Spin Quantum Number

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?

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Simon Bridge
Homework Helper
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
Mentor
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.

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