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)?
Best Answer
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.