Finding Mutual Potential Energy of 2 1S Electrons in He

[jaimon]

How can I find the mutual potential energy of two 1S electrons in helium? Each of the two 1S electrons is described by a hydrogenic wave function. I've been trying the addition theoremfor spherical harmonics but it seems not to work.
Thanks

 

[CarlB]

This is a quantum problem of course. You begin with the wave function for the two electrons. The wave functions are identical, except one is spin up and the other is spin down. Label the two wave functions "1" and "0".

You want the average value of the potential energy operator V. I guess V is proportional to $$1/r = 1/|r_1-r_0|$$ potential, but it's been 25 years and I'm not going bet my life on it. And besides, there are several choices of constants depending on which type of E&M you prefer.

The answer is then <1,0| V |1,0>. The integral will be something like:

$$\int \int \psi_1^*(r_1)\;\psi_0^*(r_0) \psi_0(r_0)\psi_1(r_1) d^3r_1 d^3r_0 /|r_1 - r_0|.$$

where r_1 and r_0 are 3-vectors and the integrals are over all space. In the above, I've been sloppy in ignoring the detail having to do with spin and statistics. If you want to get your work judged correct I suggest you be more careful and antisymmetrize and keep track of spin and all that.

Carl