Particle%20Physics%20Option

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Last Time
Spin Algebra for a spin operator J Isospin
operator I follows this same algebra Isospin is
also additive. Two particles with Isospin Ia and
Ib will give a total Isospin I Ia Ib
By defining I I1 iI2 and I- I1 - iI2 we
could Raise and Lower the third component of
isospin I-i,mgt i(i1)-m(m-1)1/2i,m-1gt
Ii,mgt i(i1)-m(m1)1/2i,m1gt NOTICE
I1/2,-1/2gt Idgt ugt (or -d-bargt) All
part of what we called SU(2)
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Much Ado about
Isospin

• Concept Developed Before the Quark Model
• Only works because M(up) ? M(down)
• Useful concept in strong interactions only
• Often encountered in Nuclear physics
• From SU(2), there is one key quantum number I3

Up quark ? Isospin 1/2 I3 1/2 Anti-up quark
? I 1/2 I3 -1/2
Down quark ? I 1/2 I3 -1/2 Anti-down
quark ? I 1/2 I3 1/2
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Graphical Method of finding all the possible
combinations
1). Take the Number of possible states each
particle can have and multiply them. This is
the total number you must have in the end. A
spin 1/2 particle can have 2 states, IF we are
combining two particles 2 X 2 4 total in
the end.
I3
1
1/2
0
-1/2
2) Plot the particles as a function of the I3
quantum numbers.
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Graphical Method of finding all the possible
combinations
Triplet
Singlet
Group B
Group A
I3
I3
I3
Sum
1
1
1
1/2
1/2
1/2
0
0
0
-1/2
-1/2
-1/2
-1
-1
-1
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Graphical Method of finding all the possible
combinations
We have just combined two fundamental
representations of spin 1/2, which is the
doublet, into a higher dimensional
representation consisting of a group of 3
(triplet) and another object, the singlet.
What did we just do as far as the spins are
concerned?
Quantum states Triplet I I, I3gt 1,1gt
1/2,1/2gt1 1/2,1/2gt2 1,0gt 1/?2 (1/2,1/2gt1
1/2,-1/2gt2 1/2,-1/2gt1 1/2,1/2gt2 ) 1,-1gt
1/2,-1/2gt1 1/2,-1/2gt2 Singlet 0,0gt 1/ ?
2 (1/2,1/2gt1 1/2,-1/2gt2 - 1/2,-1/2gt1
1/2,1/2gt2)
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Reminder u 1/2,1/2gt u-bar or d
1/2,-1/2gt
Quantum states Triplet I 1 I, I3gt 1,1gt
1/2,1/2gt1 1/2,1/2gt2 -udgt 1,0gt
1/2(1/2,1/2gt1 1/2,-1/2gt2 1/2,-1/2gt1
1/2,1/2gt2 ) 1/2(uugt - ddgt)
1,-1gt 1/2,-1/2gt1 1/2,-1/2gt2
udgt Singlet 0,0gt1/2(1/2,1/2gt1 1/2,-1/2gt2
- 1/2,-1/2gt1 1/2,1/2gt2 1/2(uugt
ddgt)
Must choose either quark-antiquark states, or q-q
states. We look for triplets with similar masses.
MESONs fit the bill! p,p0,p- and ?, ?0, ?-
(q-qbar pairs). ?0, ?0, and ?0 are
singlets. WARNING Ask about 1,0gt minus sign or
read Burcham Jobes pgs. 361 and 718
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But quarks are also in groups of 3 so wed like
to see that structure too
I3
3/2
I3
I3
1
s
a
1
1
1/2
1/2
1/2
0
0
0
-1/2
-1/2
-1/2
-1
-1
-1
-3/2
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Isospins of a few baryon and meson states