Consider a system with a state of fixed total angular momentum $l = 2$. What
are the eigenvalues of the following operators
(a)$ L_z$
(b) $3/5L_x −4/5L_y$
(c) $2L_x −6L_y +3L_z$
My problem is more to do with the definition of the angular momentum operator:
I think the angular momentum operator is $L^2=L_x^2+L_y^2+L_z^2$. I have seen many different eigenvalues this gets when applied to an eigen ket:
- $L^2|\psi\rangle=\hbar^2 k^2|\psi\rangle$
- $L^2|\psi\rangle=\hbar^2 j(j+1 )|\psi\rangle$
along with a few others. I understand that these are sort of equivilent and we are just using numbers to represent the value. However, what is the $l=2$? Is it the $k$, the $j$?
I know what to do from here on, $m$ (the quantum m=number for angular momentum along a given axis) varies from $-j$ to $+j$
Best Answer
I think the trick here is to note that the operator in (b) measures the component of angular momentum along the axis $\hat{n} = (3/5, 4/5, 0)$. It's eigenvalues must be $\{2,1,0,-1,-2\}$, the same as those of $L_z$, because you could have chosen your z-axis to lie along $\hat{n}$.
Similarly, the operator in (c) is 7 times the component of $\vec{L}$ along the normalized axis $\hat{n} = (2/7, -6/7, 3/7)$. It's eigenvalues must therefore be $\{14,7,0,-7,-14\}$, by the same reasoning.