# Help solve this specific linear schrodinger equation, please

$$\left[-\frac{1}{2}\nabla^2 - \frac{2}{r} + C\right]\phi(r) = E\phi(r)$$

where $C$ is a known function. I am just looking for some help on the strategy to solve explicitly for $\phi(r)$. One thing I am having trouble understanding is how we can solve this equation without expressing $\phi(r)$ in terms of $E$, but it seems like it is done all the time: http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/05.5%3A_Particle_in_Boxes/Particle_in_a_1-dimensional_box

Thanks!

• math.stackexchange.com – ParaH2 Nov 27 '15 at 22:12
• @Shadock thats legit, but I feel like people here will probably know the specifics of SE better – user2879934 Nov 27 '15 at 22:18
• This is not a problem, we need to use maths in chemistry and physics and in a lot of different subjects. If you want a good answer you may ask the question in ME. :-) – ParaH2 Nov 27 '15 at 22:21
• The particle in a box is a "simple" system in that the SE reduces to basically $\mathrm{d}^2\psi/\mathrm{d}x^2 = -k\psi$ and the general solution is quite obvious. When you have that $-2/r$ term, it gets a bit more irritating. Exactly what context are you trying to solve the SE in? The form of the SE that you are using makes me think the $\ce{He+}$ ion (where that function $C = 0$) - if I am not wrong you can use the power series method. There is a name for the solutions, I think spherical Bessel functions. – orthocresol Nov 27 '15 at 22:35
• @orthocresol Hey! Good question...the form of that function is actually the hartree fock mean field approximation integral, so you are spot on, I am trying to do the HF method for helium. – user2879934 Nov 27 '15 at 22:40