Measuring the x magnetic moment

[fargoth]

if [tex]L_z|Y_l^m>=m|Y_l^m>[/tex]
whats my possible values for [tex]L_x(A|Y_0^0>+B|Y_1^0>+C|Y_2^0>)[/tex]?
i thought id use [tex]L_x=\frac{L_++L_-}{2}[/tex], create a matrix that would be block diagonal (a block for every l) and then i should find the eigenvalues for every block and say these are the values id get from measuring L in the x direction... am i correct?
does this mean there are more then one Y that can give with some probability the same value of L in the x direction?
and can one Y give with some probability one value and with some probability a different value for L in the x direction?

 

[Aaron Bex]

Yes, you are correct. You can calculate the eigenvalues for each block in the matrix and these will be the possible values for L_x when measuring the state |Y_l^m>.
Yes, different Y states can give the same value of L in the x direction with some probability.
And yes, one Y state can give with some probability one value and with some probability a different value for L in the x direction.