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Axisless rotation
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Axisless rotation
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Atomic-S
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Axisless rotation
«
on:
13/05/2008 07:01:27 »
Classical geometry tells us that when a rigid sphere rotates in three dimensional
space, it is impossible for all portions of the surface to be moving with the same
speed at the same time; indeed, to be moving at all, because there always must be
an axis. The axis can lay any direction, but an axis there must be. It is related
to the principle that the divergence of the curl of a vector field is inherently
zero. Trying to get around this principle by allowing the sphere to be non-rigid
does not help a whole lot, because there still must be at least one pair of
entrance and exit whorls of opposing handedness, again related to the same vector
principle.
But given the uncertainties in quantum mechanics, it is worth asking whether in the
quantum world, what is classically impossible might be possible. That is, having a
structure which spins but in such a way that there is no axis; all directions are
equivalent. That would be analogous to a sphere all portions of whose surface are
moving about the center at the same speed at the same time.
In the case of particles of spin 1/2, it appears that this is not possible, but
that any (stationary) spin state of such a particle must have one and only one
definite axis about which the spin projection component is fully unambiguous.
States can be found which will place the axis in any direction we choose, but must
always exist and is always the same direction for the specific chosen state.
In the case of particles of spin 1, things appear more interesting. About a given
axis, a state can be chosen such that the component of spin about it is
unambiguously zero. This state will have the characteristic that the spin about any
other axis is always ambiguous. Therefore the only axis of unambiguity is an axis
of non-rotation, from whence it follows that the rotation is axisless. However, it
is not quite fully symmetrical, in that the rotation does not appear identical for
all axes. For axes at right angles to that of non-rotation, there is a 50%--50%
chance that a measurement will show spin "up" or spin "down". For axes more closely
aligned with the axis of non-rotation, the probability of "up" or "down"
diminishes, equally for both, while the probability of "no rotation" increases,
approaching 100% as the axis approaches that of total irrotation. And although it
is clear that this achieves axisless rotation, there does not seem to be a way to
generalize the state sufficiently to achieve total symmetry in all directions. The
best we can do seems to be some kind of ellipsoid of ambiguity having 3 mutually
perpendicular axes along which the probability profile is somehow special.
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LeeE
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Axisless rotation
«
Reply #1 on:
13/05/2008 16:02:09 »
I'm not really up on QM, but the first thing that comes to mind is that axisless rotation could be considered to be akin to spreading - a bit like the expansion of the universe but confined to a closed surface. In that respect, I think the axis would become the origin but the effect, when viewed from 90 deg to the axis, would appear to be rotation.
No idea if that helps or not.
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Axisless rotation
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