On Tuesday, April 24, 2018 at 1:24:14 AM UTC, Bruce wrote: > > From: <[email protected] <javascript:>> > > On Tuesday, April 24, 2018 at 12:14:06 AM UTC, [email protected] wrote: >> >> >> On Monday, April 23, 2018 at 11:54:15 PM UTC, Bruce wrote: >>> >>> From: <[email protected]> >>> >>> On Monday, April 23, 2018 at 7:38:30 PM UTC, [email protected] wrote: >>> >>>> On Monday, April 23, 2018 at 1:20:05 PM UTC, [email protected] >>>> wrote: >>>>> >>>>> >>>>> >>>>> On Monday, April 23, 2018 at 8:58:53 AM UTC, Bruce wrote: >>>>>> >>>>>> From: <[email protected]> >>>>>> >>>>>> >>>>>> On Monday, April 23, 2018 at 5:53:59 AM UTC, Bruce wrote: >>>>>>> >>>>>>> From: <[email protected]> >>>>>>> >>>>>>> >>>>>>> Let's agree that electrons A and B form a singlet entangled system. >>>>>>> Let's further agree that they are non separable. What do you do with >>>>>>> the >>>>>>> fact that when their spins are measured, they ARE in different spatial >>>>>>> locations, not even space separated in Bell experiments. How do we deal >>>>>>> with this FACT? AG >>>>>>> >>>>>>> >>>>>>> What do you want me to do with the fact? I learn to live with facts >>>>>>> that I can't do anything about. The fact that the system is non-local >>>>>>> is a >>>>>>> fact that you just have to come to terms with. >>>>>>> >>>>>>> Bruce >>>>>>> >>>>>> >>>>>> *ISTM that when you have a theory that seems correct and in some >>>>>> sense is well tested, but there are facts which contradict it, in this >>>>>> case >>>>>> a key fact right in front of your nose which contradicts it -- the fact >>>>>> that we see as plain as daylight that the subsystems as spatially >>>>>> separated >>>>>> -- invariably the theory must be wrong. AG* >>>>>> >>>>>> >>>>>> I wish you luck with your project to prove quantum mechanics wrong. >>>>>> >>>>>> Bruce >>>>>> >>>>> >>>>> *Right now I have a more modest goal. Starting from the postulates of >>>>> QM, how do you justify writing the wf of the singlet state as a >>>>> superposition of tensor product states? TIA AG * >>>>> >>>> >>>> *What it's not. It's not the SWE. It's not Born's Rule. It's not the >>>> operator correspondence with observables. AG * >>>> >>> >>> *I suppose it could be traced to the superposition principle; that the >>> state vector of the singlet state is a linear combination of the states >>> which are members of the corresponding Hilbert space of the system. But why >>> are these states tensor product states? AG* >>> >>> >>> Why try worrying these things out for yourself? The easiest thing is to >>> go and look up a text book. >>> >>> Bruce >>> >> >> *Recall when I asked whether entanglement necessarily implies non >> locality. You replied "not necessarily" and gave the classical example of >> elastic scattering of billiard balls where the momentum of its constituents >> and the whole system is known exactly. No uncertainty. In the wf for the >> singlet system you assume a definite net spin angular momentum, zero. How >> can you treat the singlet system quantum mechanically and at the same time >> assume you know its spin momentum exactly? Do you think this question could >> be answered in a text book? How could I even pose it to an inert, non >> responsive medium? AG * >> > > *I just took a quick look at chapter 15, section 4 of Merzbacher, Quantum > Mechanics (Third Edition). The tensor equation can't be copied. It appears > in the blank lines below. Immediately you can see the problem with this > kind of treatment. It doesn't explain WHY, from First Principles, the > tensor product can be used to describe the composite system. It's virtually > impossible to find an explanation from First Principles. AG* > > > 4. Quantum Dynamics in Direct Product Spaces and Multiparticle Systems. > Often the state vector space of a system can be regarded as the direct, > outer, or tensor product of vector spaces for simpler subsystems. The > direct product space is formed from two independent unrelated vector spaces > that are respectively spanned by the basis vectors /A;) and I B;) by > constructing the basis vectors > > Although the symbol @ is the accepted mathematical notation for the direct > product of state vectors, it is usually dispensed with in the physics > literature, and we adopt this practice when it is unlikely to lead to > misunderstandings. If n1 and n2 are the dimensions of the two factor > spaces, the product space has dimension nl X n2. This idea is easily > extended to the construction of direct product spaces from three or more > simple spaces. > > > Quite right. And what else are you going to use for many-particle systems > that have independent Hilbert spaces -- you multiply them together, of > course. > > Bruce >
Is this what you would characterize as a rigorous analysis? I can think of other alternatives. The answer has to be from First Principles or it's just hand waving. AG -- 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 post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
