Problem Set 3

Problem 5

The eigenvectors for the 3/2 state:

(eigenvectors[1]  = {1, 0, 0, 0}) // MatrixForm

(eigenvectors[2]  = {0, 1, 0, 0}) // MatrixForm

(eigenvectors[3]  = {0, 0, 1, 0}) // MatrixForm

(eigenvectors[4]  = {0, 0, 0, 1}) // MatrixForm

Raising operator:

(Iplus  = {{0, Sqrt[3], 0, 0} , {0, 0, 2, 0}, {0, 0, 0, 
              Sqrt[3]}, {0, 0, 0, 0}}) // MatrixForm

Check with one eigenvector:

Iplus . eigenvectors[3] // MatrixForm

Lowering Operator:

(Iminus = Transpose[Iplus]) // MatrixForm

I_z operator:

(Iz = {{3/2, 0, 0, 0}, {0, 1/2, 0, 0}, {0, 0, -1/2, 0}, {0, 0, 
              0, -3/2}}) // MatrixForm

I_x operator:

(Ix = (Iplus + Iminus)/2) // MatrixForm

I_y operator:

(Iy = (Iplus - Iminus)/2/I) // MatrixForm

I^2 operator:

(Isquare  =  Ix . Ix + Iy . Iy + Iz . Iz) // 
      MatrixForm