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## Homework Statement

Given ##\psi = AR_{21}[BY_1^1 + BY_1^{-1} + CY_1^0]##, find ##\left<L_z\right>## and ##\left<L^2\right>##. (This is not the beginning of the homework problem, but I know my work is correct up to here. I am not looking for a solution, only an answer as to whether or not my method of finding expectation values is correct, thus why I have generalized the problem with these coefficients instead of using the actual coefficients from the problem.)

Here, ##A##, ##B##, and ##C## are constants, ##Y_l^m## is the angular wave function for an electron in a hydrogen atom in the state ##\left|nlm\right>##, and similarly ##R_{nl}## is the electron's radial wave function.

## Homework Equations

##L_z f_l^m = \hbar m f_l^m##

##L^2 f_l^m = \hbar^2 l (l+1) f_l^m##

##\left<Q\right> = \left<g\right|Q\left|g\right>## for the expectation value of an observable ##Q## for a function ##g##.

## The Attempt at a Solution

Since the wave function is a linear combination of the angular wave functions, and in calculating the angular momentum the radial wave function can be ignored, let ##f_l^m## be the wave function without the radial component: $$f_l^m = A[BY_1^1 + BY_1^{-1} + CY_1^0].$$ The expectation value of ##L_z## would then be $$\left<L_z\right> = \left<f_l^m\right|\hbar m\left|f_l^m\right>$$

The problem I have is that there isn't one particular value of ##m##. We instead have three values. However, if we calculate ##\left<f_l^m | f_l^m\right>##, then even though the initial math would be horrible (creating nine terms where there previously were only three), because these angular wave functions are mutually orthogonal, any product of two wave functions that are not the same will be 0, meaning we can just square each wave function in place and remove the imaginary components.

Essentially, $$\left<L_z\right> = \left<f_l^m\right|\hbar m\left|f_l^m\right> = \left<ABY_1^1\right|\hbar m\left| ABY_1^1\right> + \left<ABY_1^{-1}\right|\hbar m\left|ABY_1^{-1}\right> + \left<ACY_1^0\right|\hbar m\left|ACY_1^0\right>=A^2 \left[B^2\left<Y_1^1\right|\hbar \left| Y_1^1\right> + B^2\left<Y_1^{-1}\right|-\hbar\left|Y_1^{-1}\right> + C^2\left<Y_1^0\right|0\left|Y_1^0\right>\right] = A^2(0) = 0$$

and ##\left<L^2\right>## would be found in a similar way. Is this an acceptable way to find the expectation value?