Physics 452

Quantum mechanics IIWinter 2012

Karine Chesnel

HomeworkPhys 452

Sign up for the QM & Research presentationsNext week, W April 6 or F April 8 Homework #24

20 pts

Tuesday Apr 3: assignment #2111.5, 11.6, 11.7

Thursday Apr 5: assignment #2211.8, 11.10, 11.11, 11.13

Tuesday April 10: assignment #2311.14, 11.18, 11.20

Class- schedule

Phys 452

Today April 2: Partial waves, phase shifts

Wed April 4 : Born approx., Compton effect

Friday April 6 : research & QM presentations I

Monday 9 :research & QM presentations II

Treats and vote for best presentation In each session

Research and QM presentationPhys 452

Template

As an experimentalistIn the lab …

…or doing simulationsor theory

Research and QM presentationPhys 452

Template

Focus onone physical principle or

phenomenoninvolved

in your research

Make a connection with a topic covered in Quantum Mechanics:

A principleAn equation

An application

Phys 452Scattering

Quantum treatment

Plane wave Spherical wave

ikr

ikz eA e fr

Relationshipwith cross-section

2dD fd

1kr

Radiation zone0V

intermediate zone

Phys 452Scattering

Partial wave analysis

, , ,mlr R r Y

Develop the solution interms of spherical harmonics,

solution to a spherically symmetrical potential

2 2 2

2 2

( 1)2 2d u l lV r u Eu

m dr m r

0V

Scattering zone

Phys 452 ScatteringPartial wave analysis

0V 0V

0V

1kr

Intermediate zone

22

2 2

( 1)d u l l u k udr r

Scattering zone

Solve the Schrödingerequation with potential V

Physical Solution

1( ) lR r h krHankel functionsGeneral Solution

1,

.

, , ,ikz ml m l l

l m

r A e c h kr Y

11, , 2 1 cosikz ll l l

l

r A e k i l a h kr P

Partial wave amplitude

Geometrical considerations

/ikrR r e r

1krRadiation zone

22

2

d u k udr

Phys 452Scattering

Partial wave analysis

Connecting intermediate and radiation zone

0V

0V

1kr

2 *' '

'

2 1 2 ' 1 cos cosl l l ll l

D f l l a a P P Differential cross-section

24 2 1 ll

D d l a Total cross-section

2 1 cosl ll

f l a P

, ,ikr

ikz er A e fr

when 1kr

with

Orthogonality ofLegendre polynomials

Phys 452Scattering

Partial wave analysis

Connecting all three regions and expressing the Global wave function in spherical coordinates

0V

0V

1kr

,ikr

ikz er A e fr

24 2 1 ll

D d l a Total cross-section

0

2 1 cosikz ll l

l

e i l j kr P

Rayleigh’s formula

1, 2 1 cosll l l l

l

r A i l j kr ika h kr P To be determined

by solving the Schrödinger equation

in the scattering region+ boundary conditions

Jl Bessel functions

Phys 452Scattering

Partial wave analysis

1, 2 1 cosll l l l

l

r A i l j kr ika h kr P

Legendre polynomial

Bessel function Hankel function

(1)1 ( ) ( ) ( )l lh x j x in x

Phys 452Scattering

Partial wave analysis

Example: Hard-sphere scattering0V

12 1 cos 0ll l l l

l

A i l j kr ika h kr P

24 2 1 ll

l a Total cross-section

, 0a Boundary conditions

V

(Pb 11.3)

Exploiting n l nlP P 1l

ll

j kaa i

kh ka

24 a 1ka

Phys 452Scattering- Partial wave analysis

Spherical delta function shell

0V

0

sin ikrkr eA akr r

0V

Pb 11.4

1kaAssumption (low energy scattering)

0 0 02 1 cos cosl ll

f l a P a P a Outside:

sin krr B

kr Inside:

Boundary conditionsContinuity of Discontinuity of ' 2

2' m a

Find a relationshipbetween a0 and (a,

f D

Phys 452Scattering

Phase - shifts

Physical representation in 1D

wall

Physical representation in 3D

ikzAe

2i kzAe

2i kzikzAe Ae

2 li kreAkr

ikzAe

Phys 452Scattering

Phase – shifts and interference effects

Physical representation in 1D

wall

ikzAe

2i kzAe

2

2(0) 1

i kzikz

i

Ae Ae

A e

2 2 2sinI A

interference

Phys 452Quiz 34

In scattering and interference processes, the phase shift generally depends on the wavelength

A. True

B. False

Phys 452ScatteringPhase - shifts

2 li kreAkr

22 1 1 cos2

li kr ikrl

l ll e eA Pi kr kr

ikzAe

2 1 1 cos2

likr ikrl l

lA e e Pikr

If 0V

If 0V

Outgoing spherical

wave

Incomingspherical

wave

Asymptotic behavior at 1r

Phys 452Scattering

Phase - shifts

Partial wave amplitude Phase shift

la l

1, 2 1 cosll l l l

l

r A i l j kr ika h kr P

21 11 sin2

l li il la e e

ik k

Connecting the asymptotic behavior at 1kr

Phys 452Scattering

Phase - shifts

1 sinlil la ek

0

1 2 1 sin coslil l

l

f l e Pk

Scatteringamplitude

22

0

4 2 1 sin ll

lk

ScatteringCross-section

Phys 452Scattering – phase shift

Reflection against a wallPb 11.5

wall

ikzAe

2i kzikzBe Ae

0V

Region 2Region 1

1) Solve the Schrödinger equation

02'

m E Vk

• In region 2

• In region 1 2mEk

2) Continuity at boundary: , '

3) Identify the phase shift

2i kzikzBe Ae

Phys 452Scattering – Phase shift

0V , 0a Boundary conditions

V

(Pb 11.3)

1l

ll

j kaa i

kh kaWe found

Pb 11.6 Hard sphere scattering

21 11 sin2

l li il la e e

ik k Express the phase shift: using

Express in terms of functions and l lj ka ln ka

Phys 452Scattering- phase shifts

Spherical delta function shell (Pb 11.4)

0V 0sin kr

Akr

0V

Pb 11.7

1kaDo NOT do the assumption

Outside:

sin krr B

kr Inside:

Boundary conditionsContinuity of Discontinuity of ' 2

2' m a

Express in terms of and 0 ka2

2m a