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I was given with this hydrogen radial wavefunction

$$ R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right) $$

and was asked to find

a) What are the expectation values of the $L^2$, and $L_z$?

b) If you measured the orbital angular momentum squared $L^2$, what values might you get? Do the same for $L_z$.

The problem is I cannot understand the difference between the answer to both a & b, aren't they referring to the same answer?

My answer for $L^2$ is: $$ L^2 \psi = l(l+1)h^2 \psi \\ = 2h^2 \psi$$ hence the expectation value for $L^2$ is $2h^2$, now what should be my answer for b)?

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If $R_{21}$ is an eigenfunction of $L_z$ then all measurements of this quantity when the wavefunction is in such a state will yield the same value always. So, in that case, the answers to a) and b) will be the same. I am currently learning QM as well so someone correct me if this is incorrect. Your wavefunction is conveniently expressed in terms of the eigenstates of both $L^2$ and $L_z$ so by acting with these operators onto the eigenstates gives the possible values immediately. – CAF Mar 17 '14 at 20:59
same thoughts here, also waiting for the veteran's confirmation. – user1824371 Mar 17 '14 at 21:01
your own lecture notes work out this exact problem on page 5, under "class excercise", but correctly distinguish the wavefunction $\psi $, the radial function R and the spherical harmonic Y. – DavePhD Mar 17 '14 at 21:22
$L^2$ should be $l(l+1)\hbar^2$ with a h-bar – Binary Funt May 4 '15 at 13:40
Or rather, $L^2$ has eigenvalues $l(l+1)\hbar^2$ I think? – Binary Funt May 4 '15 at 13:48

The two questions are slightly different. Each individual measurement of $L^2$ or $L_z$ will return an eigenvalue. In this case, you have only one possible measurement for $L^2$ (corresponding to $l=1$), but you have two possible measurements for $L_z$; 2/3 of the time you'll get $m=1$, and 1/3 of the time you'll get $m=0$.

The expectation value, on the other hand, is essentially asking, if you took a whole bunch of measurements and averaged them, what would you get? For $L^2$, since all of your measurements are the same, you'll again get the eigenvalue when you average them, but for $L_z$ you'll get something in between.

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