Page 107 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
• With the knowledge of the whole energy spectrum, i.e., all the eigen-
values and eigenvectors that form a complete basis set, we may repre-
sent an arbitrary vector as an expansion into the eigenvectors.
i

(
,
Ψ x t) = ∑ a exp   – ---E t ψ x() (3.18)

n
n
n
—
n
• The eigen-vectors are orthogonal ( ψ ψ〈 n | m 〉 = δ nm ). This means that
the “scalar product” as defined for the wavefunctions through (3.1) is
zero for n ≠ m and one for n = m .
The square moduli a n 2 denote the probability of finding the system in a
special eigenstate. The spatial probability distribution is given by
Ψ n 2 = ψ n 2 , it does not depend on time, which also holds for the
expectation values of operators that do not depend explicitly on time. All
those quantities are called conserved quantities. All conserved quantities
commute with the Hamilton operator, i.e., we may measure them at the
same time we measure the energy of the system. Nevertheless, they may
not always give the same result for their expectation values and seem not
to be uniquely defined in the case where there exist multiple energy
eigenvalues. Thus we need to know more about the system than only the
energy of the respective state or, in other words, in some situations the
energy spectrum does not contain enough information about the system
to be described uniquely.

When talking about conserved quantities, we may ask for the probability
Γ
flux through the surface of a finite volume Ω , i.e., the continuity equa-
tion for the probability density in the same way as it exists for the charge
density in electrodynamics. For this purpose, we integrate the probability
density (3.2) on the entire volume Ω and take its time derivative:

∂ 2  ∂ ψ∗ ∂ ψ 
t ∂ ∫ ( Ψ d ) =  ∫ ----------ψ + ψ∗------- d V
V
t ∂ 
t ∂
Ω Ω
(3.19)
i
—∫
ˆ
ˆ
= --- ( ( ΨHΨ∗ – Ψ∗ HΨ) V)
d
Ω
104 Semiconductors for Micro and Nanosystem Technology