Before when we talked about Einstein's derivation of E=mc^2 we talked about a Taylor's Series - a way to rewrite an equation into an easier to use form. This doesn't work well with all kinds of equations. Since the Taylor's Series is a summation of terms with the form x^n, it usually works well with equations that have a similar form. For functions which display wave properties (where the function just keeps repeating over time), this usually works well only around the point of constructing the series. The farther away you get from that point the more the Taylor's approximation breaks down.Instead you normally do something called a Fourier Analysis. It's essentially the same thing. Instead of making an infinite series of x^n terms with differing coefficients you use an infinite series of sine terms with differing coefficients, phases, and frequencies. In essence a Fourier analysis tears a wave function apart into base sine components. Think of it as a prism. The number of sine terms necessary to replicate the function you are performing the analysis on is a measure of the bandwidth. Bandwidth then is a measure of how close sine waves are in shape to your original function. In a limiting case, perform Fourier analysis on a sine wave (e.g., decompose a sine wave into a series of sine waves) and you'll clearly only get 1 term (which as the same frequency as your original wave) and the bandwidth will be very small. Use something other than a sine wave and and you'll get a wider bandwidth.
This decomposition can be done with any kind of function. Taylor's series and Fourier Analysis are just two special cases of this type of decomposition.
This is used in a lot of modern day electronics. For example, when CDs are made the music is decomposed into a set of sine waves and the coefficients, frequencies, and phases are written to the disk. They don't use an infinite set of sine waves or an infinite bandwidth and that is why there is ultimately a highest frequency that the CD stores on disk. Synthesizers work in reverse. They add sine waves together to get different types of sounds (analog synthesizers do this, digital synthesizers use a different wave - an impulse wave). More interestingly your ear does this with sound waves. The spiral cochlea allows different frequencies of the entire sound to travel farther into the ear. Each point along the cochlea is sensitive to a particular frequency. Your brain then interprets the parts of the cochlea that are 'ringing' to determine what a sound sounds like. Pretty cool.
So why am I talking about this?
Well I mentioned before that the solution to Schrodinger's equation, Psi, contains all the information regarding dynamic attributes for a quantum thing. Here's how you get it out. Step one, solve Schrodinger's equation. Depending on the type of system you are looking at (e.g., an electron around a nucleus with 1 proton and 1 neutron) you have varying inputs and boundary conditions that go into Schrodinger's equation (potentials, masses, etc.). Solving for the equation gives you Psi which will in general be a function of time and space. Psi is wave-like in it's structure and sometimes is referred to as a proxy wave or a ghost wave or an empty wave. Proxy wave is used because Psi represents the entirety of information about the quantum object. Ghost wave is used because it doesn't represent a particle or even a real thing per se but a potential of what the quantum object could be. Empty wave is used because the amplitude of waves is a measure of their energy except in this case - it is a representation of possibilities.
Now you've got Psi you extract the dynamic attributes. You do this by performing a Fourier or similar analysis on Psi - extracting the attribute you are interested in. Where Fourier analysis always uses a sine wave, in theory any wave can be used to decompose Psi. For example
Position - To extract the position of the quantum object you decompose Psi with an impulse wave. An impulse wave is an infinitely thin spike with a height of 1 or infinity depending on how you define it. To visualize this think of a heart rate monitor that blips up as your heart beats. If this was infinitely thin and peaked at 1 you'd have an impulse wave. To visualize what it means to decompose Psi with an impulse wave think about it this way. We could also 'write' Psi as an infinite number of impulse waves lined up from left to right on a graph as Psi goes left to right. Make the height of the impulse wave the same as Psi at the same point. It's like a bunch of dominos lined up with varying heights.
Now how do we get position from that? Well the amplitude of each impulse wave at each point represents the probability of finding the quantum object at that position. Actually we need to take the amplitude squared. In essense we aren't doing anything fancy here. In fact taking Psi and squaring it is actually the same thing. Psi^2 can be interpreted as being a measure of the probability of finding the quantum object at that point in space and time. That was a little too simple. Let's do something that's a little more instructional.
