While solving Schrödinger solution we use separation of variables to separate time dependent and independent parts and then write the final solution as the product of the two solutions. How can we be sure that separation of variables provide all solutions of the Schrödinger equation? Why only use separation of variable to simplify and solve the Schrödinger equation?
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Tobias Fünke
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2Why only use separation of variable to simplify and solve the Schrödinger equation? - What do you mean? We use it because it is correct (if the Hamiltonian is time-independent). Can you use some equations to be more explicit on what you are asking? – Tobias Fünke May 17 '23 at 17:59
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It is a mathematical theorem that the solutions to that separable equation can be written as a sum over the separable solutions. – march May 17 '23 at 18:17
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4Does this answer your question? Non-separable solution for the Schrödinger equation – Quillo May 17 '23 at 19:05
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It does involve (Call it the correct because we use it) division by the non-time dependent part. So generally if the wave function touches $0$ because of the position argument you can rest assured that everything will turn out to be OK. It will change it's mind and you will not get solution that touches $0$.
I once asked a professor with a book on QM and he said that these solutions are re introduced after the work is done.
