Page 103 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
x )
x –
1
0 
Ψ xt ,(
/
2 
0 ) = ψ x() = ------------------σ 12 exp   – ( --------------------- 2 (3.5)
4σ
2π)
(
/
14
2
2
For the wavefunction of type (3.5) σ = ( 〈 ∆x ˆ ) 〉 holds.The spatial
wavefunction may be represented as a superposition of plane waves, i.e.,
as an inverse Fourier transform
∞
1
2π ∫
d
ψ x() = ------ φ k()exp ( ikx) k (3.6)
– ∞
The wavevector in (3.6) is the same as found in the momentum relation
above.
Position and ψ x() we call the wavefunction in position representation, more precisely
Momentum its spatial part. ψ k() we call the wavefunction in momentum representa-
Represen-
tion. For the expectation value of the momentum we write
tations
∞ ∞
〈|
p ˆ 〈〉 = — φ kφ〉 = ∫ φ∗ k()p ˆ φ k() k = – i— ∫ ψ∗ x() ∂ ∂ x ψ x() x (3.7)
d
d
– ∞ – ∞
The important message in (3.7) is that the result of calculating an expec-
tation value does not depend on the representation of the wavefunction.
Note that when representing wavefunctions in position space the momen-
∂
tum is a differential operator p ˆ = i—------ . This implies that the momentum
∂x
operator and the position operator do not commute, i.e., applying the
chain rule for differentiation we see that
( p ˆ x ˆ – x ˆ p ˆ )ψ x() = – i—ψ x() (3.8)
x
x
holds. In other words, it makes a difference whether p ˆ x ˆ or x ˆ p ˆ x is
x
applied to ψ x() and it does not give the same result. (3.8) is called the
commutator of the operators p ˆ x and . If operators do not commute, i.e.,
x ˆ
the commutator has a non-zero value, this means that the respective
physical quantities cannot be measured simultaneously with arbitrary
precision.

100 Semiconductors for Micro and Nanosystem Technology