Reminder: Spin Algebra for a spin operator ‘J’:

[ , ji ]= ε hJiJJ kijk ‘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: 1/2 I-|i,m> = [i(i+1)-m(m-1)] |i,m-1> 1/2 I+|i,m> = [i(i+1)-m(m+1)] |i,m+1>

NOTICE: I+|1/2,-1/2> = I+|d> = |u> (or -|d-bar>)

All part of what we called SU(2) • 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 ô Isospin = 1/2; I3 = 1/2 Anti-up quark ô I = 1/2; I3 = -1/2

Down quark ô I = 1/2; I3 = -1/2 Anti-down quark ô I = 1/2; I3 = 1/2 Graphical Method of finding all the possible combinations:

1). Take the Number of possible states each I3 particle can have and multiply them. This is the total number you must have in the end. A spin 1/2 particle can have 1 2 states, IF we are combining two particles: 1/2 2 X 2 = 4 total in the end. 0

-1/2 2) Plot the particles as a function of the -1 I3 quantum numbers. Graphical Method of finding all the possible combinations: Triplet Singlet Group A Group B I I3 3 Sum I3

1 1 1

1/2 1/2 1/2

0 0 0

-1/2 -1/2 -1/2

-1 -1 -1 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, 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) 1 1 d −= , Reminder: u = |1/2,1/2> 2 2 u-bar or d = |1/2,-1/2>

Quantum states: Triplet I = 1 |I, I3> |1,1> = |1/2,1/2>1 |1/2,1/2>2 = -|ud> ⎡π + ⎤ ⎡ρ + ⎤ |1,0> = 1/2(|1/2,1/2> |1/2,-1/2> + |1/2,-1/2> |1/2,1/2> ) ⎢ 0 ⎥ ⎢ 0 ⎥ 1 2 1 2 ⎢π ⎥ ⎢ρ ⎥ = 1/2(|uu> - |dd>) ⎢ − ⎥ ⎢ − ⎥ ⎣π ⎦ ⎣ρ ⎦ |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>)

Must choose either quark-antiquark states, or q-q states. We look for triplets with similar masses. MESONs fit the bill! π+,π0,π- and ρ+, ρ0, ρ- (q-qbar pairs). w0, f0, and h0 are singlets. WARNING: Ask about |1,0> minus sign or read Burcham & Jobes pgs. 361 and 718 But quarks are also in groups of 3 so we’d like to see that structure I3 too: 3/2 I3 I3 1 a s 1 1 1/2

1/2 1/2 0

0 0 -1/2

-1/2 -1/2 -1

-1 -1 -3/2 Isospins of a few baryon and meson states:

++ 3 3 1 1 =Δ , p = , + =Σ 1,1 1 1 2 2 2 2 0 =Ξ , 0 2 2 + 3 1 1 1 =Σ 0,1 =Δ , n ,−= − 1 1 2 2 2 2 − ,−=Ξ −=Σ 1,1 2 2 0 3 1 ,−=Δ 1 1 2 2 p ,−−= 1 1 2 2 K + = , − 3 3 2 2 ,−=Δ 1 1 2 2 n = , 1 1 2 2 K 0 ,−= 2 2 + π = 1,1 1 1 K 0 −= , π 0 = 0,1 = ′ φηη000 == 0,0 2 2 1 1 π − −= 1,1 K − ,−= 2 2