Mess Tunneling

Free of charge Wave Packet

Tunneling through a Buffer

Tunneling over a Well

Model: Tunneling Microscopic lense

Canal Transistor

-particle decay

Quantum Tunneling

Evidence

Maybe in lab

a couple of

Quantum Tunneling

Time Dependent Wave Equation

2

2m

2

x

2

2

y

2

2

back button

( x, y, z ., t )

2

2m

2

times

2

( x, sumado a, z, big t ) U ( by, y, unces ) ( x, y, z, to )

a couple of

i

to

( times, y, unces, t )

( x, y, z ) (t )

a couple of

y

a couple of

2

z

2

( x, sumado a, z ) U ( x, sumado a, z ) ( times, y, unces )

i

t

(t )

E (t )

What is the assumption Jr .?

( times, y, unces, t )

E ( x, y, z )

( back button, y, z . )e

(t )

Electronic

i t

d?gge

E

i actually t

E

3

Quantum Tunneling

Free of charge Wave Box in One Dimensions

( by, y, z ., t )

2

2m

2

by

( x, y, z )e

two

2

sumado a

E

i t

2

two

z

a couple of

( x, y, z . ) U ( by, y, z . ) ( x, sumado a, z )

E ( x, sumado a, z )

For one-dimension we have

2 2

( x) U ( x) ( x) E ( x)

a couple of

2m back button

U (x ) zero or a continuous

2 a couple of

( x)

2

2m x

( x)

i

E ( x)

D?gge

2m At the

x

2

( back button, t )

Ae

ikx

Ae

2m E

times

2

at the

E

i t

Aei kx

t

Remember we makeup a trend

packet using Fourier analysis.

4

Quantum Tunneling

Potential Step Obstacle

The energy can be greater or perhaps less than the potential barrier. 2m E U 0

2

k

2m U 0 E

2

E

2 2

( x) U ( x) ( x)

2

2m x

2

E

2m

E U ( x) ( x)

2

( x)

x2

2m E U ( x )

x

2

i actually

( x)

Ae

( x ) Ce

i

( x, t )

Ae

2m E U ( times )

by

2

i actually

Be

2m E U ( x )

by

2

elizabeth

E

i actually t

E ( x)

2m U ( x ) E

times

2

D?gge

ikx we t

2m E U ( by )

back button

2

i

Be

Sobre

Beikx

2m U ( x ) E

back button

2

k 2 ( x)

Elizabeth > U0

E < U0

i actually t

5

Quantum Tunneling

Potential Step Barrier

1st we will consider At the < U0

Region We U(x)=0

Aeikx

I ( x)

Become

ikx

k

2mE

2

Region II U(x)=U0

II

( x)

Ce

by

De

by

2m U 0 E

2

Border Conditions

Enables first make an effort to guess the wavefunction!

I actually

x

( x) back button

I

2

0

( x)

x 0

( x) back button

x

zero

II

( x)

x 0

6th

Quantum Tunneling

E < U0

Potential Step Hurdle

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