On Friday, April 13, 2018 at 12:48:11 PM UTC, Bruce wrote: > > From: <agrays...@gmail.com <javascript:>> > > On Friday, April 13, 2018 at 11:02:03 AM UTC, agrays...@gmail.com wrote: > >> >> On Friday, April 13, 2018 at 3:10:52 AM UTC, Bruce wrote: >>> >>> From: <agrays...@gmail.com> >>> >>> >>> On Thursday, April 12, 2018 at 11:56:40 PM UTC, Bruce wrote: >>>> >>>> From: <agrays...@gmail.com> >>>> >>>> >>>> On Thursday, April 12, 2018 at 10:12:58 PM UTC, agrays...@gmail.com >>>> wrote: >>>>> >>>>> >>>>> On Thursday, April 12, 2018 at 9:26:53 PM UTC, Brent wrote: >>>>>> >>>>>> >>>>>> On 4/12/2018 12:44 PM, agrays...@gmail.com wrote: >>>>>> >>>>>> *Let's simplify the model. Instead of a Nitrogen molecule, consider a >>>>>> free electron at rest in some frame. Its only degree of freedom is spin >>>>>> IIUC. Is it your claim that this electron become entangled with its >>>>>> environment via its spin WF, which is a superposition of UP and DN? Does >>>>>> this spin WF participate in the entanglement? TIA, AG* >>>>>> >>>>>> >>>>>> The electron's spin dof can only become entangled with the >>>>>> environment by an interaction with the environment. >>>>>> >>>>>> Brent >>>>>> >>>>> >>>>> Does that happen spontaneously, in the absence of a measurement? AG >>>>> >>>> >>>> If entanglement of a system with the environment requires measurement, >>>> and if virtually everything in the physical world is entangled with the >>>> environment, aka "the world" -- which seems to be the prevailing belief -- >>>> what concept of measurement do we need to explain this? AG >>>> >>>> >>>> As has been explained, entanglement is the consequence of any >>>> interaction whatsoever. Measurement is just a particular kind of >>>> interaction, one that is controlled and monitored, but otherwise not >>>> special. >>>> >>> >>> Is it correct to assume that once a system becomes entangled with >>> another system, regardless of how it happens the two systems form a >>> relationship analogous to the singlet state where non-locality applies >>> between the two systems now considered non-separable? That is, does >>> entanglement necessarily imply non-locality, a point IIUC which LC made >>> earlier on this thread? AG >>> >>> >>> The systems are not necessarily non-separable. In the classical >>> situation I outlined below, The balls are separable after the interaction >>> because each has a well-defined momentum, even though this might be unknown >>> before one ball is measured. The entanglement is sufficient so that one can >>> determine the momentum of one by measuring the other, but this is not >>> particularly mysterious in the classical case. >>> >>> The quantum case is different in that the particles do not have definite >>> momentum after the interaction -- they are in a superposition of an >>> infinite number of different momentum states, so the particles are not >>> separable -- it requires both particles to specify the overall state. >>> >> >> *I suppose you mean that each particle is represented as a wave packet >> and you're treating the interaction as a scattering problem where EM and >> gravity are not involved, but rather as a "mechanical" interaction where >> momentum is preserved. If the particles become entangled due to the >> interaction, and are now not separable, what exactly does "not separable" >> mean in this context and how does it come about? TIA, AG* >> > > * I think entanglement here means that somehow, through the interaction, > the scattering process, the wf of the total system consists of sums of > tensor states, each a product of the subsystem states, analogous to the wf > of entangled singlet state. Hard to see how this comes about, and its > relation to non-locality. TIA, AG* > > > It might be easier to understand if I give some equations. In the > centre-of-mass frame, after the interaction the total momentum in any > direction is zero. Consider just a one-dimensional case. The combined wave > function is: > > |psi> = Sum_i |p_i>|-p_i>, > > where the first ket is particle 1 and the second ket particle 2 and the > sum is over possible momenta. As you say, this is a tensor product of > individual particle states. Since there is complete correlation between the > momenta of the separate particles for all possible momenta ('possible' > determined by energy conservation), this state cannot be written as a > simple product of sepa-- arate states for particles 1 and 2. Hence it is > non-separable. > > It is not hard to see how this comes about -- it is a direct consequence > of momentum conservation. And the non-locality comes about because a > measurement on particle 1 tells you the momentum of particle 2, no matter > how far apart the particles are. > > Come on Alan. This is not really so hard, you know. > > Bruce >

*I tend to be very detailed oriented, particularly in this subject, so for me it's difficult to see the final result without going through the precise details of the mathematics including, importantly in this case, the interaction Hamiltonian -- a subject I need to review. What does it look like in this case? I assume you are summing over the momenta of the wave packets, which brings up another issue. As I recall, for a free particle, the usual treatment is to assume some fixed momentum and then solve the SWE. But this is not quite right given the uncertainty principle. In reality, one must use a wave packet and it seems non trivial to determine its spread in a particular situation. 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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.