(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Stupid noob question about state vectors

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