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## Main Question or Discussion Point

I'm reading through my quantum physics lecture notes (see page 216 of the lecture notes for more details) and under the ladder operators section there is a discussion of the expectation value of ##L_x## for a state ##\psi = R(r) \left( \sqrt{ \frac{2}{3}} Y_{11} - \sqrt{ \frac{1}{3}} Y_{10} \right)## such that

[tex]\langle \psi | L_x | \psi \rangle = \frac{1}{2} \bigg\langle \sqrt{ \frac{2}{3}} Y_{11} - \sqrt{ \frac{1}{3}} Y_{10} \bigg| L_+ + L_- \bigg| \sqrt{ \frac{2}{3}} Y_{11} - \sqrt{ \frac{1}{3}} Y_{10} \bigg\rangle, [/tex] where ##L_\pm## are the angular momentum ladder operators and ##Y_{\ell m}## are the spherical harmonics.

Now, this all makes sense to me, however the next step states that

[tex]\langle \psi | L_x | \psi \rangle = \frac{1}{2} \bigg\langle \sqrt{ \frac{2}{3}} Y_{11} - \sqrt{ \frac{1}{3}} Y_{10} \bigg| \sqrt{ \frac{2}{3}} L_- Y_{11} - \sqrt{ \frac{1}{3}} L_+ Y_{10} \bigg\rangle. [/tex]

Why have these operators been assigned seemingly at random to one of the states?

My intuition suggests that

[tex]\begin{align*} (A+B) |u_1+u_2 \rangle &= A |u_1+u_2 \rangle + B |u_1+u_2 \rangle \\&= | A u_1 + A u_2 \rangle + | B u_1 + B u_2 \rangle\end{align*}[/tex]

[tex]\begin{align*}\implies \langle u_1+u_2 | (A+B) |u_1+u_2 \rangle &= \langle u_1+u_2 | A u_1 + A u_2 \rangle + \langle u_1+u_2 | B u_1 + B u_2 \rangle \\&= \langle u_1 | A u_1 \rangle + \langle u_2 | A u_1 \rangle + \langle u_1 | A u_2 \rangle + \langle u_2 | A u_2 \rangle \\&\ \ \ \ \ \ \ \ + \langle u_1 | B u_1 \rangle + \langle u_2 | B u_1 \rangle + \langle u_1 | B u_2 \rangle + \langle u_2 | B u_2 \rangle \end{align*}.[/tex]

Where am I going wrong?

Please note that this is

[tex]\langle \psi | L_x | \psi \rangle = \frac{1}{2} \bigg\langle \sqrt{ \frac{2}{3}} Y_{11} - \sqrt{ \frac{1}{3}} Y_{10} \bigg| L_+ + L_- \bigg| \sqrt{ \frac{2}{3}} Y_{11} - \sqrt{ \frac{1}{3}} Y_{10} \bigg\rangle, [/tex] where ##L_\pm## are the angular momentum ladder operators and ##Y_{\ell m}## are the spherical harmonics.

Now, this all makes sense to me, however the next step states that

[tex]\langle \psi | L_x | \psi \rangle = \frac{1}{2} \bigg\langle \sqrt{ \frac{2}{3}} Y_{11} - \sqrt{ \frac{1}{3}} Y_{10} \bigg| \sqrt{ \frac{2}{3}} L_- Y_{11} - \sqrt{ \frac{1}{3}} L_+ Y_{10} \bigg\rangle. [/tex]

Why have these operators been assigned seemingly at random to one of the states?

My intuition suggests that

[tex]\begin{align*} (A+B) |u_1+u_2 \rangle &= A |u_1+u_2 \rangle + B |u_1+u_2 \rangle \\&= | A u_1 + A u_2 \rangle + | B u_1 + B u_2 \rangle\end{align*}[/tex]

[tex]\begin{align*}\implies \langle u_1+u_2 | (A+B) |u_1+u_2 \rangle &= \langle u_1+u_2 | A u_1 + A u_2 \rangle + \langle u_1+u_2 | B u_1 + B u_2 \rangle \\&= \langle u_1 | A u_1 \rangle + \langle u_2 | A u_1 \rangle + \langle u_1 | A u_2 \rangle + \langle u_2 | A u_2 \rangle \\&\ \ \ \ \ \ \ \ + \langle u_1 | B u_1 \rangle + \langle u_2 | B u_1 \rangle + \langle u_1 | B u_2 \rangle + \langle u_2 | B u_2 \rangle \end{align*}.[/tex]

Where am I going wrong?

Please note that this is

**not**a homework problem, so full solutions are welcome. I may need each tiny step written out to understand why this is happening.
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