On Sun, Jul 27, 2025 at 3:59 PM Alan Grayson <agrayson2...@gmail.com> wrote:
*>> Your reasoning would be perfectly correct if we were dealing with > classical vectors where the components of directional vectors are real > numbers in 3D space and orthogonal components are 100% independent. But > quantum spin states are not like that, they're vectors in 2-dimensional > Hilbert space and, unlike the sort of vectors that classical physics > usually uses, they absolutely require imaginary numbers. And even though > the experiment is being performed in 3-D space the electron only has two > independent bases states not three. Regardless of if you choose to measure > up-down or right-left you're working in the same 2D Hilbert space where any > state expressible in one basis can be written in the other.* > > > *> Vector spaces have associated fields such as complex numbers, and I > never heard that Von Neumann changed the rules of vectors spaces when he > applied them to QM. Can you cite such a change? It's so radical that it > must be clearly documented. AG* > *Von Neumann didn't change the rules of vector space nor did anybody else, what changed is what physical qualities correspond to what mathematical objects. To be consistent with experimental results, when dealing with the quantum world the physical interpretation has to be different than it is in classical physics. For example:* *Classical: 3D objects spin components along x, y, z are 3 INDEPENDENT physical quantities.* *Quantum: Spin-1/2 particles such as electrons only have 2 INDEPENDENT degrees of freedom, despite having measurable components along all three spatial axes, that's why **any state expressible in one basis can be written in the other.* *Y**ou wanted a reference for all this and you can find it in any textbook on quantum mechanics, such as the one by Dirac or Sakurai.* John K Clark See what's on my new list at Extropolis <https://groups.google.com/g/extropolis> 74d > -- 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 everything-list+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/everything-list/CAJPayv0kCRq0o5Dyt2J5i2FtdWvmUpaGD76zpA8%3DTFWiHjq1rg%40mail.gmail.com.
