A Quantum Mechanics Overview and Einstein-Podolsky Pairs
Revision 1.00
jason harris (smeger)
smegtra@goodnet.com
1/23/99
(this was written on a Mac in Monaco 9 pt., so some characters may display incorrectly on other operating systems - sorry, but stuff like deltas and greater-or-equal-thans don't have standard ASCII codes. I've included a mapping table at the end that will let you do a search and replace if you have problems)
Table of Contents
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Definitions
Coupled Wave Packets
Einstein-Podolsky Pairs are Hip'n'Groovy
Mappings for platform-dependent ASCII characters
References
Definitions
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An Einstein-Podolsky pair refers to any pair of coupled quantum-mechanical wave packets.
So, a bunch of interrelated definitions are in order. I'll define the terms quantum-mechanical, Heisenberg Uncertainty Principle, wave packet, Schrodinger Wave Equation, and observable.
1. Quantum-Mechanical: Any system that obeys the Heisenberg Uncertainty Principle and is characterized by a probability amplitude wave packet that is a solution of the Schrodinger Wave Equation. Generally, these are only microscopic things such as elecrons, photons, and protons, not macroscopic things like Coke cans, toaster ovens, and pizzas, although you can attempt to use quantum mechanics on anything. However, as the size of the object approaches dimensions with which humans are comfortable, the quantum results tend to disappear and you end up getting results that look like Newtonian mechanics, but for which the math was a hell of lot more difficult.
2. Heisenberg Uncertainty Principle: Says that the multiplicative product of the uncertainties of two different observables is always greater than zero. The uncertainty of an observable is the standard deviation one would find if one tried to measure the observable's value a bunch of times. Standard deviation is a statistics term that describes the variation one will find in a bunch of measurements. For example, the uncertainty in position is referred to as Æx and the uncertainty in momentum is referred to as Æp. So, for a free particle, the uncertainty relationship is
Æx * Æp ³ 1/2 * hbar
where hbar is planck's constant divided by 2¹ (or 1.602e-34 J s). This means that the more accurate one's measurement of position is, the less accurate any measurement of momentum will become. This also means that if you sat around and measured the position of a free particle a bunch of times, and then computed the standard deviation (which is kinda like the variation) of all your measurements, it would be greater than 1/2 * hbar.
The uncertainty relation does not exist because of our measuring equipment; rather, it is an intrinsic property of quantum systems. Many pairs of observables obey uncertainty relations, but the most famous ones are position-momentum and time-energy.
3. Wave Packet: A wave packet is a mathematical construct used to represent some particle, photon, bowling ball, whatever. The wave packet is obtained by solving the Schrodinger Wave Equation. The wave packet can be manipulated to give an equation describing the probability of finding the particle, photon, or bowling ball at some particular point in space at some particular time. So, the wave packet is simply a description of the probability of finding the particle in different regions of space-time. Coupled wave packets are wave packets which are in some way dependent upon one another. A simple example could be a pair of wave packets whose sum is always one, although the individual wave packets may have any values.
4. Schrodinger Wave Equation (SWE): A postulate of quantum mechanics (i.e., some equation that some guy just made up one day while skiing, instead of deriving like all the other physicists like to do). All quantum mechanical objects (particles, photons, ski boots) obey the SWE. Just for fun (you don't need to know this), here's the equation written as a mix of words and symbols:
-(hbar * hbar)/(2 * m) * LaPlacian[psi] + V * psi
= i * hbar * (partial derivative of psi with respect to t)
where:
hbar = (planck's constant)/(2 * ¹) = 1.602e-34 J s
m: rest mass of thing being described
LaPlacian: second order vector gradiant, or gradient of gradient, or rate at which whatever's inside the parentheses changes in space
psi: the wave packet for whatever is being described
V: function describing the potential energy of the thing being described at all points in space and at all times
i: square root of -1
t: time
You don't actually need to know this, but you should be aware that it's a _motherfucker_ to solve.
5. Observable: Some quantity that can be observed or measured. For example, position, momentum, spin, polarization.
So, to rehash, an Einstein-Podolsky pair refers to any pair of coupled quantum-mechanical wave packets. This definition should make a bit more sense now. An example pair could be a pair of electrons seperated by some distance, and whose spins must always add up to zero. Another example pair could be a pair of photons seperated by some distance, and whose polarizations always add up to zero.
