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I'm given with the following problem - it's an easy one, but it's nearly 1AM, I'm tired and I need some push into the general direction to get the solution:

A particle is assumed to be in the state

$\left(-\sqrt{1 \over 3} Y_1^0(\theta, \phi), -\sqrt{2 \over 3} Y_1^1(\theta, \phi)\right)^T$

$Y$ are spherical harmonics. What are the

  • total angular momentum $\vec{j}^2$
  • total orbital angular momentum $\vec{l}^2$
  • total spin angular momentum $\vec{s}^2$
  • total angular momentum $j_z$

of the particle

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1  
Looking at your state, I'm totally confused. What the deuce does the "T" superscript refer to, and why is that comma in there? Also, the "state" given tells us nothing about the spin state of the particle - and you haven't told us what kind of particle it is - so the majority of your questions are unanswerable. –  Anonymous Coward Feb 18 '11 at 18:55
    
In the original homework the state was given as a column vector, however I was to lazy to do a TeX matrix for that, so I choose to write it as a row vector and transpose that one. –  datenwolf Feb 18 '11 at 19:59
    
So what does the column matrix refer to? Is the problem about a spin-1/2 particle, and that's the spinor state expressed in the basis of S_z eigenstates? Or what? Being explicit about this is necessary for understanding/answering the question. Maybe you could write out your state in the L m_l S m_s basis? Then it might be more straightforward to answer your questions about j^2 by having a quick look at a Clebsch-Gordan table. I'm saying this not only to have enough information so others can help you more, but also to steer you in (possibly) the right direction for solving it yourself. –  Anonymous Coward Feb 18 '11 at 21:28
    
Wish I knew. I reproduced the problem as stated in the homework in verbatim (just did that column->row vector change). –  datenwolf Feb 19 '11 at 0:43
    
-1 for totally homework problem. –  Piotr Migdal Mar 7 '11 at 13:48

1 Answer 1

up vote 4 down vote accepted

Just recall the definitions:

  • spherical harmonics $Y^m_l$ are eigenvectors of both ${ \hat L}_z$ and $\hat L^2$ with eigenvalues of $m$ and $l(l+1)$ respectivelly (with $\hbar = 1$)
  • if you understand that this is a spin $1 \over 2$ particle then it should be obvious that $\hat s^2 = s(s+1) = {3 \over 4}$. If not, recall how spin matrices look like (e.g. start with Pauli matrices) and compute $\hat s^2$ directly.
  • the total Hilbert space $H$ is a tensor product of the scalar particle system with the spin system $H_{total} = H_s \otimes H_2$. So the operators on this space are $\hat {\mathbf L}_{total} = \hat {\mathbf L} \otimes \hat {\mathbb 1}_2$, $\hat {\mathbf s}_{total} = \hat {\mathbb 1}_s \otimes \hat {\mathbf s}$ (this just means that orbital momentum is still just differential operators and spin is just matrix operator) and $\hat J_{total} = \hat L_{total} + \hat s_{total}$. The rest is direct computation
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