Are these states normalized? A) Yes

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1 QMII-. Consider two kets and their corresponding column vectors: Ψ = φ = Are these two state orthogonal? Is ψ φ = 0? A) Yes ) No Answer: A Are these states normalized? A) Yes ) No Answer: (each state has a norm of )

2 QMII-. In spin space, the basis states (eigenstates of S, S z ) are orthogonal: = 0. Are the following matrix elements zero or non-zero? S S z A) oth are zero ) Neither are zero C) The first is zero; second is non-zero D) The first is non-zero; second is zero Answer: A. Since the kets on the right are eigenstates of both operators, the eigenvalues can be pulled outside, and one is left with 0 =.

3 QMII-3. A spin ½ particle in the spin state χ = a + b. A measurement of S z is made. What is the probability that the value of S z will be +h /? A) D) S z χ ) χ C) S z E) None of these χ S z χ Answer:. Of course, this is also equal to a.

4 QMII-4. The raising operator operating on the up and down spin states: S + = h, S+ = 0 What is the matrix form of the S +? A) h ) h C) h 0 D) E) None of these. 0 h Answer:. QMII-5. Is the raising operator S + Hermitian? A) Yes, always ) No, never C) sometimes Answer:. The Hermitian conjugate of S + is S -.

5 0 a a QMII-6. Consider the matrix equation =λ 0 b b. This is equivalent to λ a 0 λ a A) = 0 λ b ) = 0 λ 0 b λ 0 a C) = 0 0 b D) λ λ λ a λ λ b = 0 E) None of these Answer: A

6 QMII-7. Suppose a spin ½ particle is in the spin state χ = = 0, the +h / eigenstate of Ŝz. Suppose we measure Sx and then immediately measure S z. What is the probability that the second measurement (S z ) will leave the particle in the S z = down state: 0 χ = =? A) zero ) non-zero Answer:. S x and S z are incompatible, so the measurement of S x will change the state to something that is not an eigenstate of S z. The new state (which is an eigenstate of S x ) will have nonzero components of both and. z z

7 QMII-8. A quantum system consists of two particles, one of spin ½, and the other with spin 3/. What is the dimension of the spin Hilbert space for this system? A) 3/4 ) 5/4 C) D) 8 E) I don t know Answer: D. Each spin Hilbert space has dimension (s+). So the tensor product space has dimension x 4 = 8.

8 QMII-9. Scandium has one electron in the 3d shell. If we measure the z- component of that electron s total angular momentum, how many possible values might we get? A) ) 5 C) 6 D) 0 E) Answer: C. This is a bit tricky. In the d shell, l=, and the electron has s=/. So the total j can be 5/ or 3/. Hence m j can be any of the six values with integer spacing between -5/ and 5/.

9 QMII-0. Consider Scandium again. If we measure S of the 3d electron, what possible values might we get? A) ± h ) h only C) E) None of these 4 3 h only D) 4 3 h or 5 h 4 Answer: C. s=/ for an electron.

10 QMII-. Consider Scandium again. If we measure J (J is the total angular momentum) of the 3d electron, what possible values might we get? A) 4 3 h ) 5 h 4 C) 4 35 h D) All of these E) and C only Answer: E. In the answer to problem 9, we said that j can be 5/ or 3/.

11 QMII-. Consider Scandium again. If we don t know anything about the outermost electron other than that it is in a 3d orbital, what is the probability that a measurement of J will produce the result 4 5 h? A) / ) /5 C) 3/7 D) /3 E) Impossible to compute without table of Clebsch-Gordan coefficients. Answer:. There are 4 states with j=3/, and 6 states with j=5/. If they are all equally probable, then the answer is 4/(4+6) = /5.

12 QMII-3. Which of the following two-electron quantum states satisfies the requirements of the Spin-Statistics Theorem? (For this problem, my m s notation for the spin states will be χ s rather than s, ms. A) () () 0 A χ0 () () () () χ A A 0 () () () () χ A + A 0 () () () () χ A + A 0 ) ( ) 0 C) ( ) 0 D) ( ) E) oth and D Answer: C. Electrons are Fermions, hence the state must be antisymmetric under exchange of the two particles. The s=0 spin state is antisymmetric under exchange, while the s= spin state is symmetric.

13 QMII-4. Consider two identical spin- particles. We want to find r r r eigenstates of the total spin S = S + S. Which one of the following statements is correct? (I have omitted all tensor product symbols.) () () () () A) s =, m = = ( ),,0 +,0, s () () () () ) s =, m = = ( ),,0,0, s () () () () C) s =, m = = ( ), 0,0 + 0,0, s () () () () D) s =, m = = ( ),,,, s E) Help, I need my Table of Clebsch-Gordan coefficients! Answer:. You can rule out C because one particle there has spin 0. You can rule out D because it would give m s =0. The state in A is actually s =, ms =.

14 QMII-5. Consider two identical spin- bosons. Which of the following two-electron quantum states satisfies the requirements of the Spin- Statistics Theorem? (For this problem, my notation for the spin states will be m χ s s rather than ms s,. () () () () A) ( ) χ A + A () () () () ) ( ) χ A A () () () () 0 C) ( ) χ A + A () () () () 0 D) ( ) χ A E) oth A and D A Answer: E. Since the particles are bosons, the state must be symmetric under exchange of the two particles. The states in the highest ladder (s= for this problem) are always symmetric under exchange. From the previous problem, you know that the states in the s= ladder are antisymmetric under exchange. All the states in a given spin ladder have the same symmetry under exchange.