On Sunday, November 11, 2018 at 5:43:00 PM UTC, [email protected] wrote: > > > > On Sunday, November 11, 2018 at 7:52:00 AM UTC, Bruno Marchal wrote: >> >> >> On 10 Nov 2018, at 01:27, [email protected] wrote: >> >> >> >> On Friday, November 9, 2018 at 12:26:52 PM UTC, Bruno Marchal wrote: >>> >>> >>> On 8 Nov 2018, at 18:25, [email protected] wrote: >>> >>> >>> >>> On Thursday, November 8, 2018 at 11:04:20 AM UTC, Bruno Marchal wrote: >>>> >>>> >>>> On 6 Nov 2018, at 12:22, [email protected] wrote: >>>> >>>> >>>> >>>> On Tuesday, November 6, 2018 at 9:27:31 AM UTC, Bruno Marchal wrote: >>>>> >>>>> >>>>> On 4 Nov 2018, at 22:02, [email protected] 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 <[email protected]> >>>>>> wrote: >>>>>> >>>>>>> >>>>>>> On 30 Oct 2018, at 14:21, [email protected] wrote: >>>>>>> >>>>>>> >>>>>>> >>>>>>> On Tuesday, October 30, 2018 at 8:58:30 AM UTC, Bruno Marchal wrote: >>>>>>>> >>>>>>>> >>>>>>>> On 29 Oct 2018, at 13:55, [email protected] wrote: >>>>>>>> >>>>>>>> >>>>>>>> >>>>>>>> On Monday, October 29, 2018 at 10:22:02 AM UTC, Bruno Marchal wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>> On 28 Oct 2018, at 13:21, [email protected] wrote: >>>>>>>>> >>>>>>>>> >>>>>>>>> >>>>>>>>> On Sunday, October 28, 2018 at 9:27:56 AM UTC, Bruno Marchal wrote: >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> On 25 Oct 2018, at 17:12, [email protected] 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. >> > > *You are mistaken. According to QM, the measured value is always an > eigenvalue of one of the eigenstates of the operator, in this case either > Up or Dn. After the measurement, the system is in the eigenstate > corresponding to the eigenvalue measured. This eigenstate can be written in > many different bases, but this does not change what has been MEASURED. If > you rotate the apparatus, the same exact situation exists. You will NEVER > measure Up + Dn or Up - Dn regardless of how the apparatus is oriented. AG* >
*Important correction; before measurement, say of spin, the wf can be written in many forms, one being a linear combination of the eigenstates Up and Dn. It's never measured in states Up + Dn or Up - Dn, since these are not eigenstates of the spin operator. This is basic QM. Just because there are many ways to express a wf, doesn't mean a measurement measures every possible expression of that state. 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 [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. >>>>> >>>>> >>>>> >>>> -- >>>> 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. >>>> >>>> >>>> >>> -- >>> 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. >>> >>> >>> >> -- >> 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. >> >> >> -- 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.
