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## Main Question or Discussion Point

In my quantum mechanical studies, I came across the information that if you know an electron's spin on one axis, then you can not know its spin on another axis. For example, if you know that an electron is spin up on the z-axis, then apparently due to the Uncertainty Principle, you can not know whether it is spin right or left on the x-axis.

However, I also found out that apparently you can still apply spin operators to eigenstates that don't correspond to that particular spin operator.

Example: For the spin operator σ on the x-axis:

σ

σ

Now if you apply this operator to the spin up eigenstate <1 , 0> (which corresponds to the z-axis), then the product of your matrix vector multiplication comes out to be:

<0 , 1> (which is the spin down state on the z-axis)

Now what exactly does this mean physically? I initially thought that the fact that you can apply the x-axis spin operator to a z-axis eigenstate and simply get a flipped z-axis eigenstate meant that spin up on the z-axis would turn into spin left on the x-axis (since spin down and spin left are both -1 eigenvalues) if you were to redo a measurement of spin on the x-axis this time. However, wouldn't that interpretation violate the uncertainty principle since I would know that spin up on the z-axis = spin left on the x-axis? If this interpretation is incorrect, then what exactly is the implication of the result of applying a spin operator to an eigenstate for another axis?

However, I also found out that apparently you can still apply spin operators to eigenstates that don't correspond to that particular spin operator.

Example: For the spin operator σ on the x-axis:

σ

_{11}and σ_{22}= 0σ

_{12}and σ_{21}= 1Now if you apply this operator to the spin up eigenstate <1 , 0> (which corresponds to the z-axis), then the product of your matrix vector multiplication comes out to be:

<0 , 1> (which is the spin down state on the z-axis)

Now what exactly does this mean physically? I initially thought that the fact that you can apply the x-axis spin operator to a z-axis eigenstate and simply get a flipped z-axis eigenstate meant that spin up on the z-axis would turn into spin left on the x-axis (since spin down and spin left are both -1 eigenvalues) if you were to redo a measurement of spin on the x-axis this time. However, wouldn't that interpretation violate the uncertainty principle since I would know that spin up on the z-axis = spin left on the x-axis? If this interpretation is incorrect, then what exactly is the implication of the result of applying a spin operator to an eigenstate for another axis?