As we know, simple harmonic oscillator can be solved only by commutation relations between creation and annihilation operators, and the Hamiltonian expression. The spin energy is either solved only using commutative relations between spin operators in axes ($J_i$) and $J^{2}$. As an another illustration, for quantizing fields (such as Real Klein-Gordon scalar field) in QFT, one approach is to postulate canonical commutation relations between field and momentum operators.
I already know quantum operators create Lie Algebra, and commutation relations are important in a Lie Algebra. However, I have a question: is it always true that commutation relations are sufficient to obtain eigenvalues and eigenvectors of a Hamiltonian in the Hilbert space? If it is true, why commutative relations are sufficient to solve a quantum mechanics problem?
[Physics] Commutation relations in quantum mechanics
commutatorhilbert-spaceoperatorsquantum mechanics
Best Answer
If your Hamiltonian belongs to a Lie algebra for which you can solve the initial value problem in the corresponding group then you can use geometric quantization to solve the corresponding Schroedinger equation. This is because the solution of the Schroedinger equations is just $\psi(t)=e^{-itH/\hbar}\psi_0$, and $e^{-itH/\hbar}$ is an element of the group generated by the Lie algebra (in the appropriate representation on the given Hilbert space). Thus the problem is reduced to group representation theory.