Why do we need the CondonShortley phase in spherical harmonics?
You don't need it: it's a sign convention and the only thing you need to do with it is to be consistent. (In particular, this means always checking that the sign and normalization conventions for $Y_{lm}$ and $P_l^m$ agree for all the sources that you're using, and correctly account for any differences there.)
The CondonShortley sign convention is built so that the spherical harmonics will play nicely with the angular momentum ladder operators: in particular, they enable you to write \begin{align} Y_l^m(\theta,\varphi) & = A_{lm} \hat{L}_^{lm} Y_l^l(\theta,\varphi), \quad \text{and} \\ Y_l^m(\theta,\varphi) & = A_{l,m} \hat{L}_+^{l+m} Y_l^{l}(\theta, \varphi), \end{align} where the $A_{lm}=\sqrt{\frac{(l+m)!}{(2l)!(l+m)!}}$ are all positive constants. This comes from Aarfken, 6th ed (2005), Eq. (12.162) p. 794, and it uses the conventions \begin{align} Y_l^m(\theta,\varphi) & = (1)^m \sqrt{\frac{2l+1}{4\pi}\frac{(lm)!}{(l+m)!}} P_l^m(\cos(\theta)) e^{im\varphi}, \text{ for} \\ P_n^m(\cos(\theta)) & = \frac{1}{2^n n!}(1x^2)^{m/2} \frac{\mathrm d^{m+n}}{\mathrm d x^{m+n}}(x^21)^n ,\text{ and with}\\ L_\pm & = L_x\pm i L_y = \pm e^{i\varphi}\left[\frac{\partial}{\partial \theta} \pm i\cot(\theta) \frac{\partial}{\partial \varphi} \right] . \end{align} Ignoring the CondonShortley phase would introduce signs into the $A_{lm}$, which can be seen as (vaguely) undesirable  you want the awkward constants in the fiddly specialfunctiony side, which is always awkward to begin with, and not in the Hilbertspace side where clean relationships between wavefunctions and vectors are much more valuable.
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ilciavo
Updated on January 21, 2020Comments

ilciavo almost 4 years
I'm confused with different definitions of spherical harmonics:
$$Y_{lm}(\theta,\phi) = (1)^m \left( \frac{(2l+1)(lm)!}{4\pi(1+m)!} \right)^{1/2} P_{lm} (\cos\theta) e^{im\phi}$$
For example here they claim, that one can decide whether to include or omit the CondonShortley phase $(1)^m$. And, also they claim this is useful in quantum mechanical operations, such as raising and lowering.
In The Theory of Atomic Spectra, Condon and Shortley state:
"If we had approached the problem through the usual form of the theory of spherical harmonics the natural tendency would have been to chose the normalizing factors with omission of the $(1)^m$ in these formulas"
So the whole point of using $(1)^m$ is that the following identity holds
$$Y_{lm}(\theta,\phi) = (1)^m Y_{lm}(\theta,\phi)^*$$
How is the CondonShortley phase used with these operations and why is this phase beneficial?