Quantum Tunneling
One of the mysteries of quantum mechanics
W. Ubachs – Lectures MNW-Quant-Tunneling
Finite Potential Well
A finite potential well has a potential of zero between x = 0 and x = l ,
but outside that range the potential is a constant U0.
The potential outside the well is no
longer zero; it falls off exponentially.
Solve in regions I, II, and III
and use for boundary conditions
Continuity:
ψ I (0) = ψ II (0)
dψ I
dψ II
( 0) =
(0)
dx
dx
ψ II (l) = ψ III (l)
dψ II
dψ III
(l ) =
(l )
dx
dx
Bound states: E < E0
Continuum states: E > E0
Finite Potential Well
If E < U0
in the “forbidden regions”
⎡ 2m(U 0 − E )⎤
−
⎥⎦ψ = 0
dx 2 ⎢⎣
h2
d 2ψ
General solution:
Region I
x<0
ψ I = CeGx
hence
D=0
should match
Finite value at
x=0
with
G2 =
2m(U 0 − E )
h2
ψ I , III = CeGx + De −Gx
and similarly for C
ψ II = A sin kx + B cos kx
exponentially decaying into the finite walls
Finite Potential Well
These graphs show the wave functions and probability
distributions for the first three energy states.
Nonclassical effects
Partile can exist in the forbidden region
Finite Potential Well
If E > U0
free particle condition
⎡ 2 m( E − U 0 ) ⎤
+
ψ =0
⎢
⎥
2
2
⎣
⎦
dx
h
d 2ψ
d 2ψ
⎡ 2mE ⎤
+
ψ =0
⎢
2
2 ⎥
⎣ h ⎦
dx
In regions I and III
In region II
In both cases oscillating free partcile
wave function:
I,III:
λ=
h
h
=
p
2 m( E − U 0 )
II:
λ=
h
h
=
p
2mE
1 2
p2
E = mv + U 0 =
+ U0
2
2m
Tunneling Through a Barrier
In region
x<0
oscillating wave
d 2ψ
λ=
⎡ 2mE ⎤
+
ψ =0
⎢
2
2 ⎥
⎣ h ⎦
dx
Also in region
h
h
=
p
2mE
x>l
Wave with same wavelength
λ=
h
h
=
p
2mE
In the barrier:
⎡ 2m(U 0 − E )⎤
−
ψ =0
⎢
⎥
2
2
⎣
⎦
h
dx
d 2ψ
ψ b = CeGx + De −Gx
Approximation: assume that the decaying
function is dominant
ψ = De −Gx
b
Transmission: T =
ψ (x = l)
ψ (x = 0)
2
2
2
(
De −Gx )
=
= e − 2Gl
D2
Tunneling Through a Barrier
The probability that a particle tunnels through a barrier
can be expressed as a transmission coefficient, T, and
a reflection coefficient, R (where T + R = 1). If T is small,
T ≈ e − 2Gl
G=
2m(U 0 − E )
h2
The smaller E is with respect to U0, the smaller the
probability that the particle will tunnel through the
barrier.