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I Griffiths' Introduction to quantum mechanics, the spherical harmonics are defined as $$Y_l^m(\theta,\phi) = \epsilon\sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi} P_l^m(\cos \theta)$$ where $\epsilon = (-1)^m$ for $m \geq 0$ and $\epsilon = 1$ for $m<0$. Plugging in the associated Legendre function: $$P_l^m(x) = \frac{1}{2^l l}(1-x^2)^{|m|/2} \left(\frac{d}{dx} \right)^{l+|m|} (x^2-1)^l$$ the spherical harmonics can be written as
\begin{align} Y_l^m(\theta,\phi) &= \frac{\epsilon}{2^l l!}\sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi} (1-\cos^2 \theta)^{|m|/2} \left(\frac{d}{d\cos \theta} \right)^{l+|m|} (\cos^2 \theta-1)^l \\ &= \frac{\epsilon (-1)^l}{2^l l!}\sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi} \sin^{|m|} \theta \left(\frac{d}{d\cos \theta} \right)^{l+|m|} \sin^{2l} \theta \end{align} My intructor's notes however, define the spherical harmonics as $$Y_{lm}(\theta,\phi) = \frac{(-1)^l}{2^l l!}\sqrt{\frac{2l+1}{4\pi} \frac{(l+m)!}{(l-m)!}}e^{im\phi} \sin^{-m} \theta \left(\frac{d}{d\cos \theta} \right)^{l-m} \sin^{2l} \theta$$ for any $l,m$. I'm unable to see how these two expressions are the same, i.e. how to get from one to the other. It is clear that these expressions are equal for $m\leq 0$, but what about for $m>0$?

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  • $\begingroup$ Which edition & equation of Griffiths? $\endgroup$ – Qmechanic Oct 15 '18 at 10:11
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They are exactly the same. Note that the prefactors are basically the same, up to some simple factor and the choice of $|m|$ instead of $m$ in one form. That absolute value is largely inconsequential because you should start with $Y_{\ell m}$ with $m\geq 0$ only, and express $Y_{\ell,-m}$ as the complex conjugate of $Y_{\ell m}$, with a possible extra sign flip that is discussed in some detail.

Otherwise the most nontrivial part of the spherical harmonic or the associated Legendre polynomials is the $(\ell+m)$-th derivative of $(x^2-1)^\ell$ or $\sin^{2\ell} \theta $ with respect to $x$ or $\cos\theta$. These higher derivatives are exactly analogous given the substitution $x=\cos\theta$. Note that $1-x^2=\sin^2\theta$ given that substitution. Some signs are treated separately.

At most, you may end up with a slightly different convention in which the two spherical harmonics will sometimes differ by an overall sign. But they will not differ in any substantial way. Up to the overall scaling, the eigenstate of $L^2$ and $L_z$ is determined uniquely for every $\ell,m$.

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I don't believe the definition for the associated Legendre function you have written is correct. Wiki says the definition is:

$$ P_{\ell }^{m}(x)={\frac {(-1)^{m}}{2^{\ell }\ell !}}(1-x^{2})^{m/2}\ {\frac {d^{\ell +m}}{dx^{\ell +m}}}(x^{2}-1)^{\ell }. $$

If you plug in that instead, you'll get an expression much closer to your instructor's notes. Also be mindful of the $\epsilon$ factor, which is the commonly-used Condon-Shortley phase notation, which can cause some confusion.

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