On Sunday, November 11, 2018 at 7:52:00 AM UTC, Bruno Marchal wrote: > > > On 10 Nov 2018, at 01:27, agrays...@gmail.com <javascript:> wrote: > > > > On Friday, November 9, 2018 at 12:26:52 PM UTC, Bruno Marchal wrote: >> >> >> On 8 Nov 2018, at 18:25, agrays...@gmail.com wrote: >> >> >> >> On Thursday, November 8, 2018 at 11:04:20 AM UTC, Bruno Marchal wrote: >>> >>> >>> On 6 Nov 2018, at 12:22, agrays...@gmail.com wrote: >>> >>> >>> >>> On Tuesday, November 6, 2018 at 9:27:31 AM UTC, Bruno Marchal wrote: >>>> >>>> >>>> On 4 Nov 2018, at 22:02, agrays...@gmail.com wrote: >>>> >>>> >>>> >>>> On Sunday, November 4, 2018 at 8:33:10 PM UTC, jessem wrote: >>>>> >>>>> >>>>> >>>>> On Wed, Oct 31, 2018 at 7:30 AM Bruno Marchal <mar...@ulb.ac.be> >>>>> wrote: >>>>> >>>>>> >>>>>> On 30 Oct 2018, at 14:21, agrays...@gmail.com wrote: >>>>>> >>>>>> >>>>>> >>>>>> On Tuesday, October 30, 2018 at 8:58:30 AM UTC, Bruno Marchal wrote: >>>>>>> >>>>>>> >>>>>>> On 29 Oct 2018, at 13:55, agrays...@gmail.com wrote: >>>>>>> >>>>>>> >>>>>>> >>>>>>> On Monday, October 29, 2018 at 10:22:02 AM UTC, Bruno Marchal wrote: >>>>>>>> >>>>>>>> >>>>>>>> On 28 Oct 2018, at 13:21, agrays...@gmail.com wrote: >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> On Sunday, October 28, 2018 at 9:27:56 AM UTC, Bruno Marchal wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>> On 25 Oct 2018, at 17:12, agrays...@gmail.com wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> On Tuesday, October 23, 2018 at 10:39:11 PM UTC, agrays...@ >>>>>>>>> gmail.com wrote: >>>>>>>>>> >>>>>>>>>> If a system is in a superposition of states, whatever value >>>>>>>>>> measured, will be repeated if the same system is repeatedly >>>>>>>>>> measured. But >>>>>>>>>> what happens if the system is in a mixed state? TIA, AG >>>>>>>>>> >>>>>>>>> >>>>>>>>> If you think about it, whatever value you get on a single trial >>>>>>>>> for a mixed state, repeated on the same system, will result in the >>>>>>>>> same >>>>>>>>> value measured repeatedly. If this is true, how does measurement >>>>>>>>> distinguish superposition of states, with mixed states? AG >>>>>>>>> >>>>>>>>> >>>>>>>>> That is not correct. You can distinguish a mixture of particles in >>>>>>>>> the up or down states with a set of 1/sqrt(2)(up+down) by measuring >>>>>>>>> them >>>>>>>>> with the {1/sqrt(2)(up+down), 1/sqrt(2)(up-down}) discriminating >>>>>>>>> apparatus. >>>>>>>>> With the mixture, half the particles will be defected in one >>>>>>>>> direction, >>>>>>>>> with the pure state, they will all pass in the same direction. >>>>>>>>> Superposition would not have been discovered if that was not the case. >>>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> *And someone will supply the apparatus measuring (up + down), and >>>>>>>> (up - down)? No such apparatuses are possible since those states are >>>>>>>> inherently contradictory. We can only measure up / down. AG* >>>>>>>> >>>>>>>> >>>>>>>> You can do the experience by yourself using a simple crystal of >>>>>>>> calcium (CaCO3, Island Spath), or with polarising glass. Or with >>>>>>>> Stern-Gerlach devices and electron spin. Just rotating (90° or 180°) >>>>>>>> an >>>>>>>> app/down apparatus, gives you an (up + down)/(up - down) apparatus. >>>>>>>> >>>>>>> >>>>>>> *I don't understand. With SG one can change the up/down axis by >>>>>>> rotation, but that doesn't result in an (up + down), or (up - down) >>>>>>> measurement. If that were the case, what is the operator for which >>>>>>> those >>>>>>> states are eigenstates? Which book by Albert? AG * >>>>>>> >>>>>>> >>>>>>> David Z Albert, Quantum Mechanics and Experience, Harvard University >>>>>>> Press, 1992. >>>>>>> >>>>>>> https://www.amazon.com/Quantum-Mechanics-Experience-David-Albert/dp/0674741137 >>>>>>> >>>>>>> Another very good books is >>>>>>> >>>>>>> D’Espagnat B. Conceptual foundations of Quantum mechanics, I see >>>>>>> there is a new edition here: >>>>>>> >>>>>>> https://www.amazon.com/Conceptual-Foundations-Quantum-Mechanics-Advanced/dp/0738201049/ref=sr_1_1?s=books&ie=UTF8&qid=1540889778&sr=1-1&keywords=d%27espagnat+conceptual+foundation+of+quantum+mechanics&dpID=41NcluHD6fL&preST=_SY291_BO1,204,203,200_QL40_&dpSrc=srch >>>>>>> >>>>>>> It explains very well the difference between mixtures and pure >>>>>>> states. >>>>>>> >>>>>>> Bruno >>>>>>> >>>>>> >>>>>> *Thanks for the references. I think I have a reasonable decent >>>>>> understanding of mixed states. Say a system is in a mixed state of phi1 >>>>>> and >>>>>> phi2 with some probability for each. IIUC, a measurement will always >>>>>> result >>>>>> in an eigenstate of either phi1 or phi2 (with relative probabilities >>>>>> applying). * >>>>>> >>>>>> >>>>>> If the measurement is done with a phi1/phi2 discriminating apparatus. >>>>>> Keep in mind that any state can be seen