### Bra Ket Notation

Just a few notes here about shiny things that caught my eye regarding bra and ket notation in the quantum mechanics II lecture last night.

Inner Products
Inner products are the Hilbert space, quantum mechanical, state vector equivalent of the dot product for more standard vectors like position or velocity.  The unit basis ket , at least in our class, is written as

where j is the index of the component.  Associating back to Cartesian coordiantes, 1 would denote x, 2 would denote y, and 3 would denote z.  The ket vector is the same symbol in a ket and when the two are applied to each other we get the inner product

.

In other words, the inner product only produces contributions from like basis vectors, just like the dot product.

So here's the cool bit, the following all accomplish about the same thing, they find a number proportional to the magnitude of the component of one vector, lying along another vector whether those  vectors are what we most typically call vectors, or what we call bras and kets, or what we call wavefunctions:

The Projection Operator
Applying the following notation, (the projection operator), had continued to confuse me until last week.

To pull out the portion of a bra vector that points along the basis vector u sub j, the above can be applied as follows:

Applying the projection operator always seemed a bit daunting to me until I saw the entire operation written in several steps utilizing the associative property and the fact that bras and kets commute with scalars as

.

Cool note:
The projection operator is an example of an outer product.  An outer product of two vectors produces a square matrix.  The trace of the matrix is equal to the dot, (or inner), product of the two vectors.

Picture of the Day:
 From 1/21/13

### 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…