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Homework Help: Stupid noob question about state vectors

  1. Aug 6, 2010 #1
    1. The problem statement, all variables and given/known data
    I got a Problem that says:
    In this Problem we consider the angular momentum, [tex]\hat{L}[/tex], which normed common eigenstates for [tex]\hat{L}^{2}[/tex] and [tex]\hat{L}_{z}[/tex] is described by [itex]\left|l,m\right\rangle[/itex], where [itex]l[/itex] and [itex]m[/itex] is the corresponding quantum numbers:

    [tex]\hat{L}^{2}\left|l,m\right\rangle = l\left(l+1\right)\hbar^{2}\left|l,m\right\rangle[/tex]

    [tex]\hat{L}_{z}\left|l,m\right\rangle = m\hbar\left|l,m\right\rangle[/tex]

    As usual we introduce the operators:

    [tex]\hat{L}_{+} = \hat{L}_{x} + i\hat{L}_{y}[/tex] and [tex]\hat{L}_{-} = \hat{L}_{x} - i\hat{L}_{y}[/tex]

    Now consider the three-dimensional subspace, [itex]\mathcal{H}[/itex], corresponding to the quantum number [itex]l = 1[/itex], ie. the states [itex]\left|1,-1\right\rangle[/itex], [itex]\left|1,0\right\rangle[/itex] and [itex]\left|1,1\right\rangle[/itex].

    Now, express the operators [tex]\hat{L}^{2}[/tex] and [tex]\hat{L}_{z}[/tex] as matrices in [itex]\mathcal{H}[/itex].


    2. Relevant equations



    3. The attempt at a solution

    Well, my problem is that I'm not sure how to interpret the states [itex]\left|1,-1\right\rangle[/itex], [itex]\left|1,0\right\rangle[/itex] and [itex]\left|1,1\right\rangle[/itex]. Usually when there are fx. 1 in one of them, it's the first basisvector etc. But what is it now ? Is it still a vector, in which case what kind ?
    I've been flipping through my entire book to see if it showed anything similar, but I haven't been able to find anything that could tell me how I have to interpret the states "vectorwise".
    So I was hoping someone could help me :)

    I know it's kinda noobish, but I have been stuck here for a while, and can't seem to figure out what the matrix is before I know how to interpret the states.


    Regards.
     
    Last edited: Aug 6, 2010
  2. jcsd
  3. Aug 7, 2010 #2

    kuruman

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    Interpret them as three basis vectors spanning a three-dimensional Hilbert space, much like i, j and k span three-dimensional Euclidean space. You can order them any way you please, but conventionally the order is |1,1>, |1,0>, |1,-1>.

    I don't know if I have answered your question, but I don't understand the source of your confusion and exactly what you are asking. If you can explain exactly what troubles you, I might be able to give you a better answer.
     
  4. Aug 8, 2010 #3
    Hmmm, I mean, if it was just |1> I know it would be a basis vector in three dimensions (This case at least), which I would write: (1,0,0).
    But how does this look like in vector-form, that's my question and problem :)
     
  5. Aug 8, 2010 #4

    kuruman

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    OK. Note that the basis states are written in the form |L,ML> where L = 1 for all three states. You can toss that out since it is the same for everybody and write the states as
    |1>, |0>, |-1>. So

    |1> = (1,0,0)
    |0> = (0,1,0)
    |-1> = (0,0,1)

    :wink:
     
  6. Aug 9, 2010 #5
    Ahhh, I see :)

    But what if L = 2, how would that look fx. ?
     
  7. Aug 9, 2010 #6

    kuruman

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    The dimensionality of the Hilbert space is n = 2*L+1. For L =2, therefore, you are in 5-dimensional Hilbert space. If you denote the basis vectors just by their ML value, you get column vectors
    |2> = (1,0,0,0,0)
    |1> = (0,1,0,0,0)
    |0> = (0,0,1,0,0)
    ....
    and so on.
     
  8. Aug 9, 2010 #7
    Great :)

    Thank you very much kuruman.
     
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