1. Limited time only! Sign up for a free 30min personal 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!

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.

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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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)

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

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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook