One-Dimensional Scattering of Waves 2006 Quantum MechanicsProf. Y. F. Chen One-Dimensional...

One-Dimensional Scattering of Waves

2006 Quantum Mechanics Prof. Y. F. Chen


in this chapter we will explore the phenomena of lD scattering to show t

hat transmission is possible even when the quantum particle has insuffic

ient energy to surmount the barrier

the transfer matrix method will be utilized to analyze the one-dimension

al propagation of quantum waves




consider a particle of energy E and mass m to be incident from the left o

n arbitrarily shaped, 1D, smooth & continuous potential

Such a problem can be solved by ：

(1) dividing the potential into a piecewise constant function

(2) using the transfer matrix method to calculate the probability of the pa

rticle emerging on the right-hand side of the barrier


The Transfer Matrix Method


)(xV

Figure 6.1 Sketch of the quantum scattering at the jth interface between 2 s

uccessive constant values of the piecewise potential & the wave propagatin

g through the constant potential until reaching the next interface at a dist

ance after crossing the jth interface




010 xx

1

0

j

ssj dxx

1 jxx

jdjV

j j~1j

1d

12 dx

00 d010 xx

1

0

j

ssj dxx

1 jxx

jdjV

j j~1j

1d

12 dx

00 d

jV

jd

the dynamics of the quantum particle is described by the Schrödinger e

q., which is given in the jth region by ：

the general solutions ：

where

& correspond to waves traveling forward and backward in jth regi

on, respectively




)()(2 2

22

xExVxd

d

m jjj

xkij

xkijj

jj eAeAx )(

)(2 j

j

VEmk

jA

jA

the relationship between the coefficients & are determined by ap

plying the boundary conditions at the interface ：

as a result, it can be found that 　 &

is referred to be the scattering matrix




1jA

jA

jj xxjxxj xx |)(|)(1 jj xxjxxj xx |)(|)(1 &

jjjj AAAA 11

j

j

jj

j

jjj A

k

kA

k

kAA

11

11&

j

j

j

j

j

j

j

j

A

A

k

k

k

kA

A

111

1

11

11

11

j

j

j

j

j

j

j

j

A

A

k

k

k

kA

A

11

1

1

1

11

11

11

→

j

jj

j

j

A

A

A

AD

1

1

11

11

11

11

2

1

j

j

j

j

j

j

j

j

j

k

k

k

kk

k

k

k

D

→→

jD

we can find that

propagation between potential steps separated by distance carries

phase information only so that

a propagation matrix is defined as

the successive operation of the scattering & propagation matrices leads

to




( ) ( )

( ) ( )

j j j j j j

j j j

i k x d i k x d i k x i k x

j j j j

x d x

A e A e A e A e

jd

j

jj

j

j

A

A

A

A~

~P

jj

jj

dki

dki

je

e

0

0P

2

2211

1

111

1

11

0

0~

~

A

A

A

A

A

A

A

ADPDPDD

for the general case of N potential steps, the transfer matrix for each

region can be multiplied out to obtain the total transfer matrix

∵ the quantum particle is introduced from the left, the initial condition is

given by

if no backward particle can be found on the right side of the total

potential →

→




N

NN

N

jjj

N

N

A

A

A

A

A

ADPDQ

1

10

0

10 A

0NA

0

1

2221

1211

0

NA

QQ

QQ

A

as a consequence, the transmission & reflection coefficients are given

by

those can be used to calculate the transmission & reflection probability

of a quantum particle through an arbitrary 1D potential




211

2

||

1||

QAT N

211

2212

0 ||

||||

Q

QAR

&

consider a particle of energy E and mass m that are sent from the left

on a potential barrier


The Potential Barrier


Lx

LxV

x

xV B

0

0

00

)(

)(xV

BV

0 L

)(xV

BV

0 L

Figure 6.2 Sketch of the quantum scattering of a 1D rectangular barrier of energy VB

