### On the Importance of Trig Identities in Quantum Mechanics

Just a few random thoughts on why I wish I'd done a better job of memorizing my trig identities in high school.

Here's an approximate high school conversation of mine regarding trigonometric identities with my teacher Mr. Tully, (who is awesome and by the way, who is also a rancher, and also by the way is not the guy pictured to the left(picture 1)... read on!)

Me:  "Why do I have to memorize these 40 or so trig identities[2]?" (yes even then, I referenced my utterances"

Mr. Tully:  "Because they'll be very important for what you want to do later." (he knew I wanted to be a physicist)

Me, (typically not thinking 'later' might be after next week):  "Yeah... I'm not seeing it..."

Fast forward a bit to grad school.  Twice in the last week, trig identities were make or break features of homework problems.  I didn't pick up on the necessary identities in electromagnetism, and I'll probably get a B instead of an A on my homework.  A B in grad school is a C in undergrad.  Even more importantly, I didn't get to see the 'cool features', (no, really, I'm sure they were cool features), of the solution because I didn't get to it.

In quantum mechanics homework tonight, I was working on a tunneling problem.  I used the following two identities and wouldn't have been able to move forward without them (picture 2):

What's tunneling?  In the world we're used to, when an object encounters a barrier like the one shown below(picture 3), it will just bounce off.

In the energy and size levels where quantum mechanics applies though, a particle can actually just move right through the barrier and turn up on the other side.  So, why's that important?

Here's one reason.  If you have an understanding of quantum tunneling, then you can do things like the guy pictured above (picture 1), Ivar Giaver, who before he even had a PhD performed an experiment that found that electrons could tunnel from a superconductor across an insulating barrier into another superconductor.  After that, he won the Nobel prize.  His discovery also enabled magnetic resonance imaging, (MRI), technology to be used as a medical diagnosis tool, which is kind of cool.

And that is why I should have memorized my trig identities.

References:
1.  http://en.wikipedia.org/wiki/Ivar_Giaever

2.  http://en.wikipedia.org/wiki/Trigonometric_identity

Picture of the Day: (picture 4)

 From 2/7/13

### 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…

### 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…

### 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…