Environmental Engineering Reference
In-Depth Information
heating of the sun. A large interproton Coulomb barrier exists and classically particles
cannot cross such a barrier, fusion could not occur
.
2.2
Schrodinger
s Equation for the Motion of Particles
A completely different approach,
quantum physics or nanophysics
, is needed to explain
the fusion events that occur in the core of the sun. In classical physics, these protons
have no chance of reacting to form deuterons as must happen, for this requires a
spacing about 2.4 f, while the closest approach possible at 1293 eV is 1113 f, from the
equality
k
c
e
2
/
r ¼
1293 eV. The situation is as shown in Figure 1.6, where the available
energy is far below the barrier height.
To repeat, the process of fusion in the sun, with proton density and temperature
approximately as outlined, has led to all of the accumulated energy on earth in the
form of deposits of coal, oil, and natural gas, as well as the continuing flow of 170 PW
(petawatts, this is 170
10
15
W) to the earth by direct radiation. It is also a prototype or
existence proof for controlled fusion on earth. So, this is a key process, worth some
thought.
The new approach needed is based on a wave aspect for all matter particles. A hint
that a wave aspect for matter particles is needed to allow the particles to cross the
barrier in which they are not classically allowed to exist, comes from optics.
An analogous evanescent wave phenomenon occurs in optics; the evanescent
wave allows light to cross or tunnel through an air gap between two high index of
refraction glass plates. In the gap region, the light intensity falls off exponentially with
spacing. Light waves obey Maxwells equations, which are second-order differential
equations.
A direct veri
cation of de Broglies predicted wave property of matter
l ¼ h
=
p
ð
2
:
4
Þ
(where
h
is Plancks constant as appeared in Equation 1.1 and
p ¼mv
is the
momentum of the particle) was found by Davisson and Germer, who observed
magic re
ection angles for a monochromatic electron beam shone on a Ni metal
crystal. Their experiment veri
ed the prediction of de Broglie, Equation 2.4.
A more familiar geometry is the two-slit experiment shown in the Figure 2.1.
The condition for a constructive interference peak behind the two slits is that the
path difference
d
sin
¼nl
, where
l¼ h
/
p
,
h
is Plancks constant as appeared in
Equation 1.1 and
p ¼mv
is the momentum of the particle.
To return to the proton in the core of the sun, we
nd
l ¼h
/
p ¼
6.6
10
34
/
(1.67
10
27
0.498
10
6
)
¼
794 f. This is much smaller than (about 3%) of the
interproton spacing 31 800 f, so we can say that the motion of the solar core protons
on the whole is as free classical particles. (When they slow down approaching contact,
this is no longer true, and the Schrodinger treatment is essential to understand the
fusion interaction of two protons). In comparison, the electrons in a metal, even
though much less dense, are completely quantum in their motion because the
