Is this superposition state an eigenstate of J^2 and L * S?

[Ed Quanta]

Show that the following superposition state in the |l,m1;ms> basis:-(2/3)^1/2|1,-1;1/2> + (1/3)^1/2|1,0;-1/2> is an eigenstate of J^2 and L * S and determine the corresponding eigenvalues.

I have no clue how to start this. Any help would be appreciated.

 

[dextercioby]

How do you determine the action of the operators
$$\hat{\vec{J}}^{2} ,\hat{\vec{L}}^{2},\hat{\vec{S}}^{2}$$

on any eigenstate...?

Daniel.

 

[quantumdude]

First note that:

$$\hat{\mathbf {J}}^2=\hat{\mathbf {J}} \cdot \hat {\mathbf {J}}=(\hat{\mathbf {L}} + \hat{\mathbf {S}}) \cdot (\hat{\mathbf {L}} + \hat{\mathbf {S}}) =\hat{\mathbf {L}}^2 + \hat{\mathbf {S}}^2+2 \hat{\mathbf {L}} \cdot \hat{\mathbf {S}}$$

Hit each term of your state vector with it.

edit: Bloody dexter, his LaTeX is faster than mine.

 

[dextercioby]

I used "hats" for operators,that's a reason for envy...

Daniel.

P.S.BTW,i resent the boldface notation of vectors...

 

[quantumdude]

Nothing a quick edit can't fix. Now my operators don't have cold heads.