Page 109 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
Gauss
theorem Applying the Gauss theorem, (see Box 3.1), we see that
d 2
t d ∫ Ψ d V = – ∫ j S (3.24)
d
holds. (3.24) has a very simple interpretation: the change of probability
density with time in the volume Ω is caused by the flux through the sur-
Γ
face .

3.2 Free and Bound Electrons, Dimensionality
Effects

Another more sophisticated point is the fact that the energy spectrum
may show a both discrete and a continuous part. This leads us to the fol-
lowing question: How does the spectrum depend on the imposed bound-
ary conditions?

To determine finally the functional form of the wavefunction, we need
information about these boundary conditions that the electronic system
has to fulfill. The resulting spectrum of observables will be discrete, con-
tinuous or mixed and allows us to talk about its dimensionality [3.1],
[3.2].

3.2.1 Finite and Infinite Potential Boxes

The One- The simplest case to study appears to be the one-dimensional potential
Dimensional box with finite potential. The physical interpretation of its solution and
Potential Box
boundary conditions, and the transition to infinite potential walls, are
very instructive. Given the potential

 0, x ∈ [ 0 a]
,
Ux() =  (3.25)
,
 U , x ∉ [ 0 a]
0

106 Semiconductors for Micro and Nanosystem Technology