The reference frame for angular momentum components

[hokhani]

TL;DR Summary

The reference axis of quantization for angular momentum components

In which coordinate system the components of angular momentum are quantized? Better to say, if we can select the coordinate system arbitrarily, how the components of angular momentum, say z-component, are always $L_z=m\hbar$?

 

[anuttarasammyak]

Any choice of z axis direction is allowed for the quantization.

 

[PeroK]

Summary:: The reference axis of quantization for angular momentum components

In which coordinate system the components of angular momentum are quantized? Better to say, if we can select the coordinate system arbitrarily, how the components of angular momentum, say z-component, are always $L_z=m\hbar$?

This provides a good example of the QM principle that dynamic quantities do not have well-defined values unless they are measured. If a system really was spinning in some definite direction about some axis, say, then the measurement of AM about the various axes would return everything from zero to the total AM. That's not what we find. Whatever axis is chosen, the measured AM is quantized about that axis. It doesn't matter that axis you choose. And, in the case of the electron's spin (intrinsic AM) it's always the same value (up to a $\pm$). This is fundamentally incompatible with the concept of well-defined pre-measurement values for all directions.

 

[Nugatory]

In which coordinate system the components of angular momentum are quantized? Better to say, if we can select the coordinate system arbitrarily, how the components of angular momentum, say z-component, are always $L_z=m\hbar$?

No matter which direction we choose as the z axis, there will be a solution of the TISE which is an eigenfunction of the the spin component in that direction, and which is a simultaneous eigenfunction of $S^2$. That's how we can choose the z-axis arbitrarily: any choice gives us a complete basis that we can use to represent any state.

However, the state that we call "positive eigenfunction of $S_z$" when we've chosen to orient the z-axis horizontally is a different state than the state we call "positive eigenfunction of $S_z$" when we've chosen to orient the z-axis vertically. The former is a possible state for a particle prepared by passing it through a horizontally oriented SG apparatus, and the latter is not. But the particle state is the same either way, all that's changed is the basis we're using when we write it down.