Explore BrainMass
Share

Explore BrainMass

    Matrix Representation and Operators

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    If the general angular momentum quantum number j is 1 there is a triplet of |j,m_j> states:
    |1 ,1>, |1,0> and |1,-1>
    In this case a matrix representation for the operators j_x j_y and j_z, can be constructed if we represent the |j,m_j> triplet by three component column vectors as follows
    |1,1> =(■(1@0@0)) |1,0> =(■(0@1@0)) |1,1> =(■(0@0@1)) (1)

    j_z can then be represented by the matrix

    j_z=(■(1&0&0@0&0&0@0&0&-1))
    Construct matrix representations for the raising and lowering operators,〖 j〗_+ and j_- acting on the eigenstates |1 ,1>,|1,0> and |1,-1> in the representation given in equation (1)
    Use the relationships

    j_x=1/2 (j_++j_- ) j_y=1/2i (j_+-j_- )

    To construct matrix representations of j_x,j_y and j_z
    Show that the matrix representations of j_x,j_y and j_z obey the commutation relation

    [j_x,j_y ]=iћj_z

    © BrainMass Inc. brainmass.com October 10, 2019, 3:02 am ad1c9bdddf
    https://brainmass.com/physics/angular-momentum/matrix-representation-operators-404437

    Attachments

    Solution Preview

    Hello and thank you for posting your question to Brainmass!
    The solution is attached below in two files. the files are identical in content, only differ in format. The first is in ...

    Solution Summary

    The expert examines matrix representations and operators.

    $2.19