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The quantum numbers serve to enumerate the solutions to the time-independent Schrodinger equation (Energy eigenstates) which are also eigenstates of the angular momentum (squared) operator and have a definite component of angular momentum in some direction labeled $z$. When solving the problem, it just turns out to be handy to label the solutions with the ...

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The answer to your question comes from math, not from physics. Consider the wave equation in 1+1 dimensions: $$\partial^{2}_{x}\phi(x,t) - \partial^{2}_{t}\phi = 0$$ One obvious way to solve this is to convert the PDE in two variables into two ODEs in one variable. This is done by assuming that the solution can be written as some sum \phi(x,t) = ...

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