these pictures it occurred to me, the Shelter Island Quantum Mechanics Conference movie could be turned into a science of Long Island mini-series.  In addition to the famous conference, Tesla's last lab is on the island, the site of the first transatlantic transmission is there, Marconi's original radio shack is ironically just a few short miles from Tesla's lab and the old RCA transmit and receive antenna farm locations are there as well.  Oh yeah, and Brookhaven is there as well.

As I mentioned before, we hung out on Long Island, fairly close to Shelter Island, at Brookhaven National Laboratory doing research for two years.  When we headed out to Long Island from New Mexico, I had no idea just how much historic science had taken place in the area.  OK, first, a map to keep everything in perspec…

The Handy Contour Integral

Suppose you have an integral that looks like the following seen during yesterday's quantum lecture:

Your professor turns, looks at you and says, "Who can do this integral?"  After a bit, no one answers and he grins and writes down that the answer is pi.  Then he waves it away as being easy as a contour integral.  Well yeah, but how?  Here's how...
We already said we're going to a contour integral, so that mystery is solved.  We're going to move the integral into the complex plane, and choose a contour that skips the pole at u equal to zero.  Something like this

The portion lableled B skips around the pole at u equals 0, the portion labeled A is on the real axis and stretches from negative infinity to infinity.  The slightly off-screen semi-circle labled C is the return path that in this case integrates out to zero at infinity, (Jordan's lemma and whatnot if you're into the details).  We kept the pole outside of the contour, so we know at the end of t…

Memorizing Integration by Parts

This is bound to come up for me in the next few days because it's one of the favorite tricks of physicsists.  Legend has it that when Landau[1], (a legendary Russian physicist and one of my favorite authors), interviewed a new student he gave them seven problem that involved integration by parts.  If they couldn't do the problems, they were out.  There's a problem though.  As you might have guessed from the xkcd funny strip[2], (picture 2), the elegance and power of the technique can serve to obfuscate a student's understanding of it, (as we often say at black-tie physics education soirees).  To it more simply, I was never able to memorize or use integration by parts until I realized it can be built from the product rule of integration.  Watch:

There are more pointers on how to use the formula once you have it.  Basically, you've got a hammer where you're looking for an integral that's the product of a easily recognizable derivative and something else.  Mo…

H-Rays and a Giant Atom Primer

Posts will be a bit terse and scattered for the next few days.  Today I was looking into Hirsch's answer to the question "How'd you get all that excess charge got up onto the surface of that superconductor anyways Hoss?", (appropriate nod to Ray Stevens and Shriners everywhere here).  Super, super, super attentive readers might remember that the upcoming h-rays experiment[7] will be looking for Bremsstrahlung radiation produced by this theoretical excess surface charge density when the superconductor quenches.


What I found through the referenced series of Hirsch  articles[1][2][3][4] relates to the effective mass of electrons I spoke about yesterday[6].  Hirsch makes the inference that since the mass of the electrons measured within superconductors corresponds exactly to the rest mass of the electron, then they must not be interacting with either the lattice ions of the superconductor, or the other electrons in the superconductor.  Hence, he coins the phrase 's…

LENR and Electron Effective Mass

Quantum mechanics makes some rather astonishing predictions about how particles behave.  One of the most astonishing to me is that an electron's wave function can interact with a periodic potential, (say from the lattice sites of a crystal), and an applied force,(from a constant electric field for example), to make it behave as though its mass is vastly different, (sometimes even negative), compared to its rest mass in free space.  Semiconductor physicists make use of this property all the time.  It is also this property that Widom and Larsen[3] utilize in their theory of LENR paper.  The energy of an electron within a crystal depends on its quasi-momentum as shown in picture 1.  The quasi-momentum multiplied by the distance between crystal lattice sites is shown on the x axis and the electron's energy is shown on the y axis.  Notice that the graph includes regions of energy called gaps that the electron does not occupy.  The resgions of energy that are allowed are called ban…

LENR and Muon Catalyzed Fusion

Reading about low energy nuclear reactions, (LENR), I came across several theoretical references to protons capturing heavy electrons and then participating in nuclear reactions as a result.  The heavy electron, because it sits in a much tighter orbit around a proton, serves to shield the proton's positive charge from other unsuspecting nuclei until the proton has crept in close enough to fuse with them via the strong force. In modern day LENR parlance, it is speculated that these sufficiently heavy electrons exist in materials, (condensed matter), as a result of the periodic potential due to crystal lattice sites, and the wave nature of the electron, (more on this later), leading to a higher effective electron mass.



