### Renormalizing Basis Components in Quantum Wave Functions

I'm studying for our pre-midterm practice exam in quantum mechanics II this weekend, hence I've developed a bit of a fascination for all things, (even trivial things), quantum.  On our last homework, we had to decompose a wave function to its eigenfunction components, (sines and cosines in the infinite potential well that we were assigned), and then write down the time evolution for those components.  It was a bit of an exercise to remember how to normalize the components once I had them, so I'm recording the process here to hopefully make it easier to remember next time.

The wave function that had to be decomposed was

As you can see, a trig identity arose to make simple work of the problem once again.  No need to do a Fourier decomposition to sines and cosines when a simple little power-reduction formula[1] is readily available.

That leaves us with two components and two leading factors, or weights.  The mechanical normalization process is to integrate our wave function over all space, then take the result and divide our original wave function by it so that if we integrate the square of our wave function again we should get one and be normalized.

Now, multiplying back through by 1 over the square of the result we get (picture 3)

Interesting Stuff
An interesting fact about the integration of sine squared over a given region came up while I was doing this.  The result is independent of the frequency of the sine wave.  So, for the following three sine squared functions, you get the same result (pictures 4, 5, and 6)

I believe this has to do with the basis vectors formed by integer multiples in a sine series all having the same unit vector length.  It's nice to see it illustrated in a concrete way though.

References:
1.  http://en.wikipedia.org/wiki/Trigonometric_identity#Power-reduction_formula

Picture of the Day:
Fog on the Organ Mountains out the back door of the old lab

 From 2/9/13

### 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…