> On 8 Nov 2018, at 18:25, agrayson2...@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 <javascript:> 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 >>>>>>> <http://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 >>>> >>>> <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 >>>> >>>> <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. 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 >>> <https://groups.google.com/group/everything-list>. >>> For more options, visit https://groups.google.com/d/optout >>> <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 >> <https://groups.google.com/group/everything-list>. >> For more options, visit https://groups.google.com/d/optout >> <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 > <mailto:everything-list+unsubscr...@googlegroups.com>. > To post to this group, send email to everything-list@googlegroups.com > <mailto:everything-list@googlegroups.com>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <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.
