### DeBroglie Waves and Propagation of Mass

This starts a new series of posts that will hopefully inspire discussion among folks taking, teaching, and/or using quantum mechanics.  If you're reading this on Google+, the equations are referenced in the album attached to the post.

Did you know that DeBroglie came up with the concept of matter waves considering relativistic invariance? I didn't until quantum I lecture yesterday. Does anyone know how the reasoning went? I can see a way to make sense of it. If you look at energy being equal to, (eq. 1)
and think about frequency as the reciprocal variable of time in a Fourier transform, then, if you begin to consider momentum, (the other three components of the energy momentum four vector of special relativity), you soon thereafter could arrive at, (eq. 2)
where k is the reciprocal variable of the x, y, and z space coordinates in a Fourier transform.

In the same lecture, it was mentioned that the simple form of a sound wave could propagate energy but not mass.  For a wave that is capable of propagating mass, we need to go to complex space and a wave that looks like, (eq. 3)
Can anyone shed more light on why this is the case?  This wave equation has a form very similar to a Fourier transform and contains both Fourier reciprocal variables mentioned above.  Is that why the complex form is required? Does the earlier comment about special relativity belie the need for a complex component ala the component of i introduced with time in the Minkowski metric?

Picture of the Day:
 From 1/16/13

### 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…