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This may be a bit of a basic question, but I am finding a lot of contradictory info online.

Basically I am trying to write a couple of quantum mechanics scripts with python. This involves doing a lot of 3d integrals on spherical functions and using the laplace operator $ ∇^2 $.

In cartesian coordinates, writing lazily: $$ ∇^2 = d/dx^2 + d/dy^2 + d/dz^2 $$

In spherical coordinates: http://mathworld.wolfram.com/images/equations/Laplacian/Inline12.gif

The issue I see in some sources (both online and in textbooks) that sometimes $\theta$ and $\phi$ are switched. I understand that this is because sometimes one is used for the azimuthal angle and sometimes not.

When I do my 3d integrals, my convention is to use the jacobian $ r^2 * \theta $ and integrate theta from 0 to $\pi$, and integrate $\phi$ from 0 to $2\pi$. I use $tan( \theta ) = y/x$ and $cos(\phi) = z/r$. My question is... which laplacian convention would that be? I think my convention is the opposite of the one in the linked image, but I'm not 100% sure. Am I mixing up my jacobian as well? Sometimes I see sources define $\phi$ and $\theta$ opposite to me (in terms of the angles with x, y, z and r) but use the same integrals (as in, which goes to $\pi$ and which goes to $2\pi$.)

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    In my training, $\theta$ was always the polar angle (measured from the north pole) and $\phi$ the azimuthal angle. I don't however think the labels should be all that important. The thing to remember what the functions mean in spherical coordinates. The Laplacian given in the link has $\theta$ as polar and $\phi$ as azimuthal, so your integration should be consistent with the link. – 2017-01-10
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    Small addition to the question: If you look here: http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/math_prelims/node12.html It seems they define $\theta$ and $\phi$ opposite to me. They use $\phi$ as the angle on the xy plane, which is NOT what I do. But their integral limits and jacobian (at the bottom) are the same as mine. Am I mixing up my integrals...? Or does it not really matter? – 2017-01-10
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    It seems that you might be reading things backwards. The NYU link does also treat $\phi$ as the azimuthal angle ranging from $0$ to $2\pi$. I would double check your integrals to make sure you're integrating the right variable with respect to the right domain. – 2017-01-10
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    Well, they integrate $\phi$ from $0$ to $2\pi$ which is the same as I do, but they define it differently in their visual of the coordinate space. They to have it as the angle between x and y and define it as as such with an arctan, whereas in that case, I have $\theta$ as the x/y angle. – 2017-01-10
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    So, everything I've seen on the NYU page is accurate. In your post it still seems that you've switched the roles of $\phi$ and $\theta$. You want $\phi = \arctan(y/x)$ and $\cos \theta = z/r$. – 2017-01-10

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