On 7/28/2025 12:46 PM, Alan Grayson wrote:
Who says it's only two? Any two independent vectors in 2D complex Hilbert space can be used as a basis.On Monday, July 28, 2025 at 1:24:46 PM UTC-6 Brent Meeker wrote: On 7/28/2025 2:43 AM, Alan Grayson wrote:On Monday, July 28, 2025 at 12:24:33 AM UTC-6 Brent Meeker wrote: 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 TY. Why are U and D orthogonal? A Hilbert space, IIUC, is like any vector space except that it's complete, meaning that it has a metric and Cauchy sequences converge. AGBecause they're modeling the physical state and the state can't be both |U> and |D>, <U|D>=0. When you measure along some axis the result must be an eigen vector of the measurement (a repeat measurement must give 100% the same). So a measurement along an axis (calling it Up and Down is arbitrary) must yield either |U> or |D> as the eigenvector. BrentISTM that the orthogonal property is the usual application of QM, nothing particularly related to Hilbert spaces, where the wf before measurement is the linear sum of all possible states of the system BEFORE measurement, and are basis states. So, you seem to be assuming that U and D are basis states which span the space so they can't be anti-parallel. But if every linear combination is a legitimate state of the electron when exiting the SG apparatus before measurement, how can the continuous properties of these states be consistent with only two, U or D, being measured? Is this one of unsolved mysteries of QM? AG
Brent -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/ecccc47d-49ba-4aeb-a539-a82f1d254527%40gmail.com.
