1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Obtain the matrix representations of J-hat (subscript y) for the state with j=1 in te

  1. Nov 19, 2011 #1
    1. The problem statement, all variables and given/known data

    The raising and lowering angular momentum operators, J-hat(subscript +), J-hat(subscript -) are defined in terms of the Cartesian components J-hat(subscript x), J-hat(subscript y), J-hat(subscript z) of angular momentum J-hat by J-hat(+)=J-hat(x)+iJ-hat(y) and J-hat(-)=J-hat(x)-iJ-hat(y).

    Obtain the matrix representation of J(subscript y) for the state with j=1 in terms of the set of eigenstates of J-hat(subscript z).

    3. The attempt at a solution

    J(subscript y)=(-i/2) (0 sqrt 2 0)
    (-sqrt 2 0 sqrt 2)
    (0 -sqrt 2 0)

    I don't know why though. And what does it mean why 'in terms of the set of eigenstates J-hat(z)?
     
  2. jcsd
  3. Nov 19, 2011 #2
    Re: Obtain the matrix representations of J-hat (subscript y) for the state with j=1 i

    I had trouble typing out the matrix properly

    it is supposed to be:
    (0 sqrt2 0)
    (-sqrt2 0 sqrt2)
    (0 -sqrt2 0)
     
  4. Nov 19, 2011 #3
    Re: Obtain the matrix representations of J-hat (subscript y) for the state with j=1 i

    its the 'in terms of the set of eigenstates of J-z' that confuse me. If it wasn't for those words at the end I might almost be able to do the question. So what do I do? Find the matrix representation of J (y) normally, and then operate with J(y) onto J(z) and then find its eigenvalue, and those eigenvalues are the eigenstates? I tried to operate with J(y) onto J(z) but got nowhere. I feel like I have no idea what I am doing. Please help.
     
  5. Nov 20, 2011 #4
    Re: Obtain the matrix representations of J-hat (subscript y) for the state with j=1 i

    I copied this somewhere from the internet:

    the ordered basis is:
    |1 1>, |1 0>,|1,-1>

    The matrix representation of the operator J(z) in the ordered basis is:

    J(z)=
    <1,1|J z|1,1> <1,1|J z|1,0> <1,1|J z|1,-1>
    <1,0|J z|1,1> <1,0|J z|1,0> <1,0|J z|1,-1>
    <1,-1|J z|1,1> <1,-1|J z|1,0> <1,-1|J z|1,-1>

    since the basis sets are eigenstates of J z
    J z=
    1 0 0
    0 0 0
    0 0 -1

    but how did they get:
    1 0 0
    0 0 0
    0 0 -1??

    please help
     
  6. Nov 20, 2011 #5
    Re: Obtain the matrix representations of J-hat (subscript y) for the state with j=1 i

    The basis (the possible values for m) for j=1 are m=-1,0,1. Write the corresponding states in vector representation as
    [itex]
    m=-1: \hspace{2cm} \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix} \\
    m=0: \hspace{2cm} \begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix} \\
    m=1: \hspace{2cm} \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix}
    [/itex]
    The Jz operator is diagonal in this representation and has these as eigenvectors. The diagonal elements will simply be the possible m-values (that is, m=-1,0,1), giving the matrix you asked about.

    In general, for spin j, the diagonal elements of Jz are the possible m-vales, going from m=j to m=-j in integer steps.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook