What does the spherical-harmonic notation $Y^{m}_l(\hat{\textbf{r}})$ mean, and how does it relate to the usual $Y^m_l(\theta, \varphi)$?

By using the plane wave expansion, the decomposition of stationary harmonic plane wave into partial waves can be given by $$e^{i\textbf{k}\cdot\textbf{r}} = e^{ikz} = e^{ikr\cos\theta} = \sum^{\infty}_{l=0}(2l+1)i^l j_l(kr) P_l(\cos\theta)$$ where $i$ is the imaginary unit, $k$ and $r$ are the wave and position vectors, $j_{l}$ are the spherical Bessel functions, and $P_l$ are Legendre Polynomials. By the relationship between the Legendre Polynomials and Spherical Harmonics, according to Wikipedia, this expression could further be expressed as $$e^{i\textbf{k}\cdot\textbf{r}} = 4\pi \sum_{l=0}^{\infty}\sum_{m=-l}^{l} i^l j_l(kr)Y^m_l(\hat{\textbf{k}})Y^{m*}_l(\hat{\textbf{r}})$$ where the Spherical Harmonics and its conjugate are a function of the unit vectors in the direction of $k$ and $r$.

My question is how is $Y^m_l(\hat{\textbf{k}})$ or $Y^{m*}_l(\hat{\textbf{r}})$ different compared to the conventional Spherical Harmonics $Y^m_l(\theta, \varphi)$ in terms of their explicit form. How do you turn a scalar SH to a vector one and if the vector ones are only functions of unit vector doesn't that mean they don't depend on the magnitude of the value you put in since all the unit vectors have a value of $1$?

My question is how is $Y^m_l(\hat{\textbf{k}})$ or $Y^{m*}_l(\hat{\textbf{r}})$ different compared to the conventional Spherical Harmonics $Y^m_l(\theta, \varphi)$

They're not different - they're exactly the same object. The notation $Y^{m}_l(\hat{\textbf{r}})$ is often used because it is cleaner to use in situations where you don't have an explicit need to introduce spherical coordinates for the vector $\mathbf r$ and using those coordinates would introduce unnecessary notational bulk. (Why? For the situation you've used, consider the fact that you've got two sets of angles, one for $\mathbf r$ and one for $\mathbf k$, so you need four symbols or the introduction of subscripts. The vector-argument notation expresses exactly the same object without requiring any new notation.) But they're exactly the same: $$Y^{m}_l(\hat{\textbf{r}}) = Y^m_l(\theta, \varphi)$$ so long as you stick to the reasonable $\hat{\mathbf r} = (\sin(\theta)\cos(\varphi), \sin(\theta)\sin(\varphi), \cos(\theta))$. (For clarity, this notation assumes that $\hat{\mathbf r} = \mathbf r/ ||\mathbf r||$; this is standard enough that it can be used without explanation.)

Vector spherical harmonics, on the other hand, are rather different objects - they are vector-valued functions, and they are useful if you have e.g. an outgoing spherical electromagnetic wave, and you want a good basis to express the spatial dependence of the vector character of the fields.

The functions you've found are simply the usual scalar spherical harmonics, with a notational shortcut thrown in to simplify life.

• Just to make sure by $\hat{\mathbf r} = (\sin(\theta)\cos(\varphi), \sin(\theta)\sin(\varphi), \cos(\theta))$ you mean $\hat{\mathbf r} = \sin(\theta)\cos(\varphi)\hat{\mathbf i} + \sin(\theta)\sin(\varphi)\hat{\mathbf j} + \cos(\theta)\hat{\mathbf k}$ right? Also so if $Y^{m}_l(\hat{\textbf{r}}) = Y^m_l(\theta, \varphi)$ then does it mean $Y^{m}_l(\hat{\textbf{k}}) = Y^m_l(\theta_{k}, \varphi_{k})$, so one of them is in $r$ space and the other is in $k$ space? – user3613025 Sep 4 '18 at 10:27
• Yes, both of those interpretations are correct. (However, just to point out: you haven't defined $\theta_k$ or $\varphi_k$. Are they angles in a list, with $k=0,1,2,\ldots$? If you mean that they are the polar and azimuth angles of the vector $\mathbf k$ in the same coordinate system as that for $\mathbf r$, then formally you need to say so explicitly. This is what I mean by "notational clutter".) – Emilio Pisanty Sep 4 '18 at 10:32
• Wait so $Y^{m}_l(\hat{\textbf{r}}) = Y^m_l(\theta, \varphi)$ doesn't mean that they're pointing towards the $\hat{r}$ direction? – user3613025 Sep 4 '18 at 10:35
• That second comment makes no sense at all. Which object is doing the "pointing"? And what, precisely, do you mean by "the $\hat{r}$ direction"? – Emilio Pisanty Sep 4 '18 at 10:36
• I guess $\hat{r}$ points anywhere in a radial direction from the origin of the sphere, made up by $\theta$ and $\phi$?(Idk if that makes sense lol) Also yes I did mean that $\theta_{k}$ and $\varphi_{k}$ are the polar and azimuthal angles of the vector $\mathbf k$, was that not self-apparent?If not then how does one usually "say so explicitly"?As in clarifying it in words or notation? – user3613025 Sep 4 '18 at 10:47

Nothing: a vector is defined by modulus and angles; but a unit vector has fixed modulus, so it's just another way to indicate the angles $(\theta, \ \varphi)$.

You can also write $Y_l^m (\Omega)$, since the solid angle is also related to them.

Note: we should name those three harmonics differently, but we the physicist like to use the same letter, altough the function is different if we introduce one angle or another.