Momentum - Momentum is extracted by decomposing Psi with a spatial sine wave (Fourier analysis). If you figure out how to make Psi with a bunch of sine waves you will have a number of frequencies and amplitudes. Let's just assume that we can rewrite Psi as being the sum of 2 sine waves - A & B. A has an amplitude squared of 0.5 and frequency 1 and B has an amplitude squared of 0.5 and frequency 2. In other words if we did a Fourier analysis on the original wave it would calculate for us those two amplitudes & two frequencies. We would then know it's comprised of two sine waves. In general it takes many sine waves to perfectly mimic the shape of Psi. We're just using a simple case here to illustrate.
The amplitudes squared represent the probability of finding the momentums related to those probabilities - in other words there is a 50% probability of finding the momentum defined by wave A and a 50% probability of finding the momentum defined by wave B. How is momentum defined by wave A and B? Take the respective frequencies and put it into this equation
p = hf
p here is momentum, h is a constant (Planck's constant) and f is the frequency. So in summary the particle we found Psi for has a 50% chance of having momentum h and a 50% chance of having momentum 2h.
Now a couple of points:
- Other attributes like energy are similarly extracted from Psi. Decompose Psi with a particular wave family, use the amplitude squared as a measure of probability and plug the other defining parameter (usually a frequency) into an equation to get the real life dynamic attribute you're trying to find.
- This process seems awfully recipe-like doesn't it? It does. There isn't a lot of intuition here. The fact is it just works. No one really has a good explanation why. Many scientists avoid the quantum reality question entirely and use quantum theory as an exceptionally versatile hammer. Not only is the application relatively straight forward, the result are typically correct to multiple decimal places. In fact quantum theory bats .1000 in this regard. It is a hammer for all occasions.
- The fact that we're dealing with probability waves leads to some interesting possibilities. Let's stick with position for now. Because waves can deconstructively interfere, you can have probabilities that drop to zero at some points (call it point P) but be positive either side of that point. What does this mean? Well if you view an electron as an ordinary object (i.e. a particle) then you have a problem. The particle has a probability of existing either side of Point P but not at Point P. How does it get from the left of Point P to the right of Point P if it can never exist at Point P? It can't. You can't think of these objects in the traditional sense of existing somewhere. Objects have a possibility of existing many places but they don't actually exist anywhere until we measure them.
- This deconstructive nature has other interesting predictive possibilities. Quantum theory predicted quarks under certain conditions would decay to 2 muons. For some reason no one could ever find this decay experimentally. Sheldon Glashow deduced that another quark must exist whose presence adds a Psi to the mix in such a way to eliminate the probability of this decay occuring. He could even figure out some of the properties of how that quark must behave to have the right kind of Psi to eliminate the decay. Eventually that quark was found. This is the strongest type of event for a theory's strength. To predict something for which there is no experimental evidence for.
- Uncertainty conjugates are related by nature of their extracting wave form. Oh that was an ugly sentence! Let me explain. Remeber we talked early about conjugate attributes and how Heisenberg showed that the accuracy of knowing one attribute is inversely related to the accuracy of knowing the other attribute? He did this analysis through the wave extraction aspect of quantum theory. Take the two attributes we used before - momentum and position. The extraction waveforms we use on Psi for these two attributes are sine wave and impulse wave respectively. Turns out these two wave forms are mathematically as far apart as possible. Meaning if took one of those waves and decomposed it with the other the bandwidth would be larger than for any other known wave form. This is true of all conjugates. This is important because there is a well known relationship between these 'opposite' waveforms. Let me explain. Take any randomly picked wave form. Now decompose it with any other waveform and its conjugate waveform. We'll get a bandwidth for both decompositions. Multiply those bandwidths for both waveforms together and it is less than one. This is where Heisenberg got his uncertainy principle from. Bandwidth is related to measurement accuracy.
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