# Hydrogen atom with discrete nonlinear Schrödinger equation

cryptist
Hi everyone,
How can I solve hydrogen atom with discrete nonlinear schrödinger equation? Could you help me with the mathematics of that, please?

## Answers and Replies

Homework Helper
Gold Member
What physics do you propose is responsible for nonlinearity? Why do you believe that the standard linear equation is insufficient?

cryptist
I want to consider all the possible forces and potential fields even gravitational forces between proton and electron, which is always neglected. Isn't there will be nonlinearity in equations in that case?

Nicodemus
I want to consider all the possible forces and potential fields even gravitational forces between proton and electron, which is always neglected. Isn't there will be nonlinearity in equations in that case?

Gravitational forces between a proton and electron? I realize that both do exert such a field, but at those distances it's a non-issue. Sounds like you want to screw in a nail.

Why should the nice linear Schrödinger equation become non-linear? If the proton is located at r=0 you simply add V=GmM/r as a potential term for the electron, that's all. This results in a tiny correction to the usual Coulomb term.

cryptist
So there is no need for nonlinearity? Then, how should be the discrete formula?

Which discrete formula? Usually you have

$$U_C(r) = -\frac{e^2}{4\pi\epsilon_0}\frac{1}{r}$$

Now you add the potential according to Newton's law of gravitation

$$U_N(r) = -Gm_em_p\frac{1}{r}$$

The total potential energy then reads

$$U(r) = -\left(\frac{e^2}{4\pi\epsilon_0} + Gm_em_p\right)\frac{1}{r} = -\frac{e^2}{4\pi\epsilon_0} \left(1 + \epsilon\right)\frac{1}{r}$$

This results in a rescaling of the energy levels due to the term

$$\epsilon = \frac{4\pi\epsilon_0Gm_em_p}{e^2}$$