Spin 1/2 Particle: Uncertainty Principle Explained

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SUMMARY

The discussion centers on the behavior of spin 1/2 particles in relation to the uncertainty principle as described in Griffiths' text. It is established that when a particle is measured in the ↑ state along the z-axis, the z-component of spin angular momentum (Sz) is defined as h bar/2, while the x-component (Sx) can take values of either h bar/2 or -h bar/2. This illustrates that Sz and Sx cannot have definite values simultaneously, emphasizing the core tenet of the uncertainty principle. The choice of the z-axis for measurement is conventional, and the uncertainty principle applies regardless of the axis chosen for measurement.

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Molar
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In my book (Griffiths) it is said :

A particle is in ↑ state.
The z-component of the particle's spin angular momentum [Sz] is h bar/2.
The x- component of the particle's spin angular momentum [Sx] can be either h bar/2 or
-h bar /2 .

He says this is for uncertainty principle...that both Sz and Sx can't have definite value.
This is where i am having difficulties.

I mean it is always strictly like this that z component would give certain values and uncertainty would follow
x-component...?
or just because we have measured z component first...ie, if we had measured x-component first then we would
have the value h bar /2 for x- component and the uncertainty principle would act on z-component...??
or because spin is always defined along Z axis...?

thanks...
 
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Molar said:
or just because we have measured z component first...ie, if we had measured x-component first then we would
have the value h bar /2 for x- component and the uncertainty principle would act on z-component...??
Right.

The particle had to have been prepared in the ↑ state with respect to the z-axis. It could have been any axis--they just like to pick on the z-axis for some reason.
 
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