Page 108 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Quantum Mechanics of Single Electrons
where we used (3.15) and the fact that for the Hamilton operator
ˆ ∗
ˆ
H = H holds. We use
—
ˆ
2
H = – -------∇ + Ux() (3.20)
2m
and
(
2
2
Ψ∇ Ψ∗ – Ψ∗ ∇ Ψ = ∇Ψ∇Ψ∗ – Ψ∗ ∇Ψ) (3.21)
which yields a continuity equation of the form
d 2
t d ∫ Ψ d V = – ∫ ∇ j V (3.22)
d
Ω Γ
Γ
where is the surface of the region Ω . Here j is the current density vec-
tor, which is identified by

i— 1
j = ------- Ψ∇Ψ∗ –( Ψ∗ ∇Ψ) = ------- Ψp ˆ ∗ Ψ∗ –( Ψ∗ p ˆ Ψ) (3.23)
2m 2m
Box 3.1. The Gauss Theorem.

The Gauss Theorem. Consider a vector field This is also true for situations where the volume is
Ar() and a volume element with a closed sur- a non simply connected region, i.e., there may be
V
n
V
face S . The normal vector of the surface holes in .
V
points outward the volume element. Then we have In components we formulate the theorem for arbi-
the following relation between the surface integral trary dimensions D as follows
and the volume integral of the vector field ∂
∫ ° ds … = ∫ d V ∂ x … (B 3.1.2)
i
∫ ° Ads = ∫ ∇ A V (B 3.1.1) S V V i
d
V
S V where the dots are to be replaced by a vector field,
This is called the Gauss theorem. Due to this or even a scalar field, while i = 12 … D . This
,,
,
equation the properties of the field inside -espe- means that the surface integral is related to the
V
cially its sources- are related to its properties at the volume integral of the derivative of the field.
surface – especially its flux.
Current This is the current density of probability or simply the current density.
Density For a given electronic system its expectation value multiplied by the unit
charge corresponds to a measurable electrical current density.

Semiconductors for Micro and Nanosystem Technology 105