On 7/27/2025 6:14 PM, Alan Grayson wrote:
On Sunday, July 27, 2025 at 4:45:42 PM UTC-6 John Clark wrote:
On Sun, Jul 27, 2025 at 6:01 PM Alan Grayson <[email protected]>
wrote:
/> What I seek is a mathematical proof that the UP state can
be written as a linear sum of LT and RT states./
*Nobody can provide a mathematical proof of that, if it was
possible mathematicians would have predicted the laws of quantum
mechanics about the year 1800, maybe earlier. That's why
physicists need to do experiments, and in this case there is a
_physical proof_. Suppose I've measured a beam of electrons and
know they are all spin up. If you're right and they are
independent qualities then that information will be of no help
whatsoever in predicting what I will get if I decide to measure
that beam and see if the electrons in it are spin left or spin
right, but it is of considerable help.*
*
*
*Thanks to that information I can predict that 50% of the
electronswill be spin left and 50% will be spin right. And I can
also predict thatif I decide to recheck the spin left particles to
make sure they are still spin up I will find that they are _NOT_,
and the same is true if I measure the spin right particles. So I
can know if an electron is spin up or spin down, _OR_ I can know
if an electron is spin left or spin right, _BUT_ I can't know
both, and I could if they were independent qualities. So you're
wrong and I'm right, it's as simple as that. *
John K Clark See what's on my new list at Extropolis
<https://groups.google.com/g/extropolis>
*I'm not disputing the experimental results, but it's hardly obvious
that this means UP (or DN) can be written as a linear sum of RT and LT
as it violates the basic rules of vector spaces. AG*
It's confusing because the linear sum is in Hilbert space not 3-space.
In Hilbert space |U> and |D> are not anti-parallel, they are orthogonal,
<U|D>=0. There are only 2-dimensions in the Hilbert space of a spin 1/2
particle. So the space is spanned by any two orthogonal vectors; so
|Left>=(|U>+|D>)/sqrt{2} and |Right=(|U>-|D>)/sqrt{2}. That's a
mathematical proof if you believe in Hilbert space.
Brent
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