The boundary conditions for solving the hydrogen Schrödinger equation, specifically for hydrogen-1, require the wave function to approach zero as the radius approaches infinity, ensuring square integrability. Additionally, the wave function must be finite, meaning the integral of Psi*Psi must be square integrable. The method of separation of variables imposes conditions that the spatial components of the solution are time-invariant, and the wave functions in polar coordinates must be orthogonal and normal. These conditions are crucial for deriving accurate energy eigenvalues and quantum numbers for the hydrogen atom.
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Integral of Psi*Psi needs to be finite (square integrable). You can have a function go to zero as r->inf. that would not be square integrable.DEvens said:The usual thing is to have the wave function go to zero as radius goes to infinity. Is that enough?
Quantum Defect said:Integral of Psi*Psi needs to be finite (square integrable). You can have a function go to zero as r->inf. that would not be square integrable.
And you can have square integrable functions which don't go to 0 when their argument goes to infinity.Quantum Defect said:Integral of Psi*Psi needs to be finite (square integrable). You can have a function go to zero as r->inf. that would not be square integrable.