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

```> On 16 Nov 2018, at 15:38, agrayson2...@gmail.com wrote:
>
>
>
> On Friday, November 16, 2018 at 10:14:32 AM UTC, scerir wrote:
>
>
>> Il 16 novembre 2018 alle 10.19 agrays...@gmail.com <javascript:> ha scritto:
>>
>>
>>
>> On Thursday, November 15, 2018 at 2:14:48 PM UTC, scerir wrote:
>>
>>
>>> Il 15 novembre 2018 alle 14.29 agrays...@gmail.com <> ha scritto:
>>>
>>>
>>>
>>> On Thursday, November 15, 2018 at 8:04:53 AM UTC, scerir wrote:
>>> 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'.
>>>
>>>
>>> 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."
>>
>> .see pages 18-19 here https://tinyurl.com/ybm56whu
>> <https://tinyurl.com/ybm56whu>
>>
>> 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```
```
Which one? There are spin operator for each direction in space. The
superposition of up and down is a precise pure state, with precise eigenvalues,
when measuring state in the complementary directions.

> 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

You add the probabilities, but you need to add the amplitudes of probabilities
instead, and then take their square. You simply dismiss the quantum formalism,
it seems to me. The states constituted a vector space: the sum (superposition)
of orthogonal states are pure state, after a change of base, and I did give you
the corresponding operator. You are not criticising an interpretation of QM,
but QM itself.

Bruno

>
>>
>>>
>>> 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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