> On 11 Nov 2018, at 18:43, [email protected] wrote:
>
>
>
> On Sunday, November 11, 2018 at 7:52:00 AM UTC, Bruno Marchal wrote:
>
>> On 10 Nov 2018, at 01:27, [email protected] <javascript:> wrote:
>>
>>
>>
>> On Friday, November 9, 2018 at 12:26:52 PM UTC, Bruno Marchal wrote:
>>
>>> On 8 Nov 2018, at 18:25, [email protected] <> wrote:
>>>
>>>
>>>
>>> On Thursday, November 8, 2018 at 11:04:20 AM UTC, Bruno Marchal wrote:
>>>
>>>> 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] <> 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
>>>
>>> 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.
That will not help. Just say “I don’t understand”.
> 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.
The question was how to distinguish a pure state like up’ = up + down from a
mixture of up and down particles. The answer is: by measuring up’ in the base
{up’, down’}, we will get the Eigen state up’ with value +1, probability one.
And we don’t get that with a mixture of up and down, as each have probability
1/2 when measure in that base, by the Born rule.
> 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
The rotation is in the Hilbert space, but for electron spin, and photon
polarisation, the apparatus is changed by a simple rotation (actually with
different angle for polarisation and spin, but that is not relevant here).
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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