Quantum Mechanics Notes

Below is my understanding for reminding myself. Not guaranteed to be correct.

Addition of Angular Momentum

Multiple items with angular momenta can be added together into a system. It doesn’t matter if this angular momentum is from orbital angular momentum or intrinsic spin for purposes of addition.

Due to how commutation relations work out, when you add together angular momentum, it is not possible to know the magnitude of the total angular momentum and the constituents’ Z-projections at the same time.

This is because when there is a definite total angular momentum magnitude, the individual Z-projections are only defined probabilistically by the Clebsch-Gordon tables. Vice versa applies.

Reference

An odd intuition

Here’s how I’m starting to think of it:

• I know the total angular momentum magnitude, total Z-projection, and constituent angular momenta magnitudes? There could still be a number of constituent Z-projections that could have produced the total. Which configuration is it actually? Who knows?
• Oh, I know the constituent angular momenta magnitudes, constituent Z-projections, and total Z-projection? Nice. But the magnitude of the total angular momentum could still be a number of things, depending on how exactly the constituent angular momenta vectors line up. I mean, they could be pointing the same direction, opposite directions, somewhere in the middle… Which is it actually? Who knows?

Fermions and Bosons, Symmetrization

Fermions have half integer spin. Bosons have integer spin.

A system of multiple identical fermions must have a wave function that is antisymmetric under interchange of any two particles. A system of multiple identical bosons must be symmetric under interchange.

For two electrons, all the triplet states are symmetric. All the singlet states are antisymmetric. So, if there are two electrons with all the same quantum numbers (other than spin), they must be in the singlet state to anti-symmetrize the wave function.