One-Dimensional Scattering of Waves 2006 Quantum Mechanics Prof. Y. F. Chen One-Dimensional Scattering of Waves
Jan 04, 2016
One-Dimensional Scattering of Waves
2006 Quantum Mechanics Prof. Y. F. Chen
One-Dimensional Scattering of Waves
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
2006 Quantum Mechanics Prof. Y. F. Chen
One-Dimensional Scattering of Waves
One-Dimensional Scattering of 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
2006 Quantum Mechanics Prof. Y. F. Chen
The Transfer Matrix Method
One-Dimensional Scattering of Waves
)(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
2006 Quantum Mechanics Prof. Y. F. Chen
The Transfer Matrix Method
One-Dimensional Scattering of Waves
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
2006 Quantum Mechanics Prof. Y. F. Chen
The Transfer Matrix Method
One-Dimensional Scattering of Waves
)()(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
2006 Quantum Mechanics Prof. Y. F. Chen
The Transfer Matrix Method
One-Dimensional Scattering of Waves
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
2006 Quantum Mechanics Prof. Y. F. Chen
The Transfer Matrix Method
One-Dimensional Scattering of Waves
( ) ( )
( ) ( )
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 →
→
2006 Quantum Mechanics Prof. Y. F. Chen
The Transfer Matrix Method
One-Dimensional Scattering of Waves
N
NN
N
jjj
N
N
A
A
A
A
A
ADPDQ
1
10
0
10 A
0NA
0
1
2221
1211
0
NA
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
2006 Quantum Mechanics Prof. Y. F. Chen
The Transfer Matrix Method
One-Dimensional Scattering of Waves
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
2006 Quantum Mechanics Prof. Y. F. Chen
The Potential Barrier
One-Dimensional Scattering of Waves
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
2006 Quantum Mechanics Prof. Y. F. Chen
The Potential Barrier
One-Dimensional Scattering of Waves
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
→
2006 Quantum Mechanics Prof. Y. F. Chen
The Potential Barrier
One-Dimensional Scattering of Waves
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
transmission probability in the case
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
2006 Quantum Mechanics Prof. Y. F. Chen
The Potential Barrier
One-Dimensional Scattering of Waves
BVE
1T
22
20
)(2sin
L
n
mVEL
VEmB
B
3,2,1n
1T
0x Lx
transmission probability in the case
(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
→
2006 Quantum Mechanics Prof. Y. F. Chen
The Potential Barrier
One-Dimensional Scattering of Waves
BVE
BVE
BVE /)(2/)(2sin LVEmLVEm BB
1
2
21
22
21
)(2sin
)(4
11limlim
LVmL
VEm
VEE
VT BB
B
B
VEVE BB
transmission probability in the case
the wave number becomes imaginary, with
→
if →
→
2006 Quantum Mechanics Prof. Y. F. Chen
The Potential Barrier
One-Dimensional Scattering of Waves
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
transmission probability in the case
2006 Quantum Mechanics Prof. Y. F. Chen
The Potential Barrier
One-Dimensional Scattering of Waves
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
2006 Quantum Mechanics Prof. Y. F. Chen
Scattering of a Wave Package State
One-Dimensional Scattering of Waves
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
2006 Quantum Mechanics Prof. Y. F. Chen
Scattering of a Wave Package State
One-Dimensional Scattering of Waves
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