1. QM1 Concept Test 9.1
Consider the following statements for the product space of a two spin-1/2 system: 𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 cannot be written as a diagonal matrix in the uncoupled representation because
𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 does not commute with the operators 𝑆 1𝑧 and 𝑆 2𝑧 whose eigenstates are the basis vectors in the uncoupled representation.
𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 is the Hamiltonian operator which can never be diagonal no matter what basis you choose.
We are dealing with a two spin-1/2 system 𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 would be diagonal if we had a two spin-one system.
(I) only b) (II) only c) (III) only d) (I) and (III) only
e) none of the above

2. QM1 Concept Test 9.2
Choose all of the following statements that are correct about 𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 for the product space of a two spin-1/2 system.
𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 is an off-diagonal matrix if the basis vectors are the simultaneous eigenstates of 𝑆 1 2 , 𝑆 1𝑧 , 𝑆 2 2 , and 𝑆 2𝑧 .
It is possible to put 𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 into a diagonal matrix form in a suitable basis but the basis vectors will not be the eigenstates of 𝑆 1 2 , 𝑆 1𝑧 , 𝑆 2 2 , and 𝑆 2𝑧 .
The basis vectors can be chosen to be simultaneous eigenstates of 𝑆 1𝑥 and 𝑆 1𝑧 .
(I) only b) (II) only c) (I) and (II) only
d) (I) and (III) only e) All of the above.

3. QM1 Concept test 9.3
Consider the product space of two spin-1/2 systems. The raising and lowering operators for each spin, e.g., for the first spin are given as 𝑆 1+ = 𝑆 1𝑥 +𝑖 𝑆 1𝑦 and 𝑆 1− = 𝑆 1𝑥 −𝑖 𝑆 1𝑦 . Choose all of the following expressions that are correct for 𝑆 1 ∙ 𝑆 These expressions will be helpful in writing 𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 in the uncoupled or coupled representation.
𝑆 1 ∙ 𝑆 2 = 𝑆 1𝑥 𝑆 2𝑥 + 𝑆 1𝑦 𝑆 2𝑦 + 𝑆 1𝑧 𝑆 2𝑧
𝑆 1 ∙ 𝑆 2 = 𝑆 1− 𝑆 𝑆 𝑆 2− 𝑆 1𝑧 𝑆 2𝑧
𝑆 1 ∙ 𝑆 2 = 𝑆 2 − 𝑆 1 2 − 𝑆
(I) only b) (I) and (II) only c) (I) and (III) only
d) (II) and (III) only e) All of the above

4. QM1 Concept test 9.4
Consider the product space of two spin-1/2 systems. The raising and lowering operators for each spin, e.g., for the first spin are given as 𝑆 1+ = 𝑆 1𝑥 +𝑖 𝑆 1𝑦 and 𝑆 1− = 𝑆 1𝑥 −𝑖 𝑆 1𝑦 . Choose all of the following expressions for 𝑆 1 ∙ 𝑆 2 that will help us recognize that 𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 is diagonal in the uncoupled representation.
𝑆 1 ∙ 𝑆 2 = 𝑆 1𝑥 𝑆 2𝑥 + 𝑆 1𝑦 𝑆 2𝑦 + 𝑆 1𝑧 𝑆 2𝑧
𝑆 1 ∙ 𝑆 2 = 𝑆 1− 𝑆 𝑆 𝑆 2− 𝑆 1𝑧 𝑆 2𝑧
𝑆 1 ∙ 𝑆 2 = 𝑆 2 − 𝑆 1 2 − 𝑆
(I) only b) (II) only c) (III) only d) Both (I) and (II) e) None of the above

5. QM1 Concept test 9.5
Consider the product space of two spin-1/2 systems. The raising and lowering operators for each spin, e.g., for the first spin are given as 𝑆 1+ = 𝑆 1𝑥 +𝑖 𝑆 1𝑦 and 𝑆 1− = 𝑆 1𝑥 −𝑖 𝑆 1𝑦 . Choose all of the following expressions for 𝑆 1 ∙ 𝑆 2 that will help us recognize that 𝐻 =𝐶 𝑆 1 ∙ 𝑆 2 is diagonal in the coupled representation.
𝑆 1 ∙ 𝑆 2 = 𝑆 1𝑥 𝑆 2𝑥 + 𝑆 1𝑦 𝑆 2𝑦 + 𝑆 1𝑧 𝑆 2𝑧
𝑆 1 ∙ 𝑆 2 = 𝑆 1− 𝑆 𝑆 𝑆 2− 𝑆 1𝑧 𝑆 2𝑧
𝑆 1 ∙ 𝑆 2 = 𝑆 2 − 𝑆 1 2 − 𝑆
(I) only b) (II) only c) (III) only d) (I) and (II) only (e) None of the above.

