> On 4 Nov 2018, at 21:32, Jesse Mazer <laserma...@gmail.com> wrote:
> 
> 
> 
> On Wed, Oct 31, 2018 at 7:30 AM Bruno Marchal <marc...@ulb.ac.be 
> <mailto:marc...@ulb.ac.be>> wrote:
> 
>> On 30 Oct 2018, at 14:21, agrayson2...@gmail.com 
>> <mailto:agrayson2...@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?

I limit myself to spin or polarisation. In some case there are selection or 
superselection rule, like with charge or other possible physical attribute, but 
not here. I use only the core formalism of quantum mechanics, where a change of 
base is a change of the experimental device, like a rotation of some 
Stern-Gerlach apparatus.

Bruno





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