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In classical mechanics when one set system coordinate, length of projection of angular momentum(L) on the axis z can take continuous value from -L to +L, so any direction of L is possible. In quantum mechanics when one set coordinate system, values of projection are finite set of numbers, and so only particular directions of L are possible. However, if one choose another coordinate system, set of possible directions change. And if one repeat this procedure many times, set of possible directions turns out to be as in classical mechanics. I see some contradiction (defining coordinate system determine directions of L) in this fact and I wonder if someone explain that there is no contradiction.

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  • $\begingroup$ The issue is that the direction of $\vec L$ in quantum mechanics can not be defined. It has no meaning. $\endgroup$ Feb 5 '20 at 9:20
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There is no contradiction, the spin space for a 2-level system is for example spanned by the Pauli matrices $\sigma_x,\sigma_y,\sigma_z$ , but any other basis transformation to, say, $\sigma_{x'}=\sigma_x+\sigma_y,\sigma_{y'}=\sigma_y+\sigma_z,\sigma_{z'}=\sigma_x+\sigma_z$ (up to a proper normalization) would work fine.

Yes, measurement will always produce a discrete result. But since Heisenberg prevents you from measuring multiple axes simultaneously, this does not lead to a paradox. There is no fundamental 'special direction', only the one you happen to be measuring and others you don't.

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