QM of a particle in a 3D infinite square well

A particle in the infinite square well potential

has stationary states that are (tensor) products of stationary states analogous 1D potential, i.e.

with energies

Proof by separation of variables

Inside the TISE reads

we look for solutions of the form

for which the TISE becomes

hence

since each of the terms are functions of , , and respectively, the only way the LHS can equal the constant RHS is if each of the terms equals a constant, i.e.

Once boundary conditions are applied, the general solutions for , , and are thus precisely those for QM of a particle in a 1D infinite square well. Let denote solutions for the 1D case. We thus have

which is already normalized.

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