Free Particle

(x) = A cos(kx) or (x) = A sin(kx) (x)= A eikx = A cos(kx) + i A sin(kx)(x)= B e-ikx = B cos(kx) - i B sin(kx)

d

dxk

2

22

d

dxik Ae kikx

2

22 2

( )

d

dxik Be kikx

2

22 2

( )

Free Particle

(x)= A eikx +B e-ikx is a solution

• A and B are constants

• hence (x,t)= (x)e-it

• = A ei(kx- t) +B e-i(kx+ t)

Travelling wave to right Travelling wave to left

Free Particle(x,t)= A ei(kx- t) is matter wave travelling to the

right(along the positive x-axis) *(x,t)= A* e-i(kx- t)

• | (x,t)|2 = (x,t) *(x,t)= AA* =|A|2

• intensity of wave is constant!

• Probability is the same everywhere

• a free particle is equally likely to be found anywhere

• P(x,t)= |(x,t)|2 is probability of finding a particle at position x at time t

• total probability of finding it somewhere is

Free Particle

2( , ) ( , ) 1P x t dx x t dx

• consider a classical point particle moving back and forth

with constant speed between two walls located at x=0 and x=8cm

• particle spends same amount of time everywhere

• P(x)=P0 if 0< x < 8 cm

• P(x)=0 if x< 0 or x> 8cm

Free Particle• Since

• hence P0 = (1/8) cm-1 ===> probability/unit length is 1/8

• probability of finding particle in length dx is (1/8)dx

• probability of finding it at x=2cm is zero! (dx=0)• Probability of finding it in some range 1.9 to 2.1 is

(1/8)x = (1/8)(2.1-1.9)= .025

2( , ) ( , ) 1P x t dx x t dx

8

0 0

0

( ) 8 1cm

P x dx P dx P cm

Free Particle• Probability of finding it between x=0 and x=8cm is

(1/8)(8-0) = 1

• intensity of wave is constant!

• Probability is the same everywhere

• a free particle is equally likely to be found anywhere

• free particle has definite energy E=(1/2)mv2 and momentum p=mv but uncertain position

R+T=1

Barrier Tunneling• consider a barrier E < U0

U0

Schrodinger Solution• Consider the three regions : left of barrier, right of barrier and in

the barrier

• left:

• right:

• inside:

U0

1 2( ) 0ikx ikxx e e x

5( ) ikxx e x L

22

1 2( ) ( ) ikx ikxP x x e e

2

5( )P x

2 2

02 2

8( ) 0

d mE U

dx h

Barrier Tunneling• Solution inside barrier has form

• since

• P(x) is smaller as U0 increases

3 4( ) 0k x k xx e e x L

02 2 ( )m U Ek

h

2 23( ) k xP x e

3 4

Tunneling

Tunneling

• Transmission coefficient T ~ e-2kL

• k={82m(U0-E)/h2}1/2 Note: E < U0

• if T=.02 then for every 1000 electrons hitting the barrier, about 20 will tunnel

• extremely sensitive to L and k

• width and height of barrier