- #1
Harry Smith
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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
\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, 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
\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.
Why have these operators been assigned seemingly at random to one of the states?
My intuition suggests that
\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*}
\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*}.
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.
\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, 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
\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.
Why have these operators been assigned seemingly at random to one of the states?
My intuition suggests that
\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*}
\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*}.
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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