4More results in one dimensions
IB Quantum Mechanics
4.4 Potential barrier
So far, the results were slightly weird. We just showed that however high energy
we have, there is still a non-zero probability of being reflected by the potential
step. However, things are really weird when we have a potential barrier.
Consider the following potential:
x
V
U
0
a
We can write this as
V (x) =
0 x ≤ 0
U 0 < x < a
0 x ≥ a
We consider a stationary state with energy
E
with 0
< E < U
. We set the
constants
E =
~
2
k
2
2m
, U − E =
~
2
κ
2
2m
.
Then the Schr¨odinger equations become
ψ
00
+ k
2
ψ = 0 x < 0
ψ
00
− κ
2
ψ = 0 0 < x < a
ψ
00
+ k
2
ψ = 0 x > a
So we get
ψ = Ie
ikx
+ Re
−ikx
x < 0
ψ = Ae
κx
+ Be
−κx
0 < x < a
ψ = T e
ikx
x > a
Matching ψ and ψ
0
at x = 0 and a gives the equations
I + R = A + B
ik(I − R) = κ(A − B)
Ae
κa
+ Be
−κa
= T e
ika
κ(Ae
κa
− Be
−κa
) = ikT e
ika
.
We can solve these to obtain
I +
κ − ik
κ + ik
R = T e
ika
e
−κa
I +
κ + ik
κ − ik
R = T e
ika
e
κa
.
After lots of some algebra, we obtain
T = Ie
−ika
cosh κa − i
k
2
− κ
2
2kκ
sinh κa
−1
To interpret this, we use the currents
j = j
inc
+ j
ref
= (|I|
2
− |R|
2
)
~k
m
for x < 0. On the other hand, we have
j = j
tr
= |T |
2
~k
m
for
x > a
. We can use these to find the transmission probability, and it turns
out to be
P
tr
=
|j
trj
|
|j
inc
|
=
|T |
2
|I|
2
=
1 +
U
2
4E(U − E)
sinh
2
κa
−1
.
This demonstrates quantum tunneling. There is a non-zero probability that the
particles can pass through the potential barrier even though it classically does
not have enough energy. In particular, for
κa
1, the probability of tunneling
decays as
e
−2κa
. This is important, since it allows certain reactions with high
potential barrier to occur in practice even if the reactants do not classically have
enough energy to overcome it.