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 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*
>

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