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## Main Question or Discussion Point

How do we know that separable solutions of Schrodinger equation (in 3d) form a complete basis? I understand that the SE is a linear PDE and therefore every linear combination of the separable solutions will also be a solution , but how do we know that the converse, i.e 'every solution can be written as a linear combination of separable solutions', is true? If we can separate out the variables to get ordinary differential equations, can we take it for granted that the solutions we'll get will be complete?

( I guess my question is not limited to the schrodinger equation alone. I have seen this done to laplace equation also in Electrostatics and I got the same doubt)

( I guess my question is not limited to the schrodinger equation alone. I have seen this done to laplace equation also in Electrostatics and I got the same doubt)