Introduction to
Time dependentTime-independent methods:
Kap. 7-lect2
),(ˆ),( trHtrt
i
),(),(ˆ trEtrH
Methods to obtain an approximate eigen energy, E and wave function:
perturbation methods
Perturbation theory Variational method Scattering theory
),()(),( trtctr nn
n
Ground/Bound states Continuum states
Non degenerate states
Methods to obtain an approximate expression for the expansion amplitudes.
Approximation methods in Quantum Mechanics
Degenerate states
GoldenRule
Scattering Theory:
• Classical Scattering:– Differential and total cross section– Examples: Hard sphere and Coulomb
scattering
• Quantal Scattering:– Formulated as a stationary problem– Integral Equation– Born Approximation– Examples: Hard sphere and Coulomb scattering
ddd
dd
d
d)sin(
0
2
0
scin
dNj
dtd
Number of scattered particles into :
insc jdj
per unit time:
dj
j
d
d
in
scDifferential Cross Section:
Total Cross Section:
Dimension: AreaInterpretation: Effective area for scattering.
Dimension: NoneInterpretation: ”Probability” for scattering into d
innj
ddscj
The Scattering Cross Section (To be corrected, see Endre Slide)
ddd
dd
d
d)sin(
0
2
0
Number of scattered particles into :
Differential Cross Section:
Total Cross Section:
innj
dd
The Scattering Cross Section
Nout
Nout
Quantal Scattering - No Trajectory! (A plane wave hits some object and a spherical wave emerges)
innj
dd
r
eCf
ikr
scattered ),(
scatteredinn
rkiinn Ce
• Solve the time independent Schrödinger equation• Approximate the solution to one which is valid far away from the scattering center• Write the solution as a sum of an incoming plane wave and an outgoing spherical wave.• Must find a relation between the wavefunction and the current densities that defines the
cross section.
Procedure:
scj
Current Density:
imm
ij *** Re)(
2
Incomming current density:
2C
m
kjin
Outgoing spherical current density:
2
22),(
mr
kfCjsc
r
eCf
ikr
sc ),(
ikzrkiin CeCe
zyx ez
fe
y
fe
x
frf
)(
eeer
fr
.......
2),(),(
rOr
eikCf
r
eCf
r
ikrikr
Example - Classical scattering:
d
b
bdbdd
ddd
dd )sin(
d
dbb
d
d
sin
b R
2
2cos
R
b
2sin
2
R
d
db
4
2R
d
d
2R
Hard Sphere scattering:
Independent of angles!= Geometrical Cross sectional area of sphere!
Example from 1D:
scatteredinn
0V
0
22
2V
m
kE
ikxikxscattered BeAeC
ikxBe
B
Af ),(
Forward scattering
Reflection
ikxinn Ce
ikxAe
In this case (since potential is discontinuous) we can find f excactly by ”gluing”
The Schrödinger equation - scattering form:
)( )( )(2
22
rErrVm
)( )()( 22 rrUrk
:get we)(2
)( and 2
with 2
22
rVm
rUm
kE
Now we must define the current densities from the wave function…
The final expression:
2
2
22
2
22
),(),(
f
mk
Cd
drfmrk
C
jd
drj
d
d
in
sc
2),(
fd
d
Summary
Then we have:
2),(
fd
d
…. Now we can start to work
)( )()( 22 rrUrk
Write the Schrödinger equation as:
)),(( r
efeC
ikrrki
Asymptotics:
Integral equation
)( )()( 22 rrUrk
)()(G ''22 rrrrk
With the rewritten Schrödinger equation we can introducea Greens function, which (almost) solves the problem for a delta-function potential:
Then a solution of:
can be written:
rdrrUrrGrr 30 )()()()()(
where we require: 0)( 022 rk
because….
rdrrUrrGkrkrk 3220
2222 )()()()()()()()(
This term is 0 This equals )( rr
Integration over the delta function gives result:
)()()()( 22 rrUrk
rdrrUrrGrr 30 )()()()()(
Formal solution:
Useless so far!
Must find G(r) in )()(G ''22 rrrrk
Note: sder rsi 33)2(
1)(
rsirsi ese 22and:
sdekssdek rsirsi 322322 )()(
sdks
erG
rsi3
222)2(
1)(
)()2(
1)()( 3
322 rrderGk rsi
r
erG
rki
4)(
Then:
The function: solves the problem!
”Proof”:
The integral can be evaluated, and the result is:
rdrrUrr
eer
rrikrki
3
01 )()(4
1)(
Inserting G(r), we obtain:
0)( 022 rk
implies that:
rkier )(0
rdrrUrr
eer
rrikrki
3)()(
4
1)(
At large r this can be recast to an outgoing spherical wave…..
The Born series:
rkier )(0
rdrrUrr
eer
rrikrki
3
12 )()(4
1)(
And so on…. Not necesarily convergent!
rdrrUrr
eer
rrikrki
3
01 )()(4
1)(
We obtains:
rdrrUrr
eer
rrikrki
3)()(
4
1)(
At large r this can be recast to an outgoing spherical wave…..
The Born series:
rkier )(0
rdrrUrr
eer
rrikrki
3
12 )()(4
1)(
And so on…. Not necesarily convergent!
)( )()( 22 rrUrk
Write the Schrödinger equation as:
)),(( r
efeC
ikrrki
Asymptotics:
SUMMARY
)1(
)21(
2
2
2/12
2
2
22
r
rrr
r
r
r
rrr
rrrrrr
The potential is assumed to have short range, i.e. Active only for small r’ :
rr
rikikrrrik eee
1)
rrr
11
ff k
p
r
rk
Asymptotics - Detector is at near infinite r
2)
),(
3)()(4
1)(
f
rkiikr
rki rdrrUer
eer f
Asymptotic excact result:
Still Useless!
The Born approximation:
rdrVem
r
eer rkki
ikrrki f
3)(
2)(
4
2)(
rdrVem
f rkki f 3)(
2)(
2),(
The scattering amplitude is then:
:) The momentum change Fourier transform of the potential!
Use incomming wave instead of )'(r Under integration sign:
Valid when:
1)( rkier
1)()(4
1 3
rdrrUrr
e rrik
Weak potentialsand/or large energies!
rdrVem
f rkkiB
q
f 3)(
2)(
4
2),(
fk
k
q
2sin2
kq
2
00
cos'
0
22
sin)(4
2)( ddedrrrV
mf iqrB
2cos
2sin2sin
dkdq
2cos
0
2sin)(
2drqrrV
q
m
Spheric Symmetric potentials:
Total Cross Section:
ddd
dd )sin(
qdqqfk
dfk
BB
22
02
0
2)(
2)sin()(2
Summary - 1’st. Born Approximation:
rdrVem
f rkkiB
q
f 3)(
2)(
4
2),(
qdqqfk
kB
22
02
)(2
2),( Bf
d
d
fkkq
Best at large energies!
Example - Hard sphere 1. Born scattering:
b R
2
4
2R
d
d
2R
Classical Hard Sphere scattering:
Quantal Hard Sphere potential:
qRRq
mrdrqrRr
q
mf B sin
2sin
22
02
RrrV )(
qRRq
m
d
d 22
2
2sin
2
Depends on angles - but roughly independent when qR << 1 Thats it!