### Separated at Birth? Quantum Mechanics and Electrical Engineering Systems Analysis

I've been working on a history project peripherally for months, I'm just recording a few notes here as I still haven't gotten to the bottom of it.  Because I haven't arrived at the answer to my research yet, the following will ramble on a bit, but I wanted to capture my notes so far.  You see, my old electrical engineering courses keep creeping into my quantum classes and vice versa.  It's not that it's just the same math, it's also the same notation.  The ultimate answer to all of this may be that both subjects pulled their notation from pure mathematics.

The latest inspiration for really looking into this came up as I was studying for my quantum midterm yesterday. I came across the following integral in Merzbacher that I felt certain I'd seen before (picture 1).  Merzbacher certainly felt it should be familiar since not a bit of explanation was given for its execution.

When I got home, I pulled out my EE systems engineering book, "Discrete Time and Continuous Time Linear Systems" by Robert Mayhan and sure enough, there was something fairly similar, but not exact, in the table of Laplace transforms.

I know that seems tenuous, but it was only the latest in a series of similarities.  In quantum mechanics, we denote the hamiltonian by H.  In EE, the response function of a circuit is denoted by h(t) in the time domain and H(omega) in the frequency domain.  What do we do with that function you might ask?  We operate on state vectors of course!

Not only does the systems book define  operations on state equations, (although never mentioning the word opeartor), it also uses the same notation.  Here's Mayhan

and here's Merzbacher on quantum mechanical operators

Both fields are interested in determining the energy in the system as indexed by frequencies.

Finally, (for the moment), what we call a Green's function in quantum mechanics is called a convolution in EE systems analysis.

Getting into the History
I finally started looking into the history all this last night and came up with the following rather disjointed points.

The first journal article reference I could find to the use of Laplace transforms in circuit analysis was in the Proceedings of the Institute of Radio Engineers[3] circa 1949 in the form of a reference to a 1942 paper by the same author (picture 6)
I found references to the subject of Laplace transform circuit analysis, (frequently called operational calculus), in books a few decades earlier[1] (picture 7),

If you're a Philadelphia Experiment fan like I am, yes that V. Bush.

I also found a portion of an electrical engineering thesis[2] that looks very similar to the sudden approximation in perturbative quantum mechanics (picture 8).

That's all I've got for now.  Does anyone else have any thoughts on how the two fields wound up with such similar notation?

References:

2.  The Principle of Equivalent Areas
10.1109/TAI.1960.6371578

3.  Clavier article
Clavier A.G. (1949). Application of Fourier Transforms to Variable-Frequency Circuit Analysis, Proceedings of the IRE, 37 (11) 1287-1290. DOI:

### 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…