I'm reading a paper (download here) which uses a polar coordinates to reach a result, but I can't understand how it uses polar coordinates at all. I try to explain all steps using photos of paper.
The system is a helium atom including two electrons and one nucleus named 1 , 2 and 0 respectively. There are three distances in this system: regarding nucleus located in the origin, $r_1$: distance of electron 1 from the origin, $r_2$ distance of electron 2 from the origin, $r_{12}$ distance between two electrons (you can recognize these in the second image easily)
First the paper defines a function called distribution function $f(r_{12})$ and then normalizes it as you can see in the following image:
After that it defines wave function of the system and normalizes it too. Then it equates them (because integration over both lead to 1), First question arises here: Why it uses $f(r_{12}) dr_{12}$ in equation 1, I mean why the integral sign is absent?!
Then It makes $d\tau_1d\tau_2$ equal to all distances and angels which is absolutely unclear to me. See the following image:
On one hand it says we are using polar coordinates. We know there is only one angle in this coordinate while here there are three angels $\theta, \phi$ and $\chi$ ! On the other hand we know integration over all space is $r^2 \sin[\theta] \;dr\;d\theta \;d\phi$ in spherical coordinates and $r dr d\theta$ in polar coordinates, so I can't understand how it reaches to such an equivalent for $d\tau_1d\tau_2$. It doesn't match what I know about integrating over all space. I will be so grateful if someone can help me to answer my questions. Thanks in advance,
Best Answer
To answer your first question,
I will just point out that right below that expression, the paper adds that
Thus the right hand side should be interpreted as an integral over a region that satisfies this description (i.e. a thin spherical shell at a distance of $r_{12}$ from particle $1$). The authors resorted to this verbal description probably because it is a bit cumbersome to explicitly write down the domain of the integral, which could be described as $r_{12}<|\boldsymbol{\tau}_1 - \boldsymbol{\tau}_2| < r_{12} + d r_{12}$. The differential $d r_{12}$ is the reason why there is a differential on the left hand side. (Note that I have made the variables $\boldsymbol{\tau}_1$ and $\boldsymbol{\tau}_2$ bold, because I want to make it clear that these are short hands for two volume integrals. For example $d\boldsymbol{\tau}_1$ can stand for $d x_1 d y_1 dz_1$ in Cartesian coordinates or $r_1^2 \sin\theta_1 dr d\theta d\phi$ in spherical coordinates.)
As for your second question concerning the coordinates, I would say that it is indeed a little different from the conventional polar (spherical) coordinates, but presumably this is to exploit the symmetries in the problem or some special properties of the integrand. Let me walk through the transformation first, then address some of the specific concerns you raised.
First, just as a sanity check, let's count the number of variables on the two sides of the transformation $$ d\boldsymbol{\tau}_1 d\boldsymbol{\tau}_2 = r_1 r_2 r_{12} \sin\theta_1 d r_1 d r_2 d r_{12} d \theta_1 d \phi_1 d \chi \ . $$ As I noted above, the left hand side is two 3D volume forms, so we are integrating over six variables. The right hand side also has six variables. Three of these are the usual spherical coordinates for particle $1$, namely $r_1$, $\theta_1$, and $\phi_1$. The other three require some explaining, but essentially they specify the position of the second particle. In any case we can see now that at least the number of variables match up.
To get to these three variables that describe the position of particle $2$, we can start from a set of spherical coordinates that takes $\boldsymbol{\tau}_1$ as the $z$-axis. In this "relative spherical coordinate system", we use the coordinates $r_2$, $\theta_{12}$, and $\chi$. The first is of course the usual radial coordinate. The second is the angle between the directions $\boldsymbol{\tau}_1$ and $\boldsymbol{\tau_2}$. The third is the angle of rotation from the $01z$ plane to the $012$ plane. To clarify, $01z$ is the plane that contains the $z$ axis and $\boldsymbol{\tau}_1$, and $012$ is the plane that contains $\boldsymbol{\tau}_1$ and $\boldsymbol{\tau}_2$. (See Figure 1 in the paper. You should think of point $2$ as hovering above the page and the triangle $012$ as folding away from the page, hinged along the $01$ line segment.) The volume form in these coordinates is just the ordinary spherical form $r_2^2 \sin\theta_{12} d r_2 d \theta_{12} d \chi$.
Next we want to get rid of the coordinate $\theta_{12}$ and use $r_{12}$ instead. The coordinate transform is given by the law of cosines, $$ r_{12} = \sqrt{r_1^2 + r_2^2 - 2 r_1 r_2 \cos\theta_{12}}\ . $$ We also need the Jacobian factor of this substition from $(r_2, \theta_{12}, \chi)$ to $(r_2, r_{12}, \chi)$, which turns out to be $r_1 r_2 \sin\theta_{12} / r_{12}$. This means that the volume form transforms as $$ r_2^2 \sin\theta_{12} d r_2 d \theta_{12} d\chi = \frac{r_2^2 \sin\theta_{12} d r_2 d r_{12} d\chi}{r_1 r_2 \sin\theta_{12}/r_{12}} = \frac{r_2 r_{12}}{r_1} d r_2 d r_{12} d\chi\ . $$ Putting this together with the first part of the volume form $r_1^2 \sin\theta_1 dr_2 d\theta_1 d \phi_1$, we get the desired $r_1 r_2 r_{12} d r_1 d r_2 d r_{12} d\theta_1 d \phi_1 d\chi$.
Finally let me address the problem with terminology that might have tripped you up in the first place. It seems that you felt uneasy because the phrase "polar angles" seems to suggest the usage of polar coordinates, which is a two dimensional coordinate system that only contains one angular variable. However, people often use "polar coordinates" to mean "spherical coordinates" when the context makes it clear that the space in question is three-dimensional. The polar angle $\theta$ and azimuthal angle $\phi$ are then referred to collectively as the "polar angles".