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Am I correct in saying that the central-potential solutions to the time-independent Schrodinger equation are all of the form $R(r)\Theta(\theta)\Phi(\phi)$? If so, do $R$, $\Theta$ and $\Phi$ have any significance when not all multiplied together? For example, would $\int \Theta\Phi\sin\theta \; d\theta d\phi$ give me the probability of finding the particle within the solid angle I integrate over?

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  • $\begingroup$ Do you know that the SE is linear, and hence the solution space is a vector space? $\endgroup$
    – Qmechanic
    Mar 25, 2020 at 9:55
  • $\begingroup$ @Qmechanic Yes. $\endgroup$
    – user113773
    Mar 25, 2020 at 10:00

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Am I correct in saying that the central-potential solutions to the time-independent Schrodinger equation are all of the form $R(r)\Theta(\theta)\Phi(\phi)$?

No. There are plenty of non-separable solutions that cannot be expressed in that way. They're a useful subset of solutions (particularly because they can be used to build a basis), but they're a thin slice of the global solution space.

For example, would $\int \Theta\Phi\sin\theta \; d\theta d\phi$ give me the probability of finding the particle within the solid angle I integrate over?

Yes, that is correct.

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