So%20as%20an%20exercise%20in%20using%20this%20notation%20let

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So as an exercise in using this notation lets
look at
The indices indicate very specific matrix or
vector components/elements. These are not
matrices themselves, but just numbers, which we
can reorder as we wish. We still have to respect
the summations over repeated indices!
And remember we just showed
?(g?) g
??
i.e.
All dot products are INVARIANT under Lorentz
transformations.
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as an example, consider rotations about the z-axis
even for ROTATIONS
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The relativistic transformations
suggest a 4-vector
that also transforms by
so
should be an invariant!
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In the particles rest frame
px ?
0
p?p? ?
mc2
m2c2
E ?
In the lab frame
-?mv
E c
?
?mc
so
?
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Limitations of Schrödingers Equation
1-particle equation
2-particle equation
mutual interaction
n?pe?e np ? np3? e- p ? e- p 6? 3g
But in many high energy reactions the number of
particles is not conserved!
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Sturm-Liouville Equations

• a class of differential equations that include
• Legendre's equation
• the associated Legendre equation
• Bessel's equation
• the quantum mechanical harmonic oscillator

i.e. a class of differential eq's to which
Schrodinger's equations all belong!
whose solutions satisfy
for different eigenfunctions, ?n
0
If we adopt the following as a definition of the
"inner product"
compare this directly to the vector "dot product"
then notice we have automatically
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Recall any linear combination of simple
solutions to a differential equation
is also a solution,
and, from previous slide
eigenvalues are REAL and different
eigenfunctions are "orthogonal"
?mn
Thus the set of all possible eigenfunctions
(basic solutions) provide an "orthonormal" basis
set and any general solution to the differential
equation becomes expressible as
where
any general solution will be a function in the
"space" of all possible solutions (the solution
set) sometimes called a Hilbert Space (as
opposed to the 3-dimensional space of geometric
points.
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What does it mean to have a matrix representation
of an operator? of
Schrödingers equation?
where n represents all distinguishing quantum
numbers (e.g. n, m, l, s, )
Hmn

since
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E1 0 0 0 0 . . . H 0 E2 0 0 0
. . . 0 0 E3 0 0 . . . 0
0 0 E4 0 . . . .
1 0 0
0 0 1
0 1 0
with the basis set
...
,
,
,
This is not general at all (different electrons,
different atoms require different
matrices) Awkward because it provides no
finite-dimensional representation
Thats why its desirable to abstract the formalism
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Hydrogen Wave Functions
1 0 0 0 0
0 1 0 0 0


0 0 1 0 0


0 0 0 1 0


0 0 0 0 1




0 0 0 0 0


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But the sub-space of angular momentum (described
by just a subset of the quantum numbers) doesnt
suffer this complication.
Angular Momentum
lmsmsgt
l 0, 1, 2, 3, ... Lzlmgt
mhlmgt for m - l, - l1, l-1, l L2lmgt
l(l1)h2lmgt Szlmgt mshsmsgt for ms -s,
-s1, s-1, s S2lmgt s(s1)h2smsgt
Of course nlmgt is ? dimensional again
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Classically
can measure all the spatial (x,y,z) components
(and thus L itself) of
Quantum Mechanically
not even possible in principal !
azimuthal angle in polar coordinates
So, for example
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Angular Momentum
nlml
Measuring Lx alters Ly (the operators change the
quantum states).
The best you can hope to do is measure
l 0, 1, 2, 3, ...
L2?lm(?,?)R(r) l(l1)h2?lm(?,?)R(r)
Lz ?lm(?,?)R(r) mh ?lm(?,?)R(r) for m -l,
-l1, l-1, l
States ARE simultaneously eigenfunctions of BOTH
of THESE operators!
We can UNAMBIGUOULSY label states with BOTH
quantum numbers
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l 2 ml -2, -1, 0, 1, 2
l 1 ml -1, 0, 1
2 1 0
1 0
L2 1(2) 2 L ?2 1.4142
L2 2(3) 6 L ?6 2.4495
Note the always odd number of possible
orientations A degeneracy in otherwise
identical states!
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Spectra of the alkali metals (here Sodium) all
show lots of doublets
1924 Pauli suggested electrons posses
some new, previously un-recognized
non-classical 2-valued property
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Perhaps our working definition of angular
momentum was too literal
too classical
perhaps the operator relations
Such Commutation Rules are recognized by
mathematicians as the defining algebra of a
non-abelian (non-commuting) group
may be the more fundamental definition
Group Theory Matrix Theory
Reserving L to represent orbital angular
momentum, introducing the more generic operator
J to represent any or all angular momentum
study this as an algebraic group
Uhlenbeck Goudsmit find actually J0, ½, 1,
3/2, 2, are all allowed!
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quarks
leptons
spin
p, n, e, ?, ?, ?e , ?? , ?? , u, d, c, s, t, b
the fundamental constituents of all matter!
spin up spin down
s h 0.866 h
ms
sz h
( )
1 0
n l m gt gt ?nlm
spinor
( )
( )
( )
the most general state is a linear expansion in
this 2-dimensional basis set
? 1 0 ? 0
1
? ?
with a 2 b 2 1
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ORBITAL ANGULARMOMENTUM
SPIN
fundamental property of an individual component
relative motion between objects
Earth orbital angular momentum rmv
plus spin angular momentum I?
in fact ALSO spin angular
momentum Isun?sun
but particle spin
especially that of truly fundamental particles of
no determinable size (electrons, quarks)
or even mass (neutrinos, photons)
must be an intrinsic property of the particle
itself
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Total Angular Momentum
l 0, 1, 2, 3, ... Lzlmgt
mhlmgt for m -l, -l1, l-1, l L2lmgt
l(l1)h2lmgt Szlmgt mshsmsgt for ms -s,
-s1, s-1, s S2lmgt s(s1)h2smsgt
nlmlsmsj
In any coupling between L and S it is the TOTAL
J L s that is conserved.
Example J/? particle 2 (spin-1/2) quarks bound
in a ground (orbital angular
momentum0) state
Example spin-1/2 electron in an l2 orbital.
Total J ?
Either 3/2 or 5/2 possible
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BOSONS FERMIONS spin 1 spin ½
?
e, m
p, n,
Nuclei (combinations of p,n) can have J 1/2, 1,
3/2, 2, 5/2,
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BOSONS FERMIONS spin 0 spin ½ spin
1 spin 3/2 spin 2 spin 5/2