as a superposition of other >>>>>> oblique >>>>>> or orthogonal states. >>>>>> >>>>> >>>>> I don't know if you're restricting the definition of phi1 and phi2 to >>>>> some particular type of eigenstate or not, but in general aren't there >>>>> pure >>>>> states that are not eigenstates of any physically possible measurement >>>>> apparatus, so there is no way to directly measure that a system is in >>>>> such >>>>> a state? >>>>> >>>> >>>> *Yes, such states exist IIUC. That's why I don't understand Bruno's >>>> claim that Up + Dn and Up - Dn can be measured with any apparatus, * >>>> >>>> >>>> Not *any*¨apparatus, but a precise one, which in this case is the same >>>> apparatus than for up and down, except that it has been rotated. >>>> >>>> >>>> >>>> >>>> *since they're not eigenstates of the spin operator, or any operator. * >>>> >>>> >>>> This is were you are wrong. That are eigenstates of the spin operator >>>> when measured in some direction. >>>> >>> >>> *If what you claim is true, then write down the operator for which up + >>> dn (or up - dn) is an eigenstate? AG * >>> >>> >>> >>> It is the operator corresponding to the same device, just rotated from >>> pi/2, or pi (it is different for spin and photon). When I have more time, I >>> might do the calculation, but this is rather elementary quantum mechanics. >>> (I am ultra-busy up to the 15 November, sorry). It will have the same shape >>> as the one for up and down, in the base up’ and down’, so if you know a bit >>> of linear algebra, you should be able to do it by yourself. >>> >>> Bruno >>> >> >> *You don't have to do any calculation. Just write down the operator >> which, you allege, has up + dn or up - dn as an eigenstate. I don't think >> you can do it, because IMO it doesn't exist. AG * >> >> >> >> If up and down are represented by the column (1 0) and (0 1) the >> corresponding observable is given by the diagonal matrix >> >> 1 0 >> 0 -1 >> >> Then the up’ = 1/sqrt(2) (1 1), and down’ = 1/sqrt(2) (1 -1), >> >> So the operator, written in the base up down, will be >> >> 0 1 >> 1 0 >> >> Here the eigenvalue +1 and -1 correspond to up (up’) or down (down’). >> >> I have no clue why you think that such operator would not exist. >> > > *Because the measured spin state is Up or Dn along some axis, never > anything in between. Up + Dn or Up - Dn is not physically realizable in > unprimed basis. AG* > > > > If the measured spin state is Up or Dn along some axis, the measured spin > state will be Up + Dn or Up - Down along the axis obtained by rotating the > measuring apparatus adequately. >

*But NOT along the original spin axis! You can't measure Up + Dn or Up - Dn along any particular spin axis that you choose. That was my point. If you rotate the axis, the same situation exists. AG * That is physically realisable with spin (by just rotating the Stern-Gerlach > apparatus) of with light polarisation (rotating the polariser or the CaCO3 > crystal). > > Bruno > > > > > > All pure state can be seen as a superposition, in the rotated base, and >> you can always build an operator having them as eigenvalues. >> >> Bruno >> >> >> >> >> >> >> >> >> >>> >>> >>> >>> >>> >>> >>>> Julian Swinger (and Townsend) showed that the formalism of (discrete, >>>> spin, qubit) quantum mechanics is derivable from 4 Stern-Gerlach >>>> experiments, using only real numbers, but for a last fifth one, you need >>>> the complex amplitudes, and you get the whole core of the formalism. >>>> >>>> Bruno >>>> >>>> >>>> >>>> >>>> *Do you understand Bruno's argument in a previous post on this topic? >>>> 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-li...@googlegroups.com. >>>> To post to this group, send email to everyth...@googlegroups.com. >>>> Visit this group at https://groups.google.com/group/everything-list. >>>> For more options, visit https://groups.google.com/d/optout. >>>> >>>> >>>> >>> -- >>> 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-li...@googlegroups.com. >>> To post to this group, send email to everyth...@googlegroups.com. >>> Visit this group at https://groups.google.com/group/everything-list. >>> For more options, visit https://groups.google.com/d/optout. >>> >>> >>> >> -- >> 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-li...@googlegroups.com. >> To post to this group, send email to everyth...@googlegroups.com. >> Visit this group at https://groups.google.com/group/everything-list. >> For more options, visit https://groups.google.com/d/optout. >> >> >> > -- > 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-li...@googlegroups.com <javascript:>. > To post to this group, send email to everyth...@googlegroups.com > <javascript:>. > Visit this group at https://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/d/optout. > > > -- 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.