To evaluate any physical observables, such as the current density, we must
specify how the scattering solutions are statistically weighted in the final
result. For the case of a continuous spectrum of states, with 
-function
normalization, the derivation of the correct expressions are rather tricky, because
we seek expressions for densities of charge, current, energy, etc., rather than total
quantities (which are of course infinite in an unbounded system). To illustrate
the procedure, let us follow through the derivation of the electron density in
a spatially uniform three-dimensional semiconductor in equilibrium. We approximate
the conduction band structure by a simple parabolic dispersion relation:
where 
is the wavevector. The probability that each state 
will be occupied by an electron is given by the Fermi-Dirac distribution function:
where 
is the Fermi level or chemical potential and 
, T
being the absolute temperature. (To avoid confusion with the transmission probability
which is also denoted by T, the absolute temperature will always be shown
multiplied by Boltzmann's constant 
.)
Let us now
make an ad hoc assumption that the semiconductor crystal is a cube with each
side of length L, and
apply periodic boundary conditions. Then the stationary quantum states are plane
waves (normalized to unit amplitude) of the form
Due to the periodic boundary conditions, 
must assume discrete values:
where 
, 
, and 
are integers. The total number of electrons in the
crystal N is just the sum over all of the states 
of the probability
that each state is occupied
where the factor of 2 comes from the two spin states. Now, because L is large,
the allowed values
of 
are very closely spaced, and the sum over 
can be well
approximated by an integral:
We can now write an expression for the density of electrons n:
Notice that the arbitrary crystal dimension L has dropped out of the final expression.
In order to evaluate densities using expressions such as (23)
it is usually more convenient to transform the integration variable to E. By expressing
in spherical coordinates and
manipulating the dispersion relation (18) one finds [9]:
We will also have occasion to use the corresponding expressions for integrals over
two- and one-dimensional 
vectors. For the two-dimensional case (still
assuming a parabolic dispersion relation):
For integration over a one-dimensional k, the definition of the group velocity (8) may be used to obtain an expression valid for any dispersion relation:
Inserting (24) into (23) leads to the usual expression
for the electron density in a semiconductor (as discussed in Chapter 1 of the present
volume) 
.
The procedure for evaluating a physical observable in an equilibrium system of infinite
extent may thus be generalized from the above discussion. The expectation
value of the observable quantity is calculated for each state, taking the scattering
states to be normalized to unit amplitude. The density of the observable is then
determined by inserting this expectation value into the sum in (23)
and evaluating the resulting integral, usually using the relations (2).
The two most important observables are the electron density 
and the current
density j (which is independent of position in one dimension and steady-state).
The expectation value of the density for a state 
is simply
The expectation value of j is simple, though the operator itself often is not.
(Link to manuscript with full details.)
If the
dispersion relation 
is not parabolic and independent of position, the form
of the operator j is not given by the simple textbook expression
. The current density operator is
instead whatever remains of the kinetic energy term of the Hamiltonian after
the application of Green's identity as in the derivation of (1),
and this obviously depends upon the form of the Hamiltonian itself.
For unit-incident-amplitude scattering states, however, the result is invariably
Of course, in equilibrium, these two currents cancel each other (by the principle of detailed balance) and there is no net current flow.
To investigate the transport properties of a quantum system one must generally evaluate the current flow through the system, and this requires that one examine systems that are out of thermal equilibrium. A common situation, in both experimental apparatus and technological systems, is that one has two (or more) physically large regions densely populated with electrons in which the current density is low, coupled by a smaller region through which the current density is much larger. It is convenient to regard the large regions as ``electron reservoirs'' within which the electrons are all in equilibrium with a constant temperature and Fermi level, and which are so large that the current flow into or out of the smaller ``device'' represents a negligible perturbation. The reservoirs represent the metallic contacting leads to discrete devices or experimental samples, or the power-supply busses at the system level. Consequently the electrons flowing from from a reservoir into the device occupy that equilibrium distribution which characterizes the reservoir. In a simple one-dimensional system with two reservoirs, the electrons flowing in from the left-hand reservoir have k > 0 and those flowing from the right-hand reservoir have k < 0.
Within this picture, the current that is injected from the left-hand reservoir is
and the current injected from the right-hand reservoir is
In order to simplify the calculation of J further, we must invoke some special properties of the system. The most useful such property is that symmetry which permits the separation of the spatial variables. The separation of variables is possible if the Hamiltonian can be separated into two parts:
(Here the notation 
and 
is defined with respect to the direction
of current transport.)
Then the wavefunction separates into a product of two factors:
and the energy can be separated into a product of two terms:
The expression for the total current density J can now be simplified to
where 
is the larger of the two asymptotic potentials (minimum energy for a
propagating state) and F is the Fermi-Dirac distribution function summed over the transverse states:
The form of the sum over 
depends upon the spatial configuration of the
tunneling system. Note that the velocity factor does not appear in (35)
because it was canceled by the density of states.
If the system in question is macroscopically large in its transverse dimensions,
the 
form a two-dimensional continuum, and
. Then using the two-dimensional analog of
(22) and (25) F can be analytically evaluated:
The current density can now be written in the form usually given for the tunneling current [10]:
Note that this expression is valid in general with respect to the dispersion relation
in the x direction, but requires a parabolic dispersion relation in the transverse
directions. The separation of variables leading to (38) is never
rigorously valid in a semiconductor heterostructure. The reason for this is that the
transverse effective mass 
will vary with semiconductor composition, which
varies in the x direction. In principle, one must do at least a two-dimensional
integral (if axial symmetry holds, otherwise a three-dimensional integral) as implied by
(2). Nevertheless, (38) is widely used to
model the current density in heterostructure devices. The error introduced by
assuming separation of variables is probably less severe than that due to the assumption
of an infinite coherence length.
If the transverse dimensions are constrained, but separation of variables is still
possible, the transverse motion of the electrons consists of a discrete set of standing
waves or normal modes. Such systems are referred to as ``one-dimensional'' systems,
quantum wires, or electron waveguides. The symbol 
is now interpreted as
an index for the discrete transverse modes, and the expression for the current
density now becomes
