Enoncé

◦7
jDj
2j+ 1
D1/2⊗ D1/2=D0⊕ D1
D(1)
1/2|+1i,|−1i D(2) 1/2|+2i,|−2i
Htot. =D(1)
1/2⊗ D(2) 1/2
~
S1= (S1,x, S1,y, S1,z)
~
S2~ S=~ S1+~ S2~
S= (Sx, Sy, Sz)
Htot. =D(1)
1/2⊗ D(2) 1/2
~
S
|J, Mi
ˆ
Sz|J, Mi=~M|J, Mi,~ S2|J, Mi=~2J(J+ 1) |J, Mi
~
S2~ S2
1,~
S2
2S1,±S2,±S1,z S2,z
|−,−i =|J= 1, M =−1i |J= 1, M = 0i,|J= 1, M =
+1iS+
|J= 0, M = 0i ∈ DJ=0
D1/2⊗ D1/2=D0⊕ D1
Htot. =D(1)
1/2⊗ D(2) 1/2
Htot. =D(1)
1/2⊗ D(2) 1/2

Htot. =D(1)
1/2⊗
D(2)
1/2ˆ
H=A.ˆ I+B.~ S1.~ S2A, B ∈R
ˆ
H
~
S2=~ S2
1+~
S2
2+ 2~
S1.~ S2ˆ Hˆ I~ S2
ˆ
H
|J, Mi
ˆ
H
∆E
λ= 21
∆E B
9192631770 T
∆E
Htot. =D(1)
1/2⊗ D(2) 1/2
ˆ
H=A.ˆ I+B.~ S1.~ S2
A, B ∈R
Dj1⊗ Dj2
ˆ
~
S2
DJ=1 ⊗ DJ=1/2
Q
Q= +1 e Q = 0
mn= 940MeV mp= 939MeV
|pi |si
Dj=1/2|pi,|ni
T= 100K kT = 10−2eV

D1/2
ˆ
Q≡base(|pi,|ni)1 0
0 0 
|pi,|ni H1/2
|pi≡|j=1
2, m =1 2i,|ni ≡ |j=1 2, m =−1 2i,
π+, π0, π−
+1,0,−1mπ+=mπ−= 139,6MeV mπ0= 135MeV
Dj=1
|π±,0i
|π+i≡|j= 1, m = 1i,|π0i ≡ |j= 1, m = 0i,|π−i ≡ |j= 1, m =−1i.
p n π
p+π−→n+π0
N+π→N0+π0σ
π+
π−
π−
π+
π−
Energie cinétique (MeV)
200100 300
100
200
Section
efficace
(mb)
π
0
p + p +
p + n +
p + p +
σ
σ(N+π→N0+π0)ˆ
U
σ(N+π→N0+π0) = 
hN0, π0|ˆ
U|N, πi
2
|N, πi H(N,π)=Dj=1/2⊗ Dj=1
Dj0

|N, πi |J, Mi
J= 3/2 3/2 1/2 3/2 1/2 3/2
m1m2M= 3/2 1/2 1/2−1/2−1/2−3/2
1 1/2 1
1−1/2p1/3p2/3
0 1/2p2/3p1/3
0−1/2p2/3p1/3
−1 1/2p1/3−p2/3
−1−1/2 1
hj1, m1, j2, m2|J, Mi
Dj1=1 ⊗ Dj2=1/2=DJ=1/2⊕ DJ=3/2
ˆ
U
|J, MiA3/2, A1/2∈C
ˆ
U≡A3/2ˆ
I40
0A1/2ˆ I2(: H(N,π)=D3/2⊕ D1/2)
A3/2, A1/2
A3/2, A1/2
A3/2 A1/2
A3/2 A1/2 A3/2=A1/2
σpπ+→pπ+= 195 mb
σpπ−→nπ0= 45 mb
σpπ−→pπ−= 23 mb
A3/2, A1/2
∆ ∆E
h= 6.5 10−22M eV.s ∆