Monday, March 18, 2013

Lorentz Contraction, Accidental Degeneracy, and Coherent States

This is just a quick note since once again I have midterms.  I found what may be my favorite issue of the American Journal of Physics ever!  Here are short summaries of the three awesome articles from the July 1965 issue.

Lorentz Contractions

I've been doing a lot of reading on time dilation and Lorentz contractions lately.  Penrose[1], Terrell, and Boas all wrote articles in the later '50s/early '60s about the fact that a sphere moving at relativistic speeds won't look contracted.  It's outline will still be that of a sphere.  In 1965, Scott and Viner[2] followed up on this work with an article that provided, (to my thinking), a far easier way of visualizing what's going on than the other three.  They showed what a piece of graph paper and a set of boxes would look like when observed moving near the speed of light.  For an open access version of the math behind the article see this excellent web page on Terrell rotations[3].  The short version of the story runs something like this.  Because the box is moving at near the speed of light to the right, it can get out of the way of light coming from its back left corner.  So, in addition to the front of the box, a stationary observer will also see the left side of the box in what appears to be a rotation.  The edges of the box that are perpendicular to the direction of the boxes motion will appear to curve.  These curves can be mathematically described by hyperbolas which works out kind of nice since Minkowski space, (four dimensional spacetime), is an example of a hyperbolic geometry.

Accidental Degeneracies
A few months ago I wrote about an article written by Fock[4] where he had pointed out that what appeared to be accidental degeneracies in the energy levels of the hydrogen atom weren't accidental at all.  They merely revealed that the potential of the hydrogen atom behaved like a four dimensional potential that had been projected into a three dimensional space.

Shibuya and Wulfman[5] point out that Keplarian orbits have an accidental degeneracy, but lie in a two dimensional plane, so that the projected potential can be modeled in three dimensions and therefore visualized.  They provide photographs of a few examples.  Since the article was written before the advent of all the nice 3D plotting packages we have now, it appears they actually built solid models of the spherical potential, (see below picture 2).

Coherent States
Finally, the article that got me to look in this issue of the journal in the first place:  Carruthers and Nieto[6] wrote a very nice article illustrating the derivation and use of coherent states.  If you've been trying to work though Merzbacher like I have and would like a different perspective on coherent states, then this is the article for you.  The authors first posit a reason for deriving coherent states at all, as a tool for demonstrating the correspondence principal for harmonic oscillators.  They then show why you can't do this with number states: the expectation value of the oscillator amplitude is always zero.  Finally they proceed to show how coherent states can give an expectation value for both the amplitude and the phase of a harmonic oscillator.

Next, they show why coherent states are called coherent, (they are wave packets that don't spread out).  It is also demonstrated that while the coherent states aren't orthogonal, they are complete.

In closing, they do a very nice derivation of how coherent states can be used to solve the forced harmonic oscillator problem.

Now, I'm off to study angular momentum and coherent states for the midterm this afternoon!

References:
1.  Penrose on Lorentz contracted spheres
http://dx.doi.org/10.1017%2FS0305004100033776
Penrose R. (1959). The apparent shape of a relativistically moving sphere, Mathematical Proceedings of the Cambridge Philosophical Society, 55 (01) 137. DOI:

2.  Scott and Viner on Lorentz contactions
http://dx.doi.org/10.1119%2F1.1971890
Scott G.D. (1965). The Geometrical Appearance of Large Objects Moving at Relativistic Speeds, American Journal of Physics, 33 (7) 534. DOI:

3.  Open access web page on Terrell rotations
http://www.math.ubc.ca/~cass/courses/m309-01a/cook/terrell1.html

4.  Fock's four dimensional potential
http://copaseticflow.blogspot.com/2012/11/fock-so4-and-azimuthal-quantum-number.html

5.  On the three dimensional projection of rotational degeneracies
http://dx.doi.org/10.1119%2F1.1971931
Shibuya T.I. (1965). The Kepler Problem in Two-Dimensional Momentum Space, American Journal of Physics, 33 (7) 570. DOI:

6.  All about coherent states
http://dx.doi.org/10.1119%2F1.1971895
Carruthers P. (1965). Coherent States and the Forced Quantum Oscillator, American Journal of Physics, 33 (7) 537. DOI:

7.  Boas on Lorentz contracted spheres
http://dx.doi.org/10.1119%2F1.1937751
Boas M.L. (1961). Apparent Shape of Large Objects at Relativistic Speeds, American Journal of Physics, 29 (5) 283. DOI:

8.  Terrell on Lorentz contracted spheres

Picture of the Day

From 3/18/13

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