Page 134 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
for an electron placed in specific state . For the reverse operation we
have λ
c λ|〉 = 0 |〉 (3.85)
λ
λ
where an electron has been moved out of the state . We must take into
account that only one fermion may occupy a quantum state that is
uniquely defined by its quantum numbers λ . Therefore, trying to put
λ
another electron in must yield a zero eigenvalue

c ( + 2 0 (3.86)
) λ|〉 =
λ
and the same must hold for twice trying to remove an electron from λ

2
c ( λ ) λ|〉 = 0 (3.87)
+ +
The action of the operator c c + c c on an arbitrary state β|〉 yields
λ λ
λ λ
 ( 0) for βλ
+
+
( c c + c c ) β|〉 =  1 + = = 1 β|〉 (3.88)
λ λ
λ λ
 ( 0 + 1) for β ≠ λ
λ
which means that, regardless of whether the state is occupied or unoc-
cupied, the resulting eigenvalue is 1. This leads us to the definition of the
anti-commutator product
+
+
+
Anti- { c c, λ } = c c + c c = 1 (3.89)
λ
λ λ
λ λ
commutator
We use curly brackets to indicate the anti-commutation relation. Note the
difference in the sign when compared to the commutator of oscillator
+
creation and destruction operators (3.55). c c = n is still the correct
λ λ
number operator. In the case of fermions, the number operator’s eigenval-
ues are either 1 or 0 and indicate whether a particle in the respective state
is present or not.

Semiconductors for Micro and Nanosystem Technology 131