While the previous paragraph describes theories of how LENR might occur in present day experiments, it's based on a set of actual observations made in the 1940s and 50s, (picture 1)[6]. In 1958, Luis Alvarez in a report to the Unitied Nations[8 open access] on hi…

NASA and LENR and Physics Central Bibliography

Yesterday's Physics Central Physics Buzz Blog[9] post about NASA and LENR, (low energy nuclear reactions), raised a lot of questions for me.  I haven't answered them all yet, so I don't have anything specific to offer, but I thought I'd pass along the following reference list of journal articles, interviews, and videos related to the post in case you wanted to learn more about the physics yourself and form your own thoughts on the matter.  If nothing else, there's a lot of interesting physics at play here including matter waves, Bloch oscillations, and beta decay.  At a minimum, I'll be covering the science behind the controversy soon.



References:
1.  NASA video with Joseph Zawodny
http://futureinnovation.larc.nasa.gov/view/articles/what/cif-safari.html
http://www.youtube.com/watch?feature=player_embedded&v=42hrCRx1JJY

2.  Forbes article on NASA LENR
http://www.forbes.com/sites/jeffmcmahon/2013/03/14/tiny-nuclear-reactions-inside-compact-fluorescent-bulbs/

3. …

Springtime in Colorado and Ice Formation

We're on the road today, so I may not get the chance to do a full post.  Elaine mentioned the other day that it had been awhile since I'd put any pictures up here, so for today's short, but hopefully pretty and interesting post, here are the lake ice formations from Lefthand Reservoir near Ward, CO.  These are springtime pictures taken in March of 2010 and the lake ice was at least a foot thick.  Towards  the end, you'll find a video of Maya the swimming super dog as a sweetener!  Don't worry, she's not swimming in the ice water, she's in Long Island Sound near Sound Beach, NY, a mere four miles from Wardenclyffe and ten or so miles form Brookhaven National Laboratory.

First, the setting.  Here's a topo map of the lake.  It sits at an altitude of about 10,600 feet roughly, 40 miles outside of Boulder, CO and immediately outside of Ward, CO.



View Larger Map

The lake itself and it's surroundings are gorgeous.  Here's a little sample.




Now for the so…

Physics, Movies, and the Columbia and MIT Radar Labs

In the comments to yesterday's post on the Lamb shift[1][2], +Bruce Elliott and I were discussing how physics history could make for great movie ideas .  This morning, it occurred to me that several of the journal articles I've read recently share a common theme, the goings on in and around the MIT and Columbia radiation, (radar), labs circa World War II. The whole thing might make a great intertwined stories movie.  So, without further ado, here's a brief summary of a few of the players, final exams are coming up, so I'll spread this out a bit over the next few weeks.

Schwinger
Schwinger (picture 1), Feynman,and Tomonaga are three of the biggest names in quantum electrodynamics, (QED).  In addition to his QED work, Schwinger was apparently a pivotal figure at the MIT radiation laboratory where he did theoretical work on radar.  The Swinger-Lippmann scattering theory[3], a sort of framework for building other scattering theories came out of waveguide work done by Schwin…

The Lamb Shift and Science Communications

Note to the reader  The following was intended to be a summary of the Lamb shift experiment and the associated QED theory.  Instead, I was so impressed by Lamb, Retherford, and Bethe's communication abilities that while I point out the very abstract highlights of both the experiment and the theory, I get a little carried away with lauding the accomplishments of Lamb, Retherford, and Bethe.

In quantum mechanics II today we studied the Lamb shift. The Dirac equation predicts that the S and P electron states of an atom such as hydrogen should have an accidental degeneracy, (an identical predicted energy state of the atom's electron for two different state of the atom-electron system). While several researchers tried to measure the energy of the two levels to determine if there were in fact degeneracies using optical techniques, they were unsuccessful.  In 1947 Willis Lamb and his graduate student Robert Retherford performed an experiment[3][4][5][6] that showed the theoretically…