Suppose we decompose the direct product of an I = 3/2 state and I = 1 state into a direct sum of...

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Suppose we decompose the direct product of an I = 3/2 state and I = 1 state into a direct sum of three irreducible representations of SU(2). What are the three missing numbers (represented by ) in the expression below? = + - , b. If one were to measure the isospin of the composite state (the state on the right hand side), what would be the probability of getting I =1/2? c.
	
Document Preview: 1- Suppose we decompose the direct product of an I = 3/2 state and I = 1 state into a direct sum of three irreducible representations of SU(2). What are the three missing numbers (represented byx1,x2,x3 ) in the expression below?32,121,x1 =v355  2 ,x2 +v115x3,12 - v1312,12 b. If one were to measure the isospin of the composite state (the state on the right hand side), what would be the probability of getting I =1/2? c. If you were to operate on the right hand side with the total isospin ladder operator I+^tot and on the left side with I+^1+I+^2 what expression would result?2- Prove that if the interaction Hamiltonian H commutes with the isospin operator I, then a. I,I3HI,I3?=dI3I3?I,I3HI,I3 b. I,I3HI?,I3?=dI3I3?dII?I,I3HI,I3andc. I,I3HI,I3=I,I3?HI,I3? (Hint: use the Hermiticity of the Isospin-operator).

Robert · Accepted Answer

1- Suppose we decompose the direct product of an I = 3/2 state and I = 1 state into a direct sum of three irreducible 
representations of SU(2).  
a. What are the three missing numbers (represented by??1,??2,??3 ) in the expression below? 
 
 
�3
2
� , �1
2
� |1, ��??1⟩ =√�
3
5
� �5  
2
 , ??2� +√
1
15
|??3, � �
1
2
� - √1
3
�1
2
�,�1
2
� 
Solution 
On using Clebsch-Gordan expansion table we can expand the product of two isospins given as follows 
We get  
x1  = 0 
x2 = ½ 
x3 = 3/2 
 
b. If one were to measure the isospin of the composite state (the state on the right hand side), what would be the 
probability of getting I =1/2?

Suppose we decompose the direct product of an I = 3/2 state and I = 1 state into a direct sum of three irreducible representations of SU(2). What are the three missing numbers (represented by ) in the...

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