I am confused because I have seen two different expressions for the radial term in polar coordinates, in Wikipedia and other documents such as this one: http://ramanujan.math.trinity.edu/rdaileda/teach/s12/m3357/lectures/lecture_3_27_2_short.pdf
The radial part of the Laplacian is defined as:
$$\nabla^2f=\partial_r^2f+\frac{1}{r}\partial_rf+[\mathrm{angular\:terms}]$$
However, when deriving the solutions for the Hydrogen atom in quantum mechanics the expression used for the polar laplacian is:
$$\nabla^2f=\frac{1}{r^2}\partial_r(r^2\partial_rf)+[\mathrm{angular\:terms}]=\partial_r^2f+2\frac{1}{r}\partial_rf+...$$
(This expression can be seen here: http://users.aber.ac.uk/ruw/teach/237/hatom.php)
Why the expression used in quantum mechanics is different than the "official" laplacian in polar coordinates?