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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