### D.11 Extension to three-dimensional solutions

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:

where the one-di­men­sion­al Hamiltonians are:

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:

or

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 the coefficients will be different for different values of and ; in other words they will be functions of and : . So, more precisely, you have

But since the -​eigenfunctions are also complete, at any given value of , you can write each as a sum of -​eigenfunctions:

where the coefficients will be different for different values of , . So, more precisely,

But since the -​eigenfunctions are also complete, you can write as a sum of -​eigenfunctions:

Since the order of doing the summation does not make a difference,

So, any wave function can be written as a sum of products of one-di­men­sion­al eigenfunctions; these products are complete.