Environmental Engineering Reference
In-Depth Information
Figure 2.1 Sketch of electron diffraction in a two-slit geometry. The de Broglie condition l¼h/p is
found to predict the angles of maxima. (Courtesy of M. Medikonda).
calculated de Broglie wavelength there exceeds the interatomic spacing. This leads
to a characteristic energy, the Fermi energy, which is larger than the classical energy
1
/
2
kT
, with many consequences. We will need to learn about this to understand solar
cells in a competent fashion.
Schrodinger found a second-order differential equation to describe matter
waves. Schrodingers equation describes in practical and accurate terms the
behavior of matter particles, protons in the sun, protons in atomic nuclei, as
well as electrons in atoms, metals, and semiconductors. The
first appearance of
thewaveaspectwastherelation
l ¼h
/
p
, the de Broglie wavelength. Here
h
is
Plancks constant and
p ¼mv
is the classical momentum of the particle. Evaluated
for the proton in the solar core, where we have found speed
v ¼
0.498
10
6
m/s,
we
nd
l¼ h
/
p ¼
6.6
10
34
/(0.498
10
6
m/s
1.67
10
27
kg)
¼
794 f. This is
small compared to the interproton spacing, but large compared to the actual
measured charge radius of the proton, which we have taken as 1.2 f. 794 f, the
proton de Broglie wavelength, is seen also to be closer to the minimum classical
spacing of two protons, 1113 f, as found for kinetic energy corresponding to
1.5
10
7
K. A second de Broglie relation gives a frequency
v¼E
/
h
(here
expressed in radians per second) to a particle of kinetic energy
E
.
The observation of electron diffraction in agreement with the de Broglie
relations (see Figure 2.1) means that a matter wave described mathematically
by
(
x
,
t
)
¼
exp(
ikx ivt
) must satisfy any more general equation as was produced
bySchrodinger.Thewavequantity
Y
Y
(
x
,
t
) predicts the location of the particle by
(
x
,
t
), where
indicates complex conjugate.
Schrodingers equation is then a statement of conservation of energy, where
k ¼
2
p
/
l
and
h
¼h
/2
p
.
(
x
,
t
)
the relation
P
(
x
,
t
)
¼Y
Y
2
k
2
½
h
=
2
mþU
h
vYðx
;
tÞ¼
0
:
ð
2
:
5
Þ
Based on this correct statement of conservation of energy, and knowing the
solution
2
Y
(
x
,
t
)
¼
exp(
ikx ivt
) in case
U¼
0, the equation has to involve
q
Y
(
x
,
t
)/
q
x
2
, to generate the
h
2
k
2
. In addition, the rst time derivative
q
Y
(
x
,
t
) is needed in order to produce the
h
v
in the statement of conservation of energy.
(
x
,
t
)/
q
t¼ivY
