If I consider the problem of for example the hydrogen atom. I.e. a central force problem with an effective potential V(r) that depends only of r, the distance between the positively charged nucleous and the negatively charged electron.(adsbygoogle = window.adsbygoogle || []).push({});

In the Schrödinger's equation, one considers the Hamiltonian operator as [itex]-\frac{\hbar }{2m} \nabla ^2 +V (\vec r, t )[/itex]. From classical mechanics we know that in a central force problem, the motion is constrained into a plane. My question is thus: why is the Laplacian taken in spherical coordinates (this assumes a 3d motion) instead of polar coordinates (this assumes a motion constrained into 2 dimensions)?

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# Hamiltonian operator in 3d

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