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21  1
5.73
Lecture
#21
revised October 21, 2002 @ 10:15 AM
1. define radial momentum
2. define orbital angular momentum
3. separate
p
2
into radial and angular terms:
4. find Complete Set of Commuting Observables (CSCO) useful for block
diagonalizing
H
5.
3DCentral Force Problems I
All
2Body, 3D problems can be reduced to
* a 2D angular part that is exactly and
universally soluble
* a 1D radial part that is
systemspecific and soluble by 1D techniques in
which you are now expert
Next 3 lectures:
Correspondence Principle
Commutation Rules
→
all matrix elements
p
r
=
r
1
q
⋅
p
i
h
()
r
L
=
r
q
×
r
p
also
L
×
L
=
i
h
L
and
L
i
,
L
j
[]
=
i
h
∑
k
ε
ijk
L
k
p
2
=
p
r
2
+
r
−2
L
2
H
,
L
2
=
H,L
i
=
L
2
,
L
i
=0
H
,
L
2
,
L
i
CSCO
L,M
L
universal basis set
separate radial
part of
H
:
H
l
(r)
=
p
r
2
2µ
+
V(
r
)
+
h
2
ll
+1
r
2
V
l
(
r
)
effective radial
potential
6. ALL Matrix Elements of Angular Momentum Components Derived from
Commutation Rules.
7. Spherical Tensor Classification of
all
operators.
⇓
8. WignerEckart Theorem
→
all angular matrix elements of all operators.
I hate differential operators.
Replace them using exclusively simple Commutation
Rule based Operator Algebra.
general definition of angular
momentum and of “vector
operators”
1D Schröd Eq.
Read:
CTDL, pages 643660 for next lecture.
Roadmap
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View Full Document21  2
5.73
Lecture
#21
revised October 21, 2002 @ 10:15 AM
Lots of derivations based on classical VECTOR ANALYSIS — much will be set aside
as NONLECTURE
Central Force Problems:
2 bodies where interaction force is along the vector
r
q
1
−
r
q
2
1
2
•
•
•
CM
r
q
12
r
q
1
r
q
2
r
q
cm
r
r
1
r
r
2
origin
r
q
2
=
r
q
1
+
r
q
12
r
q
12
=
r
q
2
−
r
q
1
=
ˆ
ix
2
−
x
1
()
+
ˆ
jy
2
−
y
1
+
ˆ
kz
2
−
z
1
r
≡
r
q
12
=
x
2
−
x
1
2
+
y
2
−
y
1
2
+
z
2
−
z
1
2
[]
1
/
2
also C.M. Coordinate system
r
r
1
=
r
q
1
−
r
q
cm
r
1
r
=
m
2
M
v
r
2
=
r
q
2
−
r
q
cm
r
2
r
=
m
1
M
H
=
H
translation
+
H
center of mass
free translation
of C of M of
system of mass
M = m
1
+ m
2
motion of fictitious
particle of mass
in coordinate system
with origin at C of M (CTDL page 713)
µ=
+
mm
12
√
√
..
H
P
HP
translation
free rotation
(no
,
dependence)
constant
=
+
+
=
µ
+
trans
CM
cm
V
Vr
2
2
2
1
2
θφ
{
LAB
BODY
free translation of
system with respect to
lab (not interesting)
motion of particle of
mass µ with respect
to origin at c. of m.
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 RobertField

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