> On 13 Nov 2018, at 19:38, agrayson2...@gmail.com wrote:
> 
> 
> 
> On Monday, November 12, 2018 at 2:32:53 AM UTC, agrays...@gmail.com wrote:
> 
> 
> On Sunday, November 11, 2018 at 5:43:00 PM UTC, agrays...@gmail.com <> wrote:
> 
> 
> On Sunday, November 11, 2018 at 7:52:00 AM UTC, Bruno Marchal wrote:
> 
>> On 10 Nov 2018, at 01:27, agrays...@gmail.com <> 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 
>>>>>>>>> <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.
>> 
>> 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
> 
> Here's another way to look at this issue: there are only two possible 
> outcomes, 1 and -1, each with probability .5, when the apparatus is oriented 
> in some arbitrary direction.

Not at all. You can have different probabilities, if you rotate a little bit 
the apparatus, you can get all probabilities that you want. You have only 
probability 1/2 when you measure the state 1/sqrt(2)(up+down) in the up/down 
apparatus.





> We know the mathematical forms of these eigenstates, and they correspond to 
> Up or Dn measurements.

The state does not determine the choice of the measurement.



> Since the total probability cannot exceed 1, there are no other eigenstates 
> possible, say for Up + Dn, or Up - Dn, for any arbitrary orientation since 
> the sum of probabilities must equal 1. AG


That does not make any sense. See the explanation in the previous post. You 
misunderstand basic QM I’m afraid.

Bruno



> 
> 
> 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 
>>>>> 
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