Thursday, January 24, 2013

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

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