[Edited]

So, I am trying to understand something. The need for the expectation value I seem to be learning - the example I have been learning involves spin.

I have the equation $$|S_{xu}>= \frac{1}{\sqrt{2}}[|S_{zu}>+|S_{zd}>]$$ with properties of

$$<S_{zu}|S_{zu}>=<S_{zd}|S_zd>=1$$

and

$$<S_{zu}|S_{zd}>= 0$$

If there is a measurement made on $$|S_{xu}>$$ it will yield for every measurement on the x-direction will be a spin up. But the z-direction is undefined since it has a superpositing in the z-direction which exhibits a spin up and a spin down in equal probability.

So the z-direction has an undefined spin, and is this the reason for the expectation value in this particular area? Since $$S_{xu}$$ does not have a particular spin direction for z, it is still possible to calculate a mean value over large periods of time. So the expectation is $$<S_{xu|\hat{S}_z|S_{xu}>$$.

What I do not understand, is if the spin direction (up or down) cannot be defined for z, why is an expectation value required, what does it do exactly?

So, I am trying to understand something. The need for the expectation value I seem to be learning - the example I have been learning involves spin.

I have the equation $$|S_{xu}>= \frac{1}{\sqrt{2}}[|S_{zu}>+|S_{zd}>]$$ with properties of

$$<S_{zu}|S_{zu}>=<S_{zd}|S_zd>=1$$

and

$$<S_{zu}|S_{zd}>= 0$$

If there is a measurement made on $$|S_{xu}>$$ it will yield for every measurement on the x-direction will be a spin up. But the z-direction is undefined since it has a superpositing in the z-direction which exhibits a spin up and a spin down in equal probability.

So the z-direction has an undefined spin, and is this the reason for the expectation value in this particular area? Since $$S_{xu}$$ does not have a particular spin direction for z, it is still possible to calculate a mean value over large periods of time. So the expectation is $$<S_{xu|\hat{S}_z|S_{xu}>$$.

What I do not understand, is if the spin direction (up or down) cannot be defined for z, why is an expectation value required, what does it do exactly?

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