> 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] <javascript:> 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, [email protected] >>>>>> <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 > > 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] <javascript:>. >> To post to this group, send email to [email protected] >> <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 [email protected] > <mailto:[email protected]>. > To post to this group, send email to [email protected] > <mailto:[email protected]>. > 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 [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.
