spin algebra for a spin operator ‘j’: ‘isospin operator ‘i’ follows this same algebra...
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Spin Algebra for a spin operator ‘J’:
‘Isospin operator ‘I’ follows this same algebraIsospin is also additive. Two particles with Isospin Ia and Ib will give a total Isospin I = Ia + Ib
Last Time:
By defining I+ = I1 + iI2 and I- = I1 - iI2 we could ‘Raise’ and ‘Lower’ the third component of isospin:
I-|i,m> = [i(i+1)-m(m-1)]1/2|i,m-1>I+|i,m> = [i(i+1)-m(m+1)]1/2|i,m+1>
NOTICE: I+|1/2,-1/2> = I+|d> = |u> (or -|d-bar>)
All part of what we called SU(2)
kijkji JiJJ ,
• 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 = 1/2Anti-up quark I = 1/2; I3 = -1/2
Down quark I = 1/2; I3 = -1/2Anti-down quark I = 1/2; I3 = 1/2
I3
0
1/2
-1/2
-1
1
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.
2) Plot the particles as a function of the I3 quantum numbers.
I3
0
1/2
-1/2
-1
1
Graphical Method of finding all the possible combinations:
I3
0
1/2
-1/2
-1
1
I3
0
1/2
-1/2
-1
1
Triplet
SingletGroup A Group B
Sum
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 anotherobject, the singlet.
What did we just do as far as the spins are concerned?
Quantum states: Triplet I = |I, I3> |1,1> = |1/2,1/2>1 |1/2,1/2>2 |1,0> = 1/2 (|1/2,1/2>1 |1/2,-1/2>2 + |1/2,-1/2>1 |1/2,1/2>2 )|1,-1> = |1/2,-1/2>1 |1/2,-1/2>2
Singlet|0,0> = 1/ 2 (|1/2,1/2>1 |1/2,-1/2>2 - |1/2,-1/2>1 |1/2,1/2>2)
Quantum states: Triplet I = 1 |I, I3> |1,1> = |1/2,1/2>1 |1/2,1/2>2 = -|ud>
|1,0> = 1/2(|1/2,1/2>1 |1/2,-1/2>2 + |1/2,-1/2>1 |1/2,1/2>2 ) = 1/2(|uu> - |dd>)
|1,-1>= |1/2,-1/2>1 |1/2,-1/2>2 = |ud>
Singlet |0,0>=1/2(|1/2,1/2>1 |1/2,-1/2>2 - |1/2,-1/2>1 |1/2,1/2>2
=1/2(|uu> + |dd>)
Reminder: u = |1/2,1/2> u-bar or d = |1/2,-1/2>
Must choose either quark-antiquark states, or q-q states. We look for triplets with similar masses. MESONs fit the bill! +,0,- and +, 0, - (q-qbar pairs). 0, 0, and 0 are singlets.WARNING: Ask about |1,0> minus sign or read Burcham & Jobes pgs. 361 and 718
2
1,
2
1d
00
But quarks are also in groups of 3so we’d like to see that structuretoo:
I3
0
1/2
-1/2
-1
1
I3
0
1/2
-1/2
-1
1
I3
0
1/2
-1/2
-1
1
3/2
-3/2
sa
Isospins of a few baryon and meson states:
2
3,
2
3
2
1,
2
3
2
1,
2
3
2
3,
2
3
0
2
1,
2
1
2
1,
2
1
n
p
2
1,
2
1
2
1,
2
1
n
p
1,1
0,1
1,10
2
1,
2
1
2
1,
2
10
1,1
0,1
1,10
0,0000
2
1,
2
1
2
1,
2
1
2
1,
2
1
2
1,
2
1
0
0
K
K
K
K