Spin Quantum Number: Why Just Z Component?

[Sheldon Cooper]

Hello everyone,
In case of hydrogen atom, when we say spin up or spin down we refer to the z component of the spin. Why are we interested only in the z component of spin and not in the x and y components?
Thanks in advance

 

[Simon Bridge]

The "z component" is the component that is in the direction of the applied magnetic field, see "Stern-Gerlach experiment".
The x-y components are those that are perpendicular to the z component. We are interested in them, they are just in superposition.

 

[jtbell]

The "z component" is the component that is in the direction of the applied magnetic field

I suspect the reason we call it the "z" component, and not "x" or "y" is that when we deal with orbital angular momentum we normally express the Schrödinger wave function $\psi$ in spherical coordinates, which uses the z-axis as the axis of the coordinate system. The math describing atoms in magnetic fields is simpler if we align the axis of the coordinate system with the magnetic field.

We carry this convention for the z-axis over to spin angular momentum for consistency, which makes it easier to add spin and orbital angular momenta.

 

[HomogenousCow]

When you solve the SE equation for a spherically symmetric Hamiltonian you want the eigenstates to be simultaneous eigenstates of L squared and L. You can only pick one of the Ls since they don't commute, we call this arbitrary direction the z direction. I'm pretty sure once you sum all the degenerate states you get a state with no bias towards any direction, correct me if I'm wrong.

 

[Chandra Prayaga]

It is correct that in a quantum system, L² and only one component of the vector L, can be measured simultaneously. We arbitrarily choose the z-component. But once we choose the z-component as the measurable quantity, L_x and L_y can no longer be specified exactly. I am not sure why you would want to add the degenerate states. Each one of them is a perfectly valid, measurable state. Of course, once you add a number of degenerate states with the same L², but different L_z, the new state no longer has a definite value of L_z.

 

[HomogenousCow]

It is correct that in a quantum system, L² and only one component of the vector L, can be measured simultaneously. We arbitrarily choose the z-component. But once we choose the z-component as the measurable quantity, L_x and L_y can no longer be specified exactly. I am not sure why you would want to add the degenerate states. Each one of them is a perfectly valid, measurable state. Of course, once you add a number of degenerate states with the same L², but different L_z, the new state no longer has a definite value of L_z.

What I mean is, in general an eigenstate in an eigenspace has no particular connection to any direction of angular momentum, it's only that when we solve the SE for rotationally invariant problems we pick an "eigenbasis" which are also Lz eigenstates.