With

the total matrix Q is given by

where &

it simplified as




N

NN

N

jjj

N

N

A

A

A

A

A

ADPDQ

1

10

0

1

0

1

0

1

0

1

0

0

1

0

1

0

1

0

1

11

11

2

1

0

0

11

11

2

11

1

k

k

k

kk

k

k

k

e

e

k

k

k

kk

k

k

k

Lik

Lik

Q

mE

k2

0 )(2

1BVEm

k

)sin(2

)cos()sin(2

)sin(2

)sin(2

)cos(

11

0

0

111

1

0

0

1

11

0

0

11

1

0

0

11

Lkk

k

k

kiLkLk

k

k

k

ki

Lkk

k

k

kiLk

k

k

k

kiLk

Q

transmission probability in the case

in terms of energy E and potential

→




BVE

12

2 2011 12

11 0 1

12

20 11

1 0

1 1cos ( ) sin ( )

| | 4

1 1 sin ( )

4

kkT k L k L

Q k k

k kk L

k k

BV

1

22 )(2

sin)(4

11

LVEm

VEE

VT B

B

B


occurs whenever ：

with

the condition corresponds to resonances in transmission that occ

ur when quantum waves back-scattered from the step change in barrier

potential at positions & interfere and exactly cancel each oth

er, resulting in zero reflection from the potential barrier




BVE

1T

22

20

)(2sin

L

n

mVEL

VEmB

B

3,2,1n

1T

0x Lx


(1) when , the transmission probability T → 1

the particles are nearly not affected by the barrier & have total

transmission

(2) in the limit case , we have

→




BVE

BVE

BVE /)(2/)(2sin LVEmLVEm BB

1

2

21

22

21

)(2sin

)(4

11limlim

LVmL

VEm

VEE

VT BB

B

B

VEVE BB


the wave number becomes imaginary, with

→

if →

→




BVE

1k 11 ik /)(21 EVm B 12

20 112

11 1 0

12

2

1 1 1 sinh ( )

| | 4

2 ( )1 1 sinh

4 ( )BB

B

kT L

Q k

m V EVL

E V E

1/)(2 LEVm B ]/)(2exp[2

1/)(2sinh LVEmLEVm BB

L

EVm

V

E

V

ET B

BB )(2

2exp116





BVE

VB = 0.1 eV

L = 5 nm

L = 2 nmL = 1 nm

E (eV)

Tra

nsm

issi

on c

oeff

icie

ntVB = 0.1 eV

L = 5 nm

L = 2 nmL = 1 nm

E (eV)

Tra

nsm

issi

on c

oeff

icie

nt

Figure 6.3 Transmission probability as a function of particle energy for eV 1.0BV

and several widths nm 5 and 2, ,1L

in terms of & , the total wave function can be given by

where is the Heaviside unit step func. , , the matrix elem

ent & are determined from

the efficient & can be found to be given by


Scattering of a Wave Package State


jA

jA

N

jjj

xxkij

xxkij

xkixkiE

xxuxxueAeA

xueAex

jjjj

11

)()(

0

)()(

)()( 00

)(xu11210 /QQA

11Q 21Q

N

N

jjj DPDQ

1

1

jA

jA

1for1

0

1

0

11

11

1

jAA

A j

ssjsjj

j

j DPD

where

and the identities & are used to express the equatio

n in a general form


Scattering of a Wave Package State


1for1

0

1

0

11

11

1

jAA

A j

ssjsjj

j

j DPD

1for11

11

2

1

11

11

1

s

k

k

k

kk

k

k

k

s

s

s

s

s

s

s

s

sD

1for0

01

s

e

ess

ss

dki

dki

sP

1for1

0

jdxj

ssj

IPD 10

10 00 d

One-Dimensional Scattering of Waves 2006 Quantum MechanicsProf. Y. F. Chen One-Dimensional...

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Transcript of One-Dimensional Scattering of Waves 2006 Quantum MechanicsProf. Y. F. Chen One-Dimensional...