(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

It's not a homework problem. I'm reading my textbook (Sakurai's Modern QM), and I'm not sure about a step (eq 3.6.6 through 3.6.8). Here it is:

We start with a wave function that's been rotated:

[tex]\langle x' + y' \delta \phi, y' - x' \delta \phi, z' | \alpha \rangle[/tex]

Now, we change the coordinate basis from Cartesian to spherical. That is,

[tex]\langle x', y', z' | \alpha \rangle \to \langle r, \theta, \phi | \alpha \rangle[/tex]

I want to show that the previous wave function transforms to:

[tex]\langle r, \theta, \phi - \delta \phi | \alpha \rangle[/tex]

2. Relevant equations

See above.

3. The attempt at a solution

Physically, I understand that we're rotating by angle [tex]\delta \phi[/tex] about the z-axis. So, it makes sense that the bra would acquire a [tex]-\delta \phi[/tex] (the ket would acquire a positive). But, I'm not sure how to mathematically show the steps (which is needed for more complicated transforms, such as Sakurai eq. 3.6.10 to 3.6.11).

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# Basis Transformation for Wave Function

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