Page 122 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
electron in the harmonic potential. This is why both phenomena may be
described in the same way.
We describe the oscillator in the q-representation. This means that the
⁄
momentum operator is given by p ˆ = – i— ddq , and the commutator is
p ˆ q ˆ – q ˆ p ˆ = – i— . The time–independent Schrödinger equation reads

  — 2  d 2 m 
2 2
  – ------- -------- + ----ω q  ψ q() = Eψ q() (3.51)
2m 
2
 dq 2 
⁄
With the coordinate transformation q = — ( mω)ξ (3.51) reads
—ω d 2 
2

------- – -------- + ξ  ψξ() = Eψξ() (3.52)
2  dξ 2 
(
2
2
–
The operator in braces in (3.52) has the form a – b = ( ab) a + b)
and thus may be written as
—ω d 2  1  d  1  d  —ω
ξ
ξ
2



------- – -------- + ξ  = —ω ------- – ------ +  ------- ------ +  + -------
2  dξ 2  2  dξ  2  dξ  2
             
b + b
(3.53)
With the definitions of the operators b + and as given by the under-
b
braces in (3.53) and shifting the zero point of the energy according to
⁄
E′ = E – —ω 2 , the Schrödinger equation reads
+
—ω b b ψ = E′ψ (3.54)
This simple form allows us to calculate the wavefunctions very easily.
We calculate the commutator of b + and , which gives
b
+
+
[ bb , + ] = bb – b b = 1 (3.55)
Take the ground–state function ψ that has the lowest energy eigenvalue
o
E ′ . Let the operator act on both sides of (3.54)
b
o

Semiconductors for Micro and Nanosystem Technology 119