# What kind of problem in quantum mechanics can have an algebraic method of solution?

For example, a harmonic oscillator can have an algebraic solution, and hydrogen potential can
also have an algebraic solution. Here the algebraic method of solution means that we can use the similar methods like $a$ and $a^\dagger$ to solve QM problem. So in general what kind of problem in QM can have an algebraic solution. For example, can infinite potential well have a algebraic solution, and so on.

• What do you mean, precisely, for "algebraic solution"? – Valter Moretti Apr 14 '14 at 6:18
• My remark/question above is due to the fact that for the harmonic oscillator it is clear what algebraic could mean: the procedure relying upon $a$ and $a^\dagger$. Also for the hydrogen potential (reducing to the previous case), but I do not understand in what sense the infinite potential case should admit an "algebraic" solution. – Valter Moretti Apr 14 '14 at 6:41
• @V.Moretti I mean that we can use similar methods like $a$ and $a^\dagger$ to solve a QM problem – 346699 Apr 14 '14 at 6:54
• Which type of QM problem? This question (v2) seems to be a list question and quite broad. – Qmechanic Apr 14 '14 at 7:42
• – Qmechanic Apr 14 '14 at 7:43

When the system is a many-body system, it can't be solved analytically. Unlike classical mechanics, for $N = 3$ or $N = 4$ (few-body systems), there do exist Faddeev equations which can be solved by iteration. There should be more factors that I don't know.