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Spin

Jr. Plender
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Joined: Dec 12, 2006
Total Topics: 3
Total Posts: 15

Posted 01/05/08 - 02:58 PM: Subject: Spin	#1
Someone explain to me, in the most simple terms, what spin is. What does it mean if a particle has a spin of 1 or a half? And how are they governed in relation to Bosi-Einstein-Statics and Fermi-Dirac-Statistics? Imagine you're explaining it to a 5 year old, so that I can get to grips with it, it would be great if anyone could explain the maths as well. Thanks.
_____________________ "Pardon one mistake and encourage the comission of many" - Publilius Syrus

derekc153
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Joined: Feb 28, 2007
Location: Arizona, United States
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Posted 01/06/08 - 04:00 PM:	#2
Spin is a fundamental property of particles, like mass and charge, which behaves like an angular momentum. You might be aware that, in classical physics, momentum is simply mass times velocity (velocity being speed in a particular direction). Angular momentum is the momentum around a particular axis, such as the momentum of a tire rotating around the axle (if you try to stop a spinning tire with your hand, you will find that it resists your attempt to stop it. This is due to the angular momentum, much like ordinary momentum makes it difficult to stop a linebacker charging down the field). You can think of spin as a particle spinning about its axis (this is how it behaves anyway), but in fact that is not strictly correct. If an electron's spin were due to the particle literally spinning about its axis, it would have to spin faster than the speed of light, which obviously violates relativity. To say that a particle has a particular spin, such as 1 or 1/2, is to specify the spin quantum number, s. The magnitude of the spin angular momentum of a particle is: \|S\| = sqrt[h_bar^2s(s+1)], where \|S\| is the magnitude of the spin angular momentum, "sqrt[...]" is the square root, h_bar is the reduced Planck's constant, x^2 is a quantity squared (in this case, the reduced Planck's constant), "" denotes multiplication, and s is the spin quantum number. Angular momentum is represented as a vector. If a ball is spinning in the xy plane, the angular momentum is said to point in the z direction. More specifically, curl you fingers in the direction in which the ball is spinning and stick your thumb out. Your thumb will point in the direction of the angular momentum (this is the right-hand-rule, or one variety of it anyway). The maximum possible value of the z-component of the spin is h_bars (the exact magnitude is h_barm, where m is the "magnetic quantum number", but I don't think we need to worry about that). The choice of z as the component to specify with s is basically arbitrary, but it is important to note that it is impossible to simultaneously specify the x, y, and z components of the spin (though the total magnitude of the spin may be known). This is because, as it so happens, specifying all three components (x, y, and z) of the spin would violate the Heisenberg uncertainty principle (one form of which forbids the possibility of simultaneously specifying both the position and the momentum of a particle). Spin can take either integer or half-integer values. Particles with integer spin (s = 0, 1, 2, etc) are known as bosons, and they are governed by Bose-Einstein statistics. An example of a boson is a photon (a particle of light). Particles with half-integer spin (s = 1/2, 3/2, etc) are called fermions, and they follow Fermi-Dirac statistics. An example of a fermion is an electron. As to explaining the specifics of Bose-Einstein and Fermi-Dirac statistics, I don't remember the math all that well, but I think I can explain some of the general ideas. For bosons (Bose-Einstein statistics) one has to symmetrize the wave-function, whereas fermions (Fermi-Dirac statistics) require that one anti-symmetrize the wave-function. The wave function* is related to the probability density of a particle, or the likelihood of measuring it to be in a given place at a given time. More specifically, the modulus-squared of the wave function is equal to the probability density (to find the modulus squared of x, multiply x by the "complex-conjugate" of x. The complex-conjugate is simply x with all the imaginary numbers i changed to -i). So what exactly does "symmetrizing/anti-symmetrizing the wave function" entail? I honestly would have to look back through my books and wrestle with it for a while. In any case, if you haven't studied this stuff, it's unlikely my explanation would help you (indeed, it might even hurt, since I'm not an expert on this stuff and I might say something wrong, which hopefully I haven't already). I can, however, tell you a little bit about some of the results for bosons and fermions. Fermions are subject to the Pauli-exclusion principle, which means that no two fermions can occupy the same energy state at the same time. If you've ever taken a chemistry course, you might be aware that there are different energy levels that an electron may occupy in an atom. The Pauli-exclusion principle prevents multiple electrons from occupying the same level. If that doesn't make any sense, a simpler example of the Pauli exclusion principle is the fact that you can't stick your hand through the wall (at least without hurting the wall or your hand). There is no such principle for bosons, however. Thus, bosons tend to all condense to the "ground state", or lowest available energy level. In practice, temperature effects often prevent that from happening (I think), but scientists have observed bose-einstein condensation in a lab (extremely low temperatures are required). I hope that was somewhat helpful. If anyone finds a mistake in my post (entirely possible), please point it out. I don't want to mislead anyone, and I'm certainly no expert. Edited by derekc153 on 01/06/08 - 04:20 PM
_____________________ "There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy." -Shakespeare's Hamlet

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