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Partial differential equation which describes the time evolution of the wavefunction of a quantum system. It is one of the first and most fundamental equations of quantum mechanics.

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What is the correct separable Schrödinger equation in spherical coordinates?

If I am not wrong, then the second one is used for (I guess) when we solve the radial part. We have to put $R(r)=u(r)/r$, so maybe in second one they have written $R(r)/r$ instead of $R(r)$ is because …
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