Page 101 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
Interference
Experiment The interference pattern on the screen shows the dependence of the inten-
sity on the acceleration bias for the electrons, which itself is a measure of
the momentum of an electron. The fact that the electron waves follow the
Bragg relation gives for the momentum and wavevector the relation
p = —k , where is the Planck constant. Thus we describe the electrons
—
as a plane wave in the same way as we did for the lattice waves (Box
2.1). These waves have a frequency ω and a wavevector k and are char-
acterized by the functions sin ( ωt – kx) , cos ( ωt – kx) , exp ( ωt – kx)
or linear combinations of them. An arbitrary linear combination Ψ x t,( , )
in general, is a complex scalar field. On the other hand, the particle’s
position x, its momentum p, charge q and many other properties that may
be determined by measurement are real quantities. Therefore, we need an
interpretation of the wavefunction.

Probability The square modulus of the wavefunction Ψ x t,( ) 2 is interpreted as the
Density probability density of finding an electron at time t at position x. Thus for

a normalized wavefunction

∫ Ψ x t) d V = ∫ Ψ∗ x t)Ψ x t) V = 〈|〉 1 (3.1)
(
(
,
,
(
2
,
ΨΨ =
d
Ω Ω
holds, because the electron must be found with probability one some-
where in the physical domain Ω . We implicitly assumed that the wave-
function is square-integrable, i.e., the integral over Ω exists. This is not
necessarily the case, as we shall later when dealing with free electrons.
〈|〉 Ψ is called the norm of Ψ and (3.1) is the standard form
ΨΨ =
for the numerical evaluation of the expectation values of operators. The
short-hand notation given by the last term in angle brackets is known as
the Dirac notation. Once we have defined the probability density
,
,
(
,
(
(
ρ xx) = Ψ∗ x t)Ψ x t) (3.2)
we calculate its moments. Remember that for a real mass density the
property corresponding to the first moment with respect to the position

98 Semiconductors for Micro and Nanosystem Technology