quarks and leptons e, m, t, u, d, c, s, t, b, n
psuedo-scalar mesons p, p-, p0, K,K-,K0
Baryon octet p, n, L
Force mediators vectorbosons g,W,Z
Baryon decupltet D, S, X, W
vector mesons r, w, f, J/y, ?
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Combining any pair of individual states j1m1gt
and j2m2gt forms the
final product state j1m1gtj2m2gt
What final state angular momenta are
possible? What is the probability of any single
one of them?
Involves measuring or calculating OVERLAPS
(ADMIXTURE contributions)
or forming the DECOMPOSITION into a new basis
set of eigenvectors.
j1j2
S
j1m1gtj2m2gt z j j1 j2m m1 m2
j m gt
j j1-j2
Clebsch-Gordon coefficients
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Matrix Representation for a selected j
J2jmgt j(j1)h2 j m gt Jzjmgt m h j m gt for
m -j, -j1, j-1, j Jjmgt ? j(j 1)-m(m1)
h j, m?1 gt
The raising/lowering operators through which we
identify the 2j1 degenerate energy states
sharing the same j.
J Jx iJy J- Jx - iJy
subtracting
adding
2Jx J J-
Jx (J J- )/2 Jy i(J- - J)/2
2iJy J - J-
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The most common representation of angular
momentum diagonalizes the Jz operator
ltjn Jz jmgt lm?mn
2 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 -1 0 0 0 0 0 -2
1 0 0 0 0 0 0 0 -1
(j1)
(j2)
Jz
Jz
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Jjmgt ? j(j 1)-m(m1) h j, m?1 gt
J- 1 1 gt
1 0 gt
lt 1 0
0 0 0 0 0 0 0
J- 1 0 gt
J-
1 -1 gt
lt 1 -1
J- 1 -1 gt
0
J 1 -1 gt
1 0 gt
lt 1 0
0 0 0 0 0 0 0
J 1 0 gt
J
1 1 gt
lt 1 1
J 1 1 gt
0
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For J1 states
a matrix representation of the angular momentum
operators
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Which you can show conform to the COMMUTATOR
relationship you demonstrated in quantum
mechanics for the differential operators of
angular momentum
Jx, Jy iJz
Jx Jy - Jy Jx
iJz
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R(?1,?2,?3)
z
z'
?1
y'
?1
y
x
x'
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R(?1,?2,?3)
z
z'
?1
z''
?2
?2
y'
y''
?1
y
x
x'
?2
x''
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R(?1,?2,?3)
z
z'
?1
z''
z'''
?2
?3
y'''
?2
?3
y'
y''
?1
y
?3
x
x'
?2
x''
x'''
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R(?1,?2,?3)
about x-axis
about y'-axis
about z''-axis
1st
2nd
3rd
These operators DO NOT COMMUTE!
Recall the generators of rotations
are angular momentum operators and
they dont commute!
but as ?n??n
Infinitesimal rotations DO commute!!