On 13/03/2005, at 9:57 AM, Dan Minette wrote:
The best place to start, I think, is spin. My old foundations of QM
teacher said that spin was probably the most QMish of all the aspects of
QM.
So, lets consider a spin 1/2 particle: the electron. Spin is intrinsic
angular momentum. It cannot be the real "spinning" of the electron without
the surface of the electron going faster than the speed of light. So, here
we have one new feature already....intrinsic angular momentum without any
observable motion.
In any given direction, a measurement of the spin of the electron gives
either +1/2 or -1/2. If one measures the spin of the electron as up in a
given direction, and then remeasures it at an angle 2x from the original
direction, one gets up again cos(x)^2 of the time and down sin(x)^2 of the
time. For example, if one measures at 180 degrees, x=90 degrees, cos(x)^2
=0 and sin(x)^2=1. This makes sense, because at 180 degrees, one should
always get down. If one measures at 90 degrees, x=45 degrees. At that
angle, cos(x)^2=.5, sin(x)^2=.5....which also makes sense.
To get this, the wave function is given as sin(x)*|d> + cos(x)*|u> ...a
superposition of two eigenstates: |d> (spin down) and |u> (spin up). The
wavefunction itself is not an observable, we only observe the eigenstates.
When this was first developed, Einstein accepted that the formalism worked,
but he thought that the indeterminacy inherent in this formalism would
eventually be replaced by a more deterministic physics. Attempts to develop
this has been labeled "hidden variable" theories, because they assume that
there are more classical variables that we don't see yet underlying QM.
But, I want to make sure that this step in the formalism is accepted first.
If this doesn't make sense, I need to clarify it before going on. (Lurkers
are encouraged to unlurk and ask questions if they need clarification.)
So far it's at just the right level for me. Not sure how long I'll be able to keep up though! You are bringing back some memories from 35 years ago, so I'm pretty rusty. I don't think I ever really got the idea of eigenstates at the time. Earlier you wrote:
Are you familiar with eigenstates and superpositions? For example, if you measure the spin in the x direction, the spin in the y (which is orthogonal to x) is a superposition of up and down. |s> = ( |+> + |->)/sqrt(2).
which is now much clearer to me than before. Sin(x)^2 and cos(x)^2 refer to probability amplitudes iirc, though why the angle used to remeasure is 2x momentarily escapes me.
Anyway, hope you and Warren manage to keep the discussion going as I am finding it most interesting. I'd like to join in some, but time seems to be at a premium.
Regards, Ray.
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