Page 117 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
condition for normalizing the eigenfunctions of a continuous spectrum.
We have chosen the expansion coefficients as φ on purpose in order to
k
show that the situation of the gaussian wavefunction expanded in plane
waves is exactly such a case. The infinitely extended momentum eigen-
functions describe ideal situations having an exact value for the momen-
tum of the electron. According to the Heisenberg principle, allowing the
transition for the momentum variance to vanish, this yields an infinite
variance of its position such that their product is larger than — ⁄ 4 . This
2
can only be a hand–waving argument to visualize the situation. Neverthe-
less, it gives a feeling for the consistency problems arising. The gaussian
wavefunction, in contrast, is an excellent example of a more realistic situ-
ation. Its interpretation is that of a moving electron having an expectation
value of momentum and position with their respective variances in accor-
dance with the Heisenberg principle. It is a more realistic situation
because the arrival of electrons at a certain position may be measured and
it coincides with an event of the measuring instrument. It is not possible
here to go deeper into this subject but nevertheless we want to draw read-
ers’ attention to the literature on the problem of measurement in quantum
mechanics.

The Suppose E > U in (3.28a) and (3.28b). Then the solutions in the three
0
Continuous regions I, II and III read
Spectrum of
the Quantum
ψ x() = a exp ( ik x) + a exp – ( ik x) (3.42a)
Box I 1 I 2 I
ψ () = b exp ( ik x) + b exp – ( ik x) (3.42b)
x
II
II
1
II
2
x
ψ () = c exp ( ik x) + c exp – ( ik x) (3.42c)
1
2
III
III
III
⁄
⁄
(
where we have k IIII = 2mE – U ) — and k II = 2mE — . In
0
,
Figure 3.3 a wavefunction for a free electron is indicated. The larger
k
kinetic energy in the box region yields a larger wavevector and thus a
wavefunction oscillating faster in space.
114 Semiconductors for Micro and Nanosystem Technology