tiger2012 said:
The eigenvalues of the HO can be obtained by counting your fingers.
Not quite. One still has to find the infinite sequence of invariant subspaces of the Hamiltonian, which gives rise to the triangular structure.
The same ladder principle is used in many books for finding the
irreducible unitary representations of the Lie algebras so(3) = su(2), though there the sequence of invariant subspaces (and hence the resulting spectrum) is finite.
In purely mathematical terms, it is the structure of
highest (or lowest) weight representations of semisimple Lie algebras, known for a long time.
- Bochner, S. (1936). Summation of derived Fourier series: an application to Fourier expansions on compact Lie groups. Annals of Mathematics, 37(2), 345-356.
The technique has been generalized by Kac (who wrote a book about it) to infinite-dimensional Lie algebras with triangular factorizations, relevant for discussing Kac-Moody algebras and the like.
- V.G. Kac, Infinite-Dimensional Lie Algebras. Cambridge University Press 1985. (3rd ed. 1994)
tiger2012 said:
https://doi.org/10.1119/5.0332743 (Editor's pick of AJP)
A hidden
upper triangular structure was noticed in the paper above. With this insight, the eigenvalues of the harmonic oscillator can be obtained immediately without calculation.
This paper has rediscovered the most elementary case of the above technique, without realizing its long history.
tiger2012 said:
The upper triangular structure also exists for the hydrogen atom.
The hydrogen atom is the special case of the Lie algebra so(4), and gives only the discrete spectrum. In the form stated in the paper, where the so(4) Lie algebra (that needs the Runge-Lenz vector) is not constructed, it even gives only the radial quantum number.
tiger2012 said:
We now really understand why the eigenstates of the harmonic oscillator is the product of a Gaussian and a polynomial.
This, and much more, is really understood for a very long time. You may wish to take it as the first rung of a long ladder that, when climbed, gives you real understanding of a lot of interesting mathematics and physics.