Maybe you have some doubt whether you really can just multiply
one-dimensional eigenfunctions together, and add one-dimensional
energy values to get the three-dimensional ones. Would a book that
you find for free on the Internet lie? OK, let’s look at the
details then. First, the three-dimensional Hamiltonian, (really just
the kinetic energy operator), is the sum of the one-dimensional ones:
To check that any product of
one-dimensional eigenfunctions is an eigenfunction of the combined
Hamiltonian , note that the partial Hamiltonians only act on
their own eigenfunction, multiplying it by the corresponding
eigenvalue:
Therefore, by definition is an eigenfunction of the three-dimensional Hamiltonian, with an eigenvalue that is the sum of the three one-dimensional ones. But there is still the question of completeness. Maybe the above eigenfunctions are not complete, which would mean a need for additional eigenfunctions that are not products of one-dimensional ones.
The one-dimensional eigenfunctions are complete, see
[40, p. 141] and earlier exercises in this book. So,
you can write any wave function at given values of and as a combination of
-eigenfunctions:
But since the -eigenfunctions are also complete, at any given value of , you can write each as
a sum of -eigenfunctions:
But since the -eigenfunctions are also complete, you can
write as a sum of -eigenfunctions:
So, any wave function can be written as a sum of products of one-dimensional eigenfunctions; these products are complete.