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−≈