Coupled Wave Packets
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As stated before, the objects in an Einstein-Podolsky pair must have coupled wave packets. The only way for wave packets to become coupled is for the two objects to interact with one another in some way. So, the objects in an Einstein-Podolsky pair must be physically close together at some point in their history. This puts a bit of a damper on doing cool things like putting one object on earth and another on Alpha Centauri and using 'em to communicate, unless you're willing to stick one of the objects on a spaceship and ship it to Alpha Centauri first.
So, an Einstein-Podolsky pair is not just any two particles anywhere in the universe. An Einstein-Podolsky pair is some two particles that have interacted at some previous time. However, this description is still not sufficient. The objects must have interacted in such a way that an observable of the first object depends on an observable of the second object, and vice versa. These two objects have coupled wave packets.
An example (completely fictitious) interaction that would produce an Einstein-Podolsky pair is as follows: Imagine some initial particle which is not spinning at all (it has zero angular momentum). This particle suddenly decomposes into a pair of new particles, both of which _are_ spinning. Angular momentum must be conserved, so the sum of the spins of the new particles must be equal to the spin of the original particle, which was zero. The new particles are an Einstein-Podolsky pair. Of course, quantum mechanical effects are not visible at macroscopic scales, so if you try this by breaking apart a bowling ball and spinning the pieces, you won't see anything too thrilling.
A realistic example is a pair of virtual particles. Virtual particles are particles that suddenly appear from nowhere, exist for a brief period of time, then merge with one another and anhilate, disappearing again. These particles are opposite in every way. For example, you may be sitting around in outer space some day, watching a small patch of vacuum very intently. Suddenly, you'll notice that two particles have appeared out of nowhere. One is an electron. The other is the opposite of an electron - an antielectron, or positron (an electron made of antimatter). An electron has a negative charge, so a positron must have a positive charge. The electron has (for example) a spin of +1/2, so the positron has a spin of -1/2. Since these particles must be opposite in every way, the values of their observables are related, so their wave packets are coupled, and they form an Einstein-Podolsky pair.
Incidentally, +1/2 and -1/2 are the only two possible spin values that particles may have. Also, the term 'spin' does not refer to anything remotely like what you see a top or a car tire doing.
Einstein-Podolsky Pairs are Hip'n'Groovy
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Okay, so at this point, we've pretty much covered the definition of an Einstein-Podolsky pair. Now, let's talk about why they're cool. Since the values of their observables are coupled, a change in the value of one observable results in a change in the value of the other observable.
In the virtual particle example above, if you flip the spin of the electron, making it -1/2, the spin of the positron will _instantly_ flip itself to +1/2. Note that I said "instantly," not "as soon as the positron 'sees' that the electron's spin has been flipped." So, there's none of this 'no faster than the speed of light' crap. If the electron and the positron are seperated by one light year, and you flip the spin of the electron at 6:15 AM Saturday, the spin of the positron will flip some infinitesimally short time later. (I'm ignoring a bunch of special relativity stuff, because it would just confuse things and doesn't help to illustrate my point)
So, this sounds pretty spiff-ass, but there's a problem. The Heisenberg Uncertainty Principle may destroy the possibility of using this for any sort of communication. Since the uncertainty relation says that you can't measure a value (such as the spin) perfectly, you _may_ not be able to transmit information using Einstein-Podolsky pairs. The problem is that when you measure a value like spin, the value may become random. So, here's an example scenario:
1. You have particle A, Cameron Diaz has particle B.
2. Cameron measures the spin of her particle as +1/2 - she sends you a telegram saying so, so you know your particle has spin -1/2.
3. Because Cameron measured her spin, it is now random. Since the spins of particles A and B must add up to zero, your particle's spin is random and opposite of Cameron's.
4. Cameron flips the spin of her particle. Your particle's spin flips instantly.
5. You read your particles spin, and say, "Kewl, it's randomly opposite the random value it had before." You go to Cameron's house and ask her what the hell she was trying to tell you.
I'm not positive that the act of measuring an observable tweaks it like this, so I'm not going to say that FTL communication is impossible using Einstein-Podolsky pairs. However, this seems to be the current thinking among physics circles. Sort of a bummer...
Hope this has been helpful to people. Please send any feedback to smegtra@goodnet.com .
Mappings for platform-dependent ASCII characters
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¹: pi
Æ: delta
³: greater-than-or-equal-to
References
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Ferry, David K. "Quantum Mechanics: An Introduction for Device Physicists and Electrical Engineers." Bristol: Institute of Physics Publishing, 1995.
ISBN 0-7503-0327-1. -- ISBN 0-7503-0328-X (pbk.)
Gribbon, John. "In Search of Schrodinger's Cat: Quantum Physics and Reality." Toronto, Bantam Books, 1984.
ISBN 0-553-34103-0