Environmental Engineering Reference
In-Depth Information
2.2.1
Time-Dependent Equation
On this basis, the Schrodinger equation in one dimension, with time-dependent
potential
U
(
x
,
t
), is
2
2
Yðx; tÞ=
q
x
2
ð
h
=
2
mÞ
q
þUðx; tÞYðx; tÞ¼i
hq
Yðx; tÞ=
q
t ¼ HY:
ð
2
:
6
Þ
, with
H
the operator
that represents the energy terms, and is seen to represent the time derivative of the
wavefunction.
The left-hand side of the equation is sometimes written
HY
2.2.2
Time-Independent Equation
In the common situation when the potential
U
is time independent, a product wave
function
Yðx
;
tÞ¼ ðxÞwðtÞ;
ð
2
:
7
Þ
when substituted into the time-dependent equation above, yields
wðtÞ¼
exp
ðiEt
=
h
Þ:
ð
2
:
8
Þ
Similarly, one obtains the
time-independent Schrodinger equation
,
2
2
mÞd
2
d
x
2
ð
h
=
ðxÞ=
þU
ðxÞ¼E
ðxÞ;
ð
2
:
9
Þ
(
x
) must satisfy the equation and
also boundary conditions, as well as physical requirements.
The physical requirements are that
(
x
) and energy
E
. The solution
to be solved for
(
x
) be continuous and have a continuous
derivative except in cases where the
U
is infinite.
(
x
) is zero where the potential
U
is
infinite.
The second requirement is that the integral of
(
x
) over the whole range of
x
(
x
)
must be
finite, so that a normalization can be found.
The solutions for the equation are traveling waves when
E
U
, as would apply for
protons free in the solar core, where we can take
U¼
0. On the other hand, for
E
>
U
,
the solutions will be real exponential functions, of the form
A
exp(
k
x
)
þ B
exp(
k
x
).
In such a case, the positive exponential solution can be rejected as nonphysical. The
decay constant can be seen to be
<
1
=
2
k
¼½
2
mðV
B
EÞ
=
h
:
ð
2
:
10
Þ
We return to the simplest possible treatment of proton fusion in the core of the
sun. Refer to Figure 1.6, from the Atzeni text on fusion. This model dates to Gamow
[12] in explaining the systematics of alpha particle decay of heavy nuclei. In the decay
process, the alpha particle is imagined as moving freely inside the nucleus, which is
treated as a square well potential. The intermediate barrier corresponds to the
Coulomb potential
k
C
(
Ze
)(2
e
)/
r
, where
r
is the spacing between the two charges.
