Quantum Mechanics for Engineers |
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© Leon van Dommelen |
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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
[41, 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.