Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.11 Extension to threedimensional solutions
Maybe you have some doubt whether you really can just multiply
onedimensional eigenfunctions together, and add onedimensional
energy values to get the threedimensional ones. Would a book that
you find for free on the Internet lie? OK, let’s look at the
details then. First, the threedimensional Hamiltonian, (really just
the kinetic energy operator), is the sum of the onedimensional ones:
where the onedimensional Hamiltonians are:
To check that any product of
onedimensional 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 threedimensional Hamiltonian, with an
eigenvalue that is the sum of the three onedimensional 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 onedimensional
ones.
The onedimensional 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 onedimensional eigenfunctions; these products are
complete.