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QM: Expectation value of raising and lowering operator

  1. Jun 26, 2015 #1
    1. The problem statement, all variables and given/known data
    Using
    [tex]
    J^2 \mid j,m_z \rangle = h^2 j(j+1) \mid j,m_z \rangle
    [/tex]
    [tex]
    J_z \mid j,m_z \rangle = hm_z \mid j,m_z \rangle
    [/tex]

    Derive that :
    [tex]
    \langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = h^2[ j(j+1) - m_z(m_z+1)]
    [/tex]

    2. Relevant equations
    [tex]
    J_- = J_x - iJ_y
    [/tex]
    [tex]
    J_+ = J_x + iJ_y
    [/tex]
    3. The attempt at a solution
    [tex]
    J_-J_+ = (J_x- iJ_y)(J_x + iJ_y) = J_x^2 + J_y^2 = J^2 - J_z^2
    [/tex]

    [tex]
    \langle j,m_z \mid J_-J_+ \mid j,m_z \rangle = \langle j,m_z \mid J^2 - J_z^2 \mid j,m_z \rangle = h^2[ j(j+1) - m_z^2]
    [/tex]

    Apparently I am missing a term here but I dont know where it should come from. I thought this should be true
    [tex]
    J_z^2 \mid j,m_z \rangle = J_zJ_z \mid j,m_z \rangle = h^2m_z^2 \mid j,m_z \rangle
    [/tex]
    (Note: h is hbar everywhere )
     
  2. jcsd
  3. Jun 26, 2015 #2

    Orodruin

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    You are assuming that ##J_x## commutes with ##J_y## when computing ##J_-J_+##. This is not the case.
     
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