PHY 711 Fall 2012 -- Lecture 3 19/3/2012

PHY 711 Classical Mechanics and Mathematical Methods

10-10:50 AM MWF Olin 103

Plan for Lecture 3:

Chapter 1 – scattering theory continued; center of mass versus laboratory reference frame.

PHY 711 Fall 2012 -- Lecture 3 29/3/2012

PHY 711 Fall 2012 -- Lecture 3 39/3/2012

Scattering geometry:

sin

d

dbb

d

d

PHY 711 Fall 2012 -- Lecture 3 49/3/2012

Relationship between scattering angle and impact parameter b for interaction potential V(r):

ErV

rb

rdrb

r )(1

/12

2

2

2

min

sin

d

dbb

d

d

0)(

1

:where

min2

min

2

E

rV

r

b

PHY 711 Fall 2012 -- Lecture 3 59/3/2012

2/sin

2/cos

1/

1sin2

:scattering Rutherford

2

1

b

b

2/sin

1

4sin 4

2

d

dbb

d

d

Example of cross section analysis

PHY 711 Fall 2012 -- Lecture 3 69/3/2012

Hard sphere scattering:

2cos

that showedyou homework your For

Db

4sin

2D

d

dbb

d

d

Example of cross section analysis

PHY 711 Fall 2012 -- Lecture 3 79/3/2012

The results above were derived in the center of mass reference frame; relationship between normal laboratory reference and center of mass:

Laboratory reference frame: Before After

u1 u2=0 v1

v2

yz

m1 m2

Center of mass reference frame: Before After

U1 U2=0V1

V2

m1 m2

PHY 711 Fall 2012 -- Lecture 3 89/3/2012

Relationship between center of mass and laboratory frames of reference

2211212211

212211

212211

mass ofcenter of Definition

vvVuu

Rrr

Rrr

R

mmmmmm

mmmm

mmmm

CM

CM

CM

CM

CM

CM mm

mm

mm

m

VVv

vvuV

11

21

22111

21

1

:caseour In

V1

VCM

v1y

PHY 711 Fall 2012 -- Lecture 3 99/3/2012

V1

VCM

v1y

Relationship between center of mass and laboratory frames of reference

211

11

11

11

/cos

sin

/cos

sintan

coscos

sinsin

mmVV

VVv

Vv

CM

CM

CM

y

yy

VVv /m/ : thatSo

: thatnote Also

and

collision elastican for that Note

211

2211

2211

mVV

UmUm

VVUVU

CM

CM

PHY 711 Fall 2012 -- Lecture 3 109/3/2012

Differential cross sections in different reference frames

y

y

y

y

cos

cos

sin

sin

d

d

d

d

d

d

d

d

d

d

d

d

LAB

CM

LAB

CM

CM

CM

LAB

LAB

22121

21

/cos/21

/coscos

: thatshow topossible isit energy, and momentum of

on conservatiand/or result previous theFrom

mmmm

mm

y

PHY 711 Fall 2012 -- Lecture 3 119/3/2012

Differential cross sections in different reference frames – continued:

yy

cos

cos

d

d

d

d

d

d

CM

CM

LAB

LAB

1cos/

/cos/21

21

2/322121

ymm

mmmm

d

d

d

d

CM

CM

LAB

LAB

21 /cos

sintan :where

mm

y

PHY 711 Fall 2012 -- Lecture 3 129/3/2012

Example: suppose m1 = m2

1cos/

/cos/21

21

2/322121

ymm

mmmm

d

d

d

d

CM

CM

LAB

LAB

21 /cos

sintan :where

mm

y

20 that note

2

1cos

sintan :case In this

y

yy

yyycos4

2

CM

CM

LAB

LAB

d

d

d

d