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## Main Question or Discussion Point

Hi Guys, I have a question about observable's and superposition's that I haven't been able to find a definitive answer to (purely for the fact that it doesn't seem to be addressed), and would greatly appreciate an answer.

When in a superposition state of an observable, you are also in a definite state of another observable. For example, suppose you have two incompatible observable's from the 1/2 spin states: X-spins and Z-spins. If you are in a definite X-up position, then you are in a superposition of Z-spin down and Z spin up. So mathematically we could express this as:

[tex](1.0)\mid S_x=+ \frac{1}{2}> = a\mid S_z=+\frac{1}{2}> + b\mid S_x=-\frac{1}{2}>[/tex]

Where: [tex]\mid a \mid ^2 +\mid b \mid ^2 =1[/tex]

In the MWH (Many Worlds Hypothesis), we have a problem known as the 'basis problem'. Essentially the crux of the problem is due to the fact that when we perceive the 'collapse of the wavefunction' and thus the splitting of the superposition component states into different branches (or worlds), it all is all dependent on the observable we choose to use, so is the mathematical representation of the splitting a real ontological split or just part of the mathematical eigenbasis representation we chose? That is the 'basis problem' in the MWH.

Now that makes sense to me, but it made me think about superpositions and choosing an observable to measure in general and what it actually means (regardless if we adhere to the MWH or not).

So my question going back to equation (1.0) is: if the system is in a definite state of X spin-up, does that mean X spin-up

I know the distinction can be hard to grasp so I'm trying to make an effort to spell it out. What I'm trying to say is, because the mathematical Hilbert space representation says X spin-up

When in a superposition state of an observable, you are also in a definite state of another observable. For example, suppose you have two incompatible observable's from the 1/2 spin states: X-spins and Z-spins. If you are in a definite X-up position, then you are in a superposition of Z-spin down and Z spin up. So mathematically we could express this as:

[tex](1.0)\mid S_x=+ \frac{1}{2}> = a\mid S_z=+\frac{1}{2}> + b\mid S_x=-\frac{1}{2}>[/tex]

Where: [tex]\mid a \mid ^2 +\mid b \mid ^2 =1[/tex]

In the MWH (Many Worlds Hypothesis), we have a problem known as the 'basis problem'. Essentially the crux of the problem is due to the fact that when we perceive the 'collapse of the wavefunction' and thus the splitting of the superposition component states into different branches (or worlds), it all is all dependent on the observable we choose to use, so is the mathematical representation of the splitting a real ontological split or just part of the mathematical eigenbasis representation we chose? That is the 'basis problem' in the MWH.

Now that makes sense to me, but it made me think about superpositions and choosing an observable to measure in general and what it actually means (regardless if we adhere to the MWH or not).

So my question going back to equation (1.0) is: if the system is in a definite state of X spin-up, does that mean X spin-up

*is*a superposition of the Z-spin? So when we have a definite state of X-spin up, are we are also observing a superposition of the Z-spins (a superposition of Z spins*is*the definite state of X spin up) or is it just Z-spins are in a superposition, when we have a definite X spin-up?I know the distinction can be hard to grasp so I'm trying to make an effort to spell it out. What I'm trying to say is, because the mathematical Hilbert space representation says X spin-up

*equals*the superposition of z-spin down and z-spin up, does that mean they are identical things? Being in an X-spin up state*is*a superposition of z-spin down and z-spin up? Or is it just the case that when a particle is in a definite state of X spin-up, the other observable Z-spins are in a superposition, but they are not identical things?
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