6. QM1 Concept test 9.6
Define the raising and lowering operators for a single spin-1/2 system as 𝑆 + = 𝑆 𝑥 +𝑖 𝑆 𝑦 and 𝑆 − = 𝑆 𝑥 −𝑖 𝑆 𝑦 . Which one of the following gives the correct values for 𝑆 𝑥 and 𝑆 𝑦 ?
𝑆 𝑥 = ( 𝑆 𝑆 − )/2, 𝑆 𝑦 = ( 𝑆 + − 𝑆 − )/2i
𝑆 𝑥 = ( 𝑆 + − 𝑆 − )/2, 𝑆 𝑦 = ( 𝑆 + − 𝑆 − )/2i
𝑆 𝑥 = ( 𝑆 𝑆 − )/2, 𝑆 𝑦 = ( 𝑆 𝑆 − )/2i
𝑆 𝑥 = ( 𝑆 + − 𝑆 − )/2, 𝑆 𝑦 = ( 𝑆 𝑆 − )/2i
None of the above

7. QM1 Concept test 9.7
Consider the product space of two spin-1/2 systems. Which one of the following is correct?
𝑆 1− 𝑆 ↑ ↑ 2 =( 𝑆 1− ↑ 1 )( 𝑆 ↑ 2 )=0
𝑆 1− 𝑆 ↑ ↑ 2 =( 𝑆 1− ↑ 1 )( 𝑆 ↑ 2 )= ℏ 2 ↑ ↑ 2
𝑆 1− 𝑆 ↑ ↑ 2 =( 𝑆 1− ↑ 1 )( 𝑆 ↑ 2 )= 2ℏ 2 ↑ ↑ 2
𝑆 1− 𝑆 ↑ ↑ 2 =( 𝑆 1− ↑ 1 )( 𝑆 ↑ 2 )= ℏ 2 ↓ ↑ 2
None of the above

8. QM1 Concept test 9.8
Consider the product space of two spin-1/2 systems. Which one of the following is correct?
𝑆 1− 𝑆 ↓ ↓ 2 =( 𝑆 1− ↓ 1 )( 𝑆 ↓ 2 )=0
𝑆 1− 𝑆 ↓ ↓ 2 =( 𝑆 1− ↓ 1 )( 𝑆 ↓ 2 )= ℏ 2 ↓ ↓ 2
𝑆 1− 𝑆 ↓ ↓ 2 =( 𝑆 1− ↓ 1 )( 𝑆 ↓ 2 )= ℏ 2 ↓ ↑ 2
𝑆 1− 𝑆 ↑ ↑ 2 =( 𝑆 1− ↑ 1 )( 𝑆 ↑ 2 )= 2ℏ 2 ↓ ↓ 2
None of the above

9. QM1 Concept test 9.9
Consider the product space of two spin-1/2 systems. Which one of the following is correct?
𝑆 1− 𝑆 ↑ ↓ 2 =( 𝑆 1− ↑ 1 )( 𝑆 ↓ 2 )=0
𝑆 1− 𝑆 ↑ ↓ 2 =( 𝑆 1− ↑ 1 )( 𝑆 ↓ 2 )= ℏ 2 ↓ ↑ 2
𝑆 1− 𝑆 ↑ ↓ 2 =( 𝑆 1− ↑ 1 )( 𝑆 ↓ 2 )= 2ℏ 2 ↓ ↑ 2
𝑆 1− 𝑆 ↑ ↓ 2 =( 𝑆 1− ↑ 1 )( 𝑆 ↓ 2 )= 2ℏ 2 ↑ ↓ 2
None of the above

10. QM1 Concept test 9.10
Consider the product space of two spin-1/2 systems. Which one of the following scalar products is correct?
↓ ↑ 2 ( 𝑆 1− 𝑆 2+ ) ↑ ↓ 2 = ( ↓ 𝑆 1− ↑ 1 )( ↑ 𝑆 ↓ 2 )=0
↓ ↑ 2 ( 𝑆 1− 𝑆 ) ↑ ↓ 2 = ( ↓ 𝑆 1− ↑ 1 )( ↑ 𝑆 ↓ 2 )= ℏ 2 ↓ ↑ 2
↓ ↑ 2 ( 𝑆 1− 𝑆 2+ ) ↑ ↓ 2 = ( ↓ 𝑆 1− ↑ 1 )( ↑ 𝑆 ↓ 2 )= ℏ 2
↓ ↑ 2 ( 𝑆 1− 𝑆 2+ ) ↑ ↓ 2 = ( ↓ 𝑆 1− ↑ 1 )( ↑ 𝑆 ↓ 2 )= 2ℏ 2
None of the above.

11. QM1 Concept Test 9.11
Consider the product space of two spin-1/2 systems. Which one of the following scalar products is correct?
↑ ↓ 2 ( 𝑆 1− 𝑆 2+ ) ↑ ↓ 2 =0
↑ ↓ 2 ( 𝑆 1− 𝑆 ) ↑ ↓ 2 = ℏ 2
↑ ↓ 2 ( 𝑆 1− 𝑆 ) ↑ ↓ 2 = ℏ 2 ↑ ↓ 2
↑ ↓ 2 ( 𝑆 1− 𝑆 2+ ) ↑ ↓ 2 = 2ℏ 2
None of the above.