Cant understand integral tranasition to spherical coordinates

[nhrock3]

there is a function $$\Psi =\frac{c}{\sqrt{r}}e^{\frac{-r}{b}}$$

find the probaility in $$\frac{b}{2}<r<\frac{3b}{2}\\$$ region

the rule states $$\int_{-\infty}^{+\infty}|\Psi|^2dv=1\\$$


$$1=\int_{-\infty}^{+\infty}|\frac{c}{\sqrt{r}}e^{\frac{-r}{b}}|^2dv$$



then they develop it as



$$c^2\int _{all space}\frac{1}{r}e^{\frac{-2r}{b}}2\pir^2dr=4\pic^2\int_{0}^{+\infty}re^{\frac{-2r}{b}}dr\\$$


they as it because of spherical coordinates

but i can't see here the jacobian of spherical coordinates.

i can't see here the x,y,z transition to r ,theta,phi



i can't see it in the last equation

 

[tiny-tim]

Welcome to PF!

Hi nhrock3! Welcome to PF!

(have a pi: π and an infinity: ∞ and an integral: ∫ and try using the X² and X₂ tags just above the Reply box )

I'm a little confused by what you've written, but basically you start with

∫ c²/r e^-2r/b dxdydz

and because it's spherically symmetric, you can divide the region into spherical shells of radius r and volume 4πr²dr,

which gives you ∫₀^∞ c²/r e^-2r/b 4πr²dr

= 4πc² ∫₀^∞ r e^-2r/b dr