�
�
�
�
323
5.5. Emission with elevated temperature and field
This integral applies for any lower limit x 0 . In the following we
choose x 0 far to the left of the barrier. The solution u + (x <
x 1 ) is the right-propagating incident wave, u − (x < x 1 ) is the
left-propagating reflected wave, and u + (x > x 2 ) is the rightpropagating transmitted wave. While this approximation breaks
down at the classical turning points x 1 and x 2 , the function w(x)
is well-behaved everywhere.
In order to calculate the transmission probability D(W ), we must
calculate the probability current j for the incident, reflected, and
transmitted waves. In general
j =
ih ¯ [ u(x) ¯
u
� (x) − u ¯(x) u
� (x) ].
(5.98)
2m
We notice in (5.96) that u + (x) = u ¯ − (x) apart from constants. The
current j is thus proportional to the conserved Wronskian, and is
therefore conserved with respect to the coordinate x as required.
Far from the classical turning points, and apart from constants, it
is easily shown in this approximation that
−1/2 iw
u = p
e
−1/2 −iw ¯
u ¯ = p ¯
e
m
i
�
−5/2 U
�
1/2
iw
u =
p
+ p
e
2
h ¯
m
i
�
−5/2 U
� −
1/2
−iw ¯
u ¯ =
p ¯
p ¯
e ,
(5.99)
2
h ¯
where we notice that p and w are both either pure real or pure
imaginary. After some algebra we arrive at a general expression
for the probability current j as
p + ¯
p i (w−w ¯)
j =
e
.
(5.100)
2m | p |
This is identical with the result obtained by Kemble [52]. Substituting, we arrive at the probability current as follows:
| a + |
2
j + (x < x 1 ) =
m