Klein Paradox

Contents

An Overview

Klein Paradox and Klein Gordon equation

Klein Paradox and Dirac equation

Further investigation

An Overview

Assumptions

We deal with a plane wave solution for

KG, Dirac equation of a single particle

The particles has an energy E (kinetic

+rest energy) before and after the barrier

A step-function potential

Probability Current or charge current

conserved

No particle flux in Region II may come

from the positive direction (causality

requirement)

Klein Paradox from KG eqn

point of view

In case of a potential, V

22

0

222 )( cmcpE

tihVE

2

0

222][ cmcVt

ih

KG

equation

Solutions in Region I:

).().(

RzpEt

h

izpEt

h

i

I ee

)'.( zpEth

i

II Te

In region II:

From KG eqn,

Weak potential

422

422

)(' cmVEp

cmEp

2mcEV

P’ is real, Only

positive value

allowed

Intermediate

potential22 mcEVmcE

P’ is imaginary

[Evanescent wave]

Strong potential

2mcEV

P’ is real

A non-classical

Behavior !!

What about charge

current conservation?

To get the values of R,T

apply continuity conditions of

Φ and its derivative at z=0:

')1(

1

TpRp

TR

Solving for R,T :

'

'

'

2

pp

ppR

pp

pT

Conservation of Charge

Current Charge current is defined by:

This is interpreted as charge current not

probability current since we have no positive

definite conserved probability in KG equation

)1(

) R() R(2

) R)( R(2

2

).(*

).().().(

).().().(*

).(

Rm

pj

eeh

ipee

im

h

eh

ipe

h

ipee

im

hj

I

zpEth

izpEt

h

izpEt

h

izpEt

h

i

zpEth

izpEt

h

izpEt

h

izpEt

h

i

I

imaginary is ',0

is ','

])'

('

[2

2

)'*.(*

)'.()'.()'*.(*

pj

realpTm

pj

eTh

ipTeTe

h

ipeT

im

hj

II

II

zpEth

izpEt

h

izpEt

h

izpEt

h

i

II

m

pjincident

Transmission coefficient (T)= 2'T

p

p

j

j

incident

II

2

'

'

pp

pp

j

jj

incident

incidentI

[ Logical result ]

incident

R

j

jReflection coefficient ( R )=

In all cases T+R=1

For weak potential

R= , T= , T+R=1

2

'

'

pp

pp2)'(

'4

pp

pp

For an intermediate potential

R=1 , T= 0 , T+R=1

For a strong potential

R= , T= , T+R=12)'(

'4

pp

pp

2

'

'

pp

pp

R>1 , T is

negativeThis is Klein paradox

Reflected current is bigger than incident

current !!!!!!

Transmitted current is opposite in charge to

incident current !!!!!

Extra particles supplied by the potential ?

Another type of particle of opposite charge

supplied by the potential ?

Particle anti-particle pair production

Dirac Equation and Klein

Paradox

For z<0 :

Incident wave:

( having spin up)

For z>0 :

Reflected spinor:

Transmitted spinor:

Consider the Case of a Strong

Potential: By applying the continuity conditions of the

spinors at the boundary:

No spin flip

The Probability Currents:

Where r =

=

R=

T=

>1

Again reflected current is greater than incident current.

Hole Theory Explanation The potential energy raised a negative energy

electron to a positive energy state creating a positive

hole (positron) behind it. The hole is attracted towards

the potential while the electron is repelled far from it

!! This process is stimulated by the incoming electron

Question about this

interpretation:

How can energy conservation be

guaranteed?

Any experimental evidence !!!???

Should we reinterpret the probability

current as charge current?

Further investigationTo construct a wave packet of several

momentum components and study its

transmission and reflection behavior from a

potential step.

To use second quantization representation of dirac

spinors

The End