I’ve gradually realized that spin is used everywhere in theoretical physics and also mathematics. Let alone in particle physics, where all the material particles are described by spin-1/2, even in condensed matter physics, people are often interested in quantum mechanical spin living on lattice. It occurs to me that I should study this again by reviewing the corresponding lectures in 8.05x that I took about two years ago.
The angular momentum is covered from lecture 20 to lecture 23, and it is used to solve the central potential problem. I understand a few points that confused me a lot when I first learn them.
Set of commuting variables
The reason why we want to study (orbital) angular momentum for the central potential problem is that we want to understand the degeneracy structure of the spectrum. In other words, we want to find a nice way to label the states uniquely. For a central potential problem, we can find the set of commuting variable as $\{H, \vec{L}^2, \hat{L}_z\}$. If we use the language of group theory, the total angular moment operator $\vec{L}^2$ labels different sectors, and the z component of the angular momentum labels different basis in a given sector.
When I first saw this exposition, I did not understand how we are sure that we cannot add other operators in this set. This can be justified a posteriori, after we get the radial equation successfully. Since we have a theorem saying that there is no degeneracy of bound states in 1D problem, we are sure that we can specify the state uniquely, so no further commuting variables are possible.
Interesting properties of isotropic 3D harmonic oscillator and hidden symmetry in hydrogen atom
I didn’t appreciate the beauty of these two examples two years ago. I understood how to get the spectra mathematically quite well then, but the patterns of the spectra didn’t get my attention.
Generally speaking, energies in different total angular momentum sector are totally unrelated, as can be seen from the infinite square well example. However, there are unexpected degenerates for different values of total angular momentum. To understand what this means, let’s compare the energy spectra of 1) infinite spherical square well, 2) 3D isotropic oscillator, and 3) hydrogen atom. Their energy spectrum is shown as below, organized according to different total angular moment sectors that are labeled by $l$.(taken directly from Prof. Zwiebach’s lecture notes that can be found here).
The infinite spherical square well represents the ordinary behavior of a central potential problem. However, for other two examples, there are unexpected degenerates between different total angular momentum sectors. The so-called hidden symmetry can explain such accident. Specifically, it means we can construct operators that commute with the Hamiltonian and are able to transform the state in one total angular moment sector to another one. It is an interesting topic, and I should work the details out in the future. Also, in chapter VII of Zee’s group theory book, the hidden symmetry of hydrogen atom is discussed.
