### A.4 Density of States

When going from small electron systems, such as atoms and molecules, to large
electron systems, solids for instance, the number of available electron states reaches
high values so that it is best expressed in terms of a density per energy and volume.
In solid state theory the assumption of periodic boundary conditions is frequently
employed and delivers simple first-order approximation of the density of states
(DOS). However, in a few cases, such as tunneling, DOS decomposed in
one energy component perpendicular () and one parallel () to a
certain plain is required. The corresponding derivation will be outlined in the
following.

In each direction with periodic boundary conditions, each quantum number is
directly related to one wavevector :

If the length of the periodicity is increased, the single quantum states narrows in
the space and becomes an continuous quantity. Then the summation over the
single states can be replaced by an integral. Making use of the relation one obtains the number of states in the -direction For periodic boundaries in two dimensions, an integral transformation to polar
coordinates is carried out in order to obtain the number of states in the -plane.
with being the energy in the -plane. Combining both solutions yields
where an integral transformation from to with the constraint
is performed. The split one/two dimensional DOS is defined as
while the commonly known three dimensional DOS for a free electron gas reads
which is usually found in textbooks [129].
For the case that the electrons are confined in the -direction, the number of states
are counted in the following way:

Here, denotes the quasi-bound states with the quantum number . In order
to refer to the unit volume, one must introduce the the square of the
wavefunction. When is multiplied with and the same integral transformation as
for the derivation of is performed, one obtains the DOS for an
one-dimensionally confined electronic system: Note that in this derivation the spin degeneracy introducing a factor has been
neglected.