Page 124 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
b
Considering the commutator for the operators b
+ + + + and we obtain
′
—ω b bb ψ = ( E + —ω)b ψ (3.62)
o o o
+
+
Obviously ψ = b ψ o is an eigenvector of the operator —ω b b with
1
′
eigen-value E + —ω .
o
Number Repeating this procedure n times we obtain the n-th eigen-vector ψ n
Operator with eigen-value E = n—ω (remember E = 0 ). Thus we have
′
′
n
o
⁄
E = ( n + 12)—ω . The quantum number n characterizes the oscillator
n
+
and is the eigenvalue of the quantum number operator b b . The normal-
ized wavefunction for an arbitrary excited state with quantum number n
reads
1 + n 1 + n
Excited State ψ = --------- b ( ) ψ = n |〉 = --------- b ( ) 0|〉 (3.63)
n
o
Wave- n! n!
Function
where n|〉 indicates the Dirac notation of a quantum state with excited
n
oscillator quanta and 0|〉 describes the oscillator vacuum with no quanta
+
present. Hence we have, from b , the number of oscillator quanta incre-
mented by one, and applying on n|〉 results in
+
|
b n|〉 = n + 1 n + 1〉 (3.64)
b
while applying yields
|
–
bn|〉 = nn 1〉 (3.65)
Bosons Note that the oscillator quantum number can be arbitrarily large for a
given state. This is the case for particles that follow Bose statistics (see
Chapter 5) and therefore these particles are called bosons.
Minimum The quantum mechanical harmonic oscillator states apply for the descrip-
Uncertainty tion of various different phenomena and not only in semiconductors. In
addition, it has a very important property because the ground–state wave-
function fulfills the equality in (3.13), i.e., the wavefunction is a quantum
state of minimum uncertainty. This is easily shown by calculating the

Semiconductors for Micro and Nanosystem Technology 121