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4) A Spin-3/2 Particle (25 marks): Consider a particle with spin angular momentum j = 3/2. There are four sublevels with this value of j, but different eigenvalues of J_z, |m = 3/2>, |m = 1/2>, |m = -1/2>, |m = -3/2>. a) Show that the raising operator J_+ in this 4-dimensional space is J_+ = h(sqrt(3)|3/2><1/2| + 2|1/2><-1/2| + sqrt(3)|-1/2><-3/2|) b) What is the lowering operator J_-? c) What are the matrix representations of J_+, J_-, J_x, J_y, J_z and J^2 in the J_z basis? State and show which of these operators are Hermitian. d) Check that the state |psi> = 1/(2*sqrt(2))(sqrt(3)|3/2> + |1/2> - |-1/2> - sqrt(3)|-3/2>) is an eigenstate of J_x with eigenvalue h/2.

          4) A Spin-3/2 Particle (25 marks): Consider a particle with spin angular momentum j = 3/2. There are four sublevels with this value of j, but different eigenvalues of J_z, |m = 3/2>, |m = 1/2>, |m = -1/2>, |m = -3/2>.

a) Show that the raising operator J_+ in this 4-dimensional space is

J_+ = h(sqrt(3)|3/2><1/2| + 2|1/2><-1/2| + sqrt(3)|-1/2><-3/2|)

b) What is the lowering operator J_-?

c) What are the matrix representations of J_+, J_-, J_x, J_y, J_z and J^2 in the J_z basis? State and show which of these operators are Hermitian.

d) Check that the state

|psi> = 1/(2*sqrt(2))(sqrt(3)|3/2> + |1/2> - |-1/2> - sqrt(3)|-3/2>)

is an eigenstate of J_x with eigenvalue h/2.

4) A Spin-3/2 Particle (25 marks): Consider a particle with spin angular momentum j = 3/2. There are four sublevels with this value of j, but different eigenvalues of Jz, |m = 3/2>, |m = 1/2>, |m = -1/2>, |m = -3/2>.

a) Show that the raising operator J+ in this 4-dimensional space is

J+ = h(sqrt(3)|3/2><1/2| + 2|1/2><-1/2| + sqrt(3)|-1/2><-3/2|)

b) What is the lowering operator J-?

c) What are the matrix representations of J+, J-, Jx, Jy, Jz and J^2 in the Jz basis? State and show which of these operators are Hermitian.

d) Check that the state

|psi> = 1/(2*sqrt(2))(sqrt(3)|3/2> + |1/2> - |-1/2> - sqrt(3)|-3/2>)

is an eigenstate of Jx with eigenvalue h/2.

Added by Sandra S.

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