Post on 25-Feb-2018
Quantum Tunneling
One of the mysteries of quantum mechanics
W. Ubachs – Lectures MNW-Quant-Tunneling
Finite Potential Well
)()( ll IIIII ψψ =
)()( lldx
ddx
d IIIII ψψ=
)0()0( III ψψ =
A finite potential well has a potential of zero between x = 0 and x = , but outside that range the potential is a constant U0.
l
The potential outside the well is no longer zero; it falls off exponentially.
Solve in regions I, II, and IIIand use for boundary conditionsContinuity:
)0()0(dx
ddx
d III ψψ=
Bound states: E < E0Continuum states: E > E0
Finite Potential Well
( )202 2h
EUmG −=( ) 02
20
2
2=⎥⎦
⎤⎢⎣⎡ −
− ψψh
EUmdxd
GxGxIIII DeCe −+=,ψ
If E < U0
with
in the “forbidden regions”
General solution:
Region I 0<x hence 0=D and similarly for C
GxI Ce=ψ should match kxBkxAII cossin +=ψ
Finite value at 0=x exponentially decaying into the finite walls
Finite Potential WellThese 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
( )02 UEmh
ph
−==λ
( ) 022
02
2=⎥⎦
⎤⎢⎣⎡ −
+ ψψh
UEmdxd
0222
2=⎥⎦
⎤⎢⎣⎡+ ψψh
mEdxd
If E > U0
In regions I and III
free particle condition
In region II
In both cases oscillating free partcilewave function:
I,III:
II: mEh
ph
2==λ
0
2
02
221 U
mpUmvE +=+=
Tunneling Through a Barrier
l>x
0<x
mEh
ph
2==λ
( ) 0220
2
2=⎥⎦
⎤⎢⎣⎡ −
− ψψh
EUmdxd
0222
2=⎥⎦
⎤⎢⎣⎡+ ψψh
mEdxd
Also in region
In region oscillating wave
mEh
ph
2==λWave with same wavelength
In the barrier:
GxGxb DeCe −+=ψ
Approximation: assume that the decayingfunction is dominant Gx
b De−=ψ
( )( )
( ) ll GGx
eD
Dex
xT 2
2
2
2
2
0−
−==
=
==ψ
ψTransmission:
Tunneling Through a Barrier
( )202h
EUmG −=
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,
The smaller E is with respect to U0, the smaller the probability that the particle will tunnel through the barrier.
GleT 2−≈