Can the Angular Dependence Be Ignored in the Hydrogen-Atom Radial Solution?

[aaaa202]

Hi,
in an assignment I have to include the radial solution of the schrödinger-equation. That is, that you solve it for the radial dependence only. Now, I do get, that the force between the electron and nuclues depends only on r, but what I still don't get is why you can supress the angular dependence. Isn't the kinetic energy somehow affected by the angles? And if not, why is it either way, that you can assume that the wave function is a product of a radial dependent function and an angular dependent?
Now, I've seen that work in 1D separating the time dependence from the wave function in a stationary state, but this is not equivalent to that is it?

 

[nickjer]

I am assuming it wants the radial solution for hydrogen atom. If so, then yes you can do separation of variables for the 3D Schrödinger equation of a hydrogen atom. You can find the solution standard in most QM textbooks or on the web.

 

[aaaa202]

Yes, I get that but you didn't answer my question.
WHY is it, that you can use separation of variables? Isn't the kinetic energy-operator dependent on the angular function?

 

[nickjer]

The kinetic energy operator is a Laplacian and is dependent on all 3-dimensions as well. Please look up how to write a Laplacian in different coordinates. You can find the Laplacian in spherical coordinates here: http://en.wikipedia.org/wiki/Laplace_operator#Three_dimensions

If you use separation of variables on the wavefunction, you will also find you can separate the differential form of the kinetic energy operator as well. You can find this as I said before in a standard QM textbook.

Just googling it I found a nice step-by-step guide, where he shows how he separates the Schrödinger equation: http://www.physics.gatech.edu/frog/lectures/ModernPhysicsLectures/MP14HydrogenAtom.ppt

 

[vela]

Hi,
in an assignment I have to include the radial solution of the schrödinger-equation. That is, that you solve it for the radial dependence only. Now, I do get, that the force between the electron and nuclues depends only on r, but what I still don't get is why you can supress the angular dependence. Isn't the kinetic energy somehow affected by the angles?

You don't ignore the angular dependence. There is a kinetic term that depends on the angular momentum, but it won't be part of the radial equation after separation.

And if not, why is it either way, that you can assume that the wave function is a product of a radial dependent function and an angular dependent?

Because it turns out the method works. You assume the form of the solution and try it out. If it works, great. If it doesn't, you toss the assumption and try something else. Fortunately, it works out in this case.

Now, I've seen that work in 1D separating the time dependence from the wave function in a stationary state, but this is not equivalent to that is it?

It's pretty much the same method. You assume the spatial and time dependence can be cleanly separated and try it out. It turns out it works, and you get a solution for the time dependence and the time-independent Schrödinger equation.

To isolate the radial equation, the mathematics is a bit more involved, so it's probably a good idea to consult a QM textbook, as nickjer suggested, just to see how the basic argument goes and then work it out yourself.