Quantum Mechanics, Localization, and Electrical Engineering

In electrical engineering circles, it's common knowledge that any stimulus in the time domain can be decomposed into a Fourier series, or a Fourier transform in the frequency domain.  In other words, you can build an arbitrary signal in the time domain using sine/cosine waves whose frequencies and amplitudes are specified by the Fourier transform.  In physics, a similar concept arises in quantum mechanics.  Objects that live in the space we're familiar with, position space, can be described as waves in momentum space where the wave number, (analogous to frequency), k, of a given wave is described by the momentum, p of an object as:

.
This is the principle behind De Broglie wave descriptions of electrons for example.

An illustration of what the terms local vs. non-local mean in quantum mechanics led to a much better understanding of the uncertainty principal and wave functions for me anyway.  Our professor mentioned that in momentum space, a potential that is specified as a function of position, V(x), is non-local.  In momentum space, the potential has to be specified as a Fourier transform built out of momentum waves with associated wave numbers and amplitudes.  In other words, the potential which had an exact value at an exact position x in position space is now a Fourier transform that is integrated over all of momentum space.  The potential in momentum space is called non-local because of this dependence on multiple wave numbers.

This brings us to the uncertainty principal. and back to electrical engineering.  Any electrical engineer worth their salt will tell you that the more exacting your signal reproduction requirements, the larger the number of different sine wave frequencies you'll need.  So in other words, if you want to create a waveform that looks like this

you'll need to use many more frequencies than if you relaxed your requirements a bit and let some rounded edges into your signal like this

In other words, the more exact you want your signal to be in time, the more numerous frequency waves you'll need to use to construct it, and the more your signal will be spread out over frequency space making it non-local.

If you want to create a delta function pulse, (an infinitely narrow pulse that happens at an exact moment in time), then you'll actually need to use every frequency of sine wave to create it.  An exact signal at an exact point in time space is spread out over every possible value of frequency space.  That's why we like to use delta pulses to find out the frequency response of an electrical circuit.  You get all the available frequencies in one idealized signal.

The exact same process occurs in quantum mechanics.  Once we make the conceptual leap that matter can in fact be described by momentum waves, considerations about uncertainty and locality just fall into place.  Want to locate an object exactly in position space?  You'll need to construct it using every available wave number from momentum space.  In other words, you won't be able to say with any certainty what it's momentum is.  If you specify the momentum of an object exactly, you'll need to describe it with every available wavelength in position space.  You won't be able to say anything about where it is with such a precise definition of momentum.

One of our favorite shapes for wave packets in physics is the Gaussian distribution

Why?  It turns out that when you perform a Fourier transform from position to momentum space, the distribution of values used in momentum space looks like

and vice versa.  The distribution has the very nice mathematical property that it requires the same distribution of values in both spaces.

Picture of the Day:

 From 1/24/13

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…

Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…

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…