Formation of stars and non-conservation of angular momentum

[TRB8985]

Good morning all,

Recently in a modern physics course of mine, my professor was covering the topic of energy levels and ionization energies and it included a diagram very similar to this one:

While it is interesting to learn that these diagrams correspond to a very specific and strict set of energies in which the angular momentum is quantized, I'm left with a bigger question that's a little reminiscent of Max Planck's historical perspective.

Why is a continuum of energies not allowed? Perhaps more interestingly, assuming the angular momentum were NOT quantized, could life still exist somehow? Or is the conservation of angular momentum in quantum systems crucial for the formation of structures which spring forth life?

 

[vanhees71]

In quantum mechanics the possible values an observable can take is given by the spectrum of the self-adjoint operators describing them. The Hamiltonian of a system should be bounded from below in order to have a stable ground state. Otherwise it's pretty unrestricted. Consequently all possible cases of spectra occur: You have the case of only discrete energy values (e.g., the harmonic oscillator), both discrete and continuous energy values (e.g., the hydrogen atom), or only continuous energy values (the free particle). All of the mentioned examples are exactly solvable in non-relativistic quantum theory.

Angular momentum takes always discrete values. More precisely you can only have the magnitude (squared) of the angular momentum $\vec{J}^2$ and one component in some direction determined. Usually one chooses $J_3$. This is, because the angular-momentum operators do not commute, but any component commutes with $\vec{J}^2$. For both operators the eigenvalues form a completely discrete spectrum. The possible values are $\hbar j(j+1)$ with $j \in \{0,1/2,1,\ldots \}$ and $j_z \hbar$ with $j_z \in \{-j,-j+1,\ldots,j-1,j \}$ for each given $j$. The total angular momentum of a closed system is always conserved.

What all this has to do with the possibility for life, is not clear to me. There's some remarkable observation, sometimes called the "Anthropic Principle". One can speculate about what happens if the physics is the same (i.e., quantum theory and general relativity) but the fundamental constants of nature like the fine-structure constant (strength of the electromagnetic interaction), the gravitational constant, etc. were different. It turns out that the formation of life as we know it is pretty much dependent on the values of these fundamental constants as we observe.