# Re: Measuring a system in a superposition of states vs in a mixed state

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On Saturday, November 17, 2018 at 4:22:35 PM UTC, agrays...@gmail.com wrote:
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> On Friday, November 16, 2018 at 4:39:42 PM UTC, scerir wrote:
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>> Il 16 novembre 2018 alle 15.38 agrays...@gmail.com ha scritto:
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>> On Friday, November 16, 2018 at 10:14:32 AM UTC, scerir wrote:
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>> Il 16 novembre 2018 alle 10.19 agrays...@gmail.com ha scritto:
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>> On Thursday, November 15, 2018 at 2:14:48 PM UTC, scerir wrote:
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>> Il 15 novembre 2018 alle 14.29 agrays...@gmail.com ha scritto:
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>> On Thursday, November 15, 2018 at 8:04:53 AM UTC, scerir wrote:
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>> Imagine a spin-1/2 particle described by the state psi = sqrt(1/2)
>> [(s+)_z + (s-)_z] .
>>
>> If the x-component of spin is measured by passing the spin-1/2 particle
>> through a Stern-Gerlach with its field oriented along the x-axis, the
>> particle will ALWAYS emerge 'up'.
>>
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>> *Why?  Won't the measured value be along the x axis in both directions,
>> in effect Up or Dn? AG*
>>
>> "Hence we must conclude that the system described by the |+>x state is
>> not the
>> same as a mixture of atoms in the |+> and !-> states. This means that
>> each atom in the
>> beam is in a state that itself is a combination of the |+> and |->
>> states. A superposition
>> state is often called a coherent superposition since the relative phase
>> of the two terms is
>> important."
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>> .see pages 18-19 here *https://tinyurl.com/ybm56whu
>> <https://tinyurl.com/ybm56whu>*
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>> *Try answering in your own words. When the SG device is oriented along
>> the x axis, now effectively the z-axix IIUC, and we're dealing with
>> superpositions, the outcomes will be 50-50 plus and minus. Therefore,
>> unless I am making some error, what you stated above is incorrect. AG *
>>
>> sqrt(1/2) [(s+)_z +(s-)_z]  is a superposition, but since sqrt(1/2)
>> [(s+)_z +(s-)_z]  =  (s+)_x the particle will always emerge 'up'
>>
>>
>> I'll probably get back to on the foregoing. In the meantime, consider
>> this; I claim one can never MEASURE Up + Dn or Up - Dn with a SG apparatus
>> regardless of how many other instruments one uses to create a composite
>> measuring apparatus (Bruno's claim IIUC). The reason is simple. We know
>> that the spin operator has exactly two eigenstates, each with probability
>> of .5. We can write them down. We also know that every quantum measurement
>> gives up an eigenvalue of some eigenstate. Therefore, if there existed an
>> Up + Dn or Up - Dn eigenstate, it would have to have probability ZERO since
>> the Up and Dn eigenstates have probabilities which sum to unity. Do you
>> agree or not, and if not, why? TIA, AG
>>
>> I think the question should rather be how to prepare a superposition
>> state like  sqrt(1/2) [(s+)_z +(s-)_z] . But when you have this specific
>> state, and when you orient the SG along "x", you always get "up".
>>
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> *If the SG field is oriented perpendicular to z axis, the usual situation
> for a measurement along z, you get Up or Dn along z axis. If field is along
> x axis, which is perpendicular to z axis, the device blocks the stream of
> electrons, so no measurement is possible.*
>```
```

*Correction; the SG device doesn't block stream of elections when its field
is oriented along x axis. But what has this to do with whether one can
measure Up + Dn, or Up - Dn along z axis, or any axis? Does it show Up + Dn
can be measured along x axis? AG*

>
> *Also, note that your simulation uses only Up or Dn, as I did above, to
> show it's impossible to measure Up + Dn, or Up - Dn. Can you respond to my
>
>>
>>
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>> In fact (s+)_z = sqrt(1/2) [(s+)_x + (s-)_x]
>>
>> and (s-)_z = sqrt(1/2) [(s+)_x - (s-)_x]
>>
>> (where _z, _x, are the z-component and the x-component of spin)
>>
>> so that psi = sqrt(1/2)[(s+)_z +(s-)_z] = (s+)_x.   (pure state, not
>> mixture state)..
>>
>> AGrayson2000 asked "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?"
>>
>> Does Everett's "relative state interpretation" show how to interpret a
>> real superposition (like the above, in which the particle will always
>> emerge 'up') and how to interpret a mixture (in which the particle will
>> emerge 50% 'up' or 50% 'down')?
>>
>>
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