Page 132 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
dc ()
m
i—---------------- =
c t()exp
dt t ∑ V mn n – ( iω mn t) (3.76)
n
For the following we assume that the perturbation is switched on instan-
taneously at t = 0 and is subsequently constant. Then the c m are given
at t = 0 by the initial condition. We choose c 0() = 1 and c 0() = 0
n
k
for n ≠ k . For very short times after t = 0 the situation has not changed
very much so that we insert the initial condition in (3.76):
dc ()
t
m
c 0()exp
i—---------------- = V mk k – ( iω t) = V mk exp – ( iω t) (3.77)
mk
mk
dt
(3.77) can be easily integrated to give
t
i
--- V
d
t
c () = – —∫ exp – ( iω t′) t′
m mk mk
0 (3.78)
1
–
= – -------------V mk ( exp – ( iω t) 1)
mk
—ω
mk
The change in the probability of finding the system in a state m is given
by c () 2 . This is the squared modulus of (3.78), which reads
t
m
⁄
4 sin ( ω t 2)
mk
c () 2 = ----- V 2 ------------------------------- (3.79)
t
m 2 mk 2
— ω
mk
Transition (3.79) is the probability density of finding the system at time in ψ if it
t
m
Rate has been found in ψ at time t = 0 . The transition rate from to m is
k
k
given by
c () 2
t
m
Γ = ------------------ (3.80)
mk
t
In the long time limit
⁄
sin ( ω t 2)
mk
-----δω(
lim ------------------------------- → πt mk ) (3.81)
2
t → ∞ ω mk 2
Semiconductors for Micro and Nanosystem Technology 129