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Showing posts from November, 2012

### Matrix Operations and Ladder Operators

The short version.  If you're curious about how the x matrix for a harmonic oscillator moves an element within the oscillator state column matrix, this post may hurry things along for you.  I was curious and spent bits and pieces of a few days trying to work it all out.  Here's what I found hastily jotted down so I won't forget and so I can get back to studying for finals.

We're rapidly plowing towards finals week, so this may be rather disjointed, but I wanted to capture a few notes on actually using matrix operations before they escaped.  In Hecht's book when calculating the perturbtion of an harmonic oscillator due to an x ubed, or an x to the fourth potential he makes a rather light reference to using the 'known' matrix elements of the x operator of the harmonic oscillator.  I've seen the references quite frequenlty and decide for once rather than just using the elelments the book wrote down, seeminly from nowhere, that I'd go calculate them.  I…

### End of Days Ham Radio Special Event Station

What with finals coming up, and midterms only recently behind me, I had completely forgotten about the Mayan calendar doomsday predictions for this December 21st.  I found out yesterday, though, that the Brazos Valley Amateur Radio Club is all over it!  They've setup an end of days special event station to celebrate our impending doom.  The call sign will be N0D, (Now 0 Days), and they'll be operating the day before, the day of, and the day after, (in case the predictions are somehow wrong), the biggest day of all.

Picture of the Day: Sunset over Sound Beach, NY

### Fock, SO(4), and the Azimuthal Quantum Number

Our quantum professor worked out the Kepler-Coulomb equation earlier this week and ended with a cryptic remark about SO(4). A little bit of digging led to a presentation about the details from the University of Minnesota math department[pdf]. It turns out that Fock did a Fourier transform of Schrodinger's equation into momentum space and saw that the resulting integral kernel looked like the projection of a sphere onto a 4 dimensional hyperplane. He then saw that Schrodinger's equation is simply the Laplace equation in this space.

I hope in the future to write something more complete on the whole matter, but with finals coming up, I'll just add a few notes and references.

Fock's original article on all this is written in German.  However, the author of the above mentioned presentation, Jonas Karlsson was kind enough to point me to Linearity, Symmetry, and Prediction in the Hydrogen Atom (Undergraduate Texts in Mathematics).  Where the entire article is translated i…

### Heisenberg, Dirac, and Matrix Quantum Mechanics

I've been reading an excellent quantum mechanics book by K.T. Hecht this week.  The book is very complete, and heavy on worked examples.  The level of complexity is much higher than Griffiths, but there are many, many worked examples.  You don't run into the key bit of information you need being encrypted into a homework problem.  If you've read Griffiths, you know what I mean.

The coolest thing in the book so far is a historical reference to Heisenberg.  Hecht points out in less than a page how Heisenberg arrived at the matrix formulation of quantum mechanics.  Starting out with the model of the Bohr atom's energy levels, he first noted that the energy was dependent on the frequency of the emitted radiation.  This suggested a model similar to a Fourier series.  Heisenberg then noted that there were two indices, n and m, and posited that the Fourier model at the atomic scale would utilize a matrix of coefficients as opposed to a list.  The rest was just (lots and lots …

### GIS Day and the Copasetic Flows Mapping Applications

The Geography department and the map library here at Texas A&M sponsored Geospatial Information Systems day here this week. I entered the associated poster competition with a poster about all the mapping applications that are available here on Copasetic Flows. This is a recap of said applications.

Satellite Mapping
Find out which amateur radio satellites will be passing your location soon, where and when they'll pass by.

APRS Fly/Drive bys
APRS is an amateur radio packet data system used for among other things, tracking the locations of vehicles belonging to ham radio operators.  This application allows you to select a recent flight or driving path of an APRS user and fly or drive along the same path on Google Earth.

QSO Mapper
The application that started all the mapping here on Copasetic Flows.  Enter your callsign and the callsign of the ham you just QSO'd and get a map linking your two locations.  You can even log your calls right on the site.

### Orthogonal Matrices: Things I Hadn't Noticed

Believe it or not, I'd never really grasped that orthogonal matrices amounted to a projection into a new orthogonal coordinate system.  I have one question left about this.  If the rows of the matrix are orthogonal unit vectors, does that guarantee that the columns are?

### The Similarity Transform: Things I hadn't noticed

As soon as I got back into graduate physics, I started noticing transforms of matrix operators that looked like this:
A is the original matrix operator A prime is the matrix operator transformed by gamma.  Gamma is any kind of vector transformation.  It might be a rotation, or a change of coordinate system, (from Cartesian to polar for example)..  Presented in this manner, the origins of the transform, A acting on gamma and the product acted upon by the inverse of gamma didn't make any sense to me.  I found an article, (I'll try to get a reference up here soon), that gave a very detailed very academic explanation, but it was still no good for me.  Recently, a professor finally went through the steps that arrive at the above.  It was short concise, and made sense!  Here they are.
Gamma is a matrix that transforms a vector into another vector, say... x prime into x.  I mentioned that already.
The inverse of gamma will convert an x vector into an x prime vector.
A is defined to b…

### Things I Hadn't Noticed Video Series

I'm coming across several things a week in my homework and classes that are suddenly crystal clear.  These are topics that, had I gone to school in a more linear fashion, should have been clear to me three or four years ago, but now will do.  Here's the first of them, a better interpretation of matrix multiplies with column vectors.