Page 104 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Quantum Mechanics of Single Electrons
Uncertainty
Principle This leads directly to the uncertainty principle for operators, which is one
of the central points of quantum theory that has always been subject of
many discussions from the moment of its formulation. For the gaussian
wavefunction (3.5) centered at x = 0 , the variances of the particles
0
position and momentum are
∞
1  x 2
12 ∫
( 〈 ∆x ˆ) 〉 = ------------------σ exp – --------- x xd = σ 2 (3.9)
 2
2
( 2π) /  2σ 2
– ∞
and
∞
2 h 2
π ∫
( 〈 ∆p ˆ ) 〉 = hσ --- exp – ( k σ )k kd = --------- 2 (3.10)
2
2
2
2
4σ
– ∞
respectively.
One- Take a one-dimensional system with wavefunction ψ x() in its spatial
Dimensional representation, that is defined in the entire interval –( ∞ ∞) . Suppose
,
System
that x ˆ 〈〉 = 0 and p ˆ 〈〉 = 0 hold. This assumption is not necessary but
taking arbitrary values for the expectation values of position and momen-
tum only makes the calculation more complicated without increasing the
understanding. The relation

∞ dψ 2
∫ αxψ + d x x d
– ∞
(3.11)
∞
  dψ∗ dψ dψ∗ dψ  
2 2
2
=   ∫ α x ψ + x----------ψ + xψ∗------- + ----------------- d ≥ 0
x
dx dx 
 dx dx 
– ∞
is evident since the square modulus is positive in –( ∞ ∞) . Here is an
α
,
arbitrary real constant. Talking into account (3.1), (3.3) and (3.7) and per-
forming integration by parts we obtain
1
2
α 〈 ( ∆x ˆ) 〉 – α + --- ( 〈 ∆p ˆ ) 〉 ≥ 0 (3.12)
2
2
—
Semiconductors for Micro and Nanosystem Technology 101