- #1
fluidistic
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In solving some PDEs such as the heat/diffusion equation or the wave equation, when the equation itself, as well as its associated boundary conditions, are independent of some variable (for example the azimuthal angle), we often use the trick to assume that the solution (and eigenfunctions) are independent of that variable too. We use no justification whatsoever and since we find "the" solution, it doesn't matter which way we obtained it, it is the only solution. That is perfectly fine to me.
However why does such a trick fail for example with Schrödinger equation? There, for example for the H-atom problem where the potential is central, there is no angle dependence anywhere. However the eigenfunctions involve spherical harmonics, and so the solution (the wavefunction) is not angle independent.
My question is thus, how can we justify or get to know beforehand whether this trick will work, without having to go through the whole process of checking whether the trick actually works?
However why does such a trick fail for example with Schrödinger equation? There, for example for the H-atom problem where the potential is central, there is no angle dependence anywhere. However the eigenfunctions involve spherical harmonics, and so the solution (the wavefunction) is not angle independent.
My question is thus, how can we justify or get to know beforehand whether this trick will work, without having to go through the whole process of checking whether the trick actually works?