Elements of Modern Physics
80
In analogy with the transmission and reflection of classical electromagnetic
waves, transmission and reflection coefficients can be defined as:
T =
2
b
q
p a
+
+
=
2
4
(
)
pq
p q
+
R =
2
a
a
−
+
=
2
p q
p q
−
+
(3.81)
In terms of the refractive index n,
n =
p
q
(3.82)
Eqs. (3.81) can be written as
T =
2
4
( 1)
n
n +
R =
2
1
1
n
n
−
+
(3.83)
which are the same as the classical transmission and reflection coefficients for
electromagnetic waves.
Case (ii) For E < V, q is imaginary. Writing
q = iα
(3.84)
where
α =
1
[2m (V – E)]
1/2
(3.85)
the solution for x > 0, is
φ r (x) = b + e
–
α
x
+ b – eα
x
(3.86)
In order to keep the probability finite as x → ∞,
b – = 0
(3.87)
The continuity equations (3.77) and (3.78) then imply that
b + =
2 p a
p i
+
+ α
a – =
p i a
p i
+
− α
+ α
(3.88)
80
In analogy with the transmission and reflection of classical electromagnetic
waves, transmission and reflection coefficients can be defined as:
T =
2
b
q
p a
+
+
=
2
4
(
)
pq
p q
+
R =
2
a
a
−
+
=
2
p q
p q
−
+
(3.81)
In terms of the refractive index n,
n =
p
q
(3.82)
Eqs. (3.81) can be written as
T =
2
4
( 1)
n
n +
R =
2
1
1
n
n
−
+
(3.83)
which are the same as the classical transmission and reflection coefficients for
electromagnetic waves.
Case (ii) For E < V, q is imaginary. Writing
q = iα
(3.84)
where
α =
1
[2m (V – E)]
1/2
(3.85)
the solution for x > 0, is
φ r (x) = b + e
–
α
x
+ b – eα
x
(3.86)
In order to keep the probability finite as x → ∞,
b – = 0
(3.87)
The continuity equations (3.77) and (3.78) then imply that
b + =
2 p a
p i
+
+ α
a – =
p i a
p i
+
− α
+ α
(3.88)