### Bound States in Potential Wells... The French Connection

While working on QM homework yesterday, I came across a very nice article[1] in the American Journal of Physics about a new perspective on solving for the energies of the bound states in a potential well.  The article points out that waves reflecting back and forth in a potential well have to remain self consistent (picture 1).

Another way to say this is that they have to catch their own tails, or wind up back where they started after a round-trip transit of the well.  Using this as a basis, the following equations are quickly derived (picture 2)

Equation 4 describes the particles wave function after traversing the well once immediately after the reflection.  Rho is the coefficient of reflectivity, the first exponent is the phase picked up by the waveform after travelling distance L, and the second exp is the wave function itself.  Equation five is the wave function immediately after the second bounce.  Note the extra reflection coefficient and phase gain.  Equation six points out the condition that must be met if equation five is to remain consistent with the original wave function.

The authors then proceed to derive the necessary reflection and transmission coefficients for a number of helpful examples including the non-symmetric square well potential, (the problem of interest on my homework).  Without giving too much away, there derivations are very easy to understand and conceptualize do to a clever representation of the depth of the potential well in terms of the k's of the wave functions.

Folks who are used to WKB and/or the Bohr-Sommerfeld quantization rule will find this article very easy reading.  On the subject of WKB, if you want to know a lot more about it and how it relates to complex analysis, check out his 1947 article from W. H. Furry in the +American Physical Society's Physical Review[2].

Super Cool Stuff About the Article's Author JF Bloch
No, he's not related to Felix Bloch as far as I know, so that's not the cool part.  Here it is.  Dr. Bloch works at the Grenoble Institute of Technology's Institute for Stationery and Graphics.  That's right, he gets to work on all the pretty stuff!  I quote, loosely translated by Google from their web site:

"From fiber to paper, screen printed"
Located in the heart of the university campus of Grenoble, the French School Stationery and Graphic Industries is the only engineering school in France which forms in three years future managers stationery, processing of paper and paperboard, and printing .

I had no idea, although it makes perfect sense, that you could work on the physics of paper!  Recently he's been working on advances in paper based capacitative touch pads[3].

References:
[1] http://ajp.aapt.org/resource/1/ajpias/v69/i11/p1177_s1

[2] http://prola.aps.org/abstract/PR/v71/i6/p360_1

http://www.ncbi.nlm.nih.gov/pubmed/22539155
Mazzeo A.D., Kalb W.B., Chan L., Killian M.G., Bloch J.F., Mazzeo B.A. & Whitesides G.M. (2012). Paper-based, capacitive touch pads., Advanced materials (Deerfield Beach, Fla.), PMID:

Picture of the Day (from the mountains north of Grenoble!)

 From 2/8/13

### Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

### The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

### Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…