# Non-local potential

I read that the Schrodinger equation for a non-local potential is given by $$-\frac{\hbar^2}{2m}\nabla^2\psi(x)+\int V(x,x')\psi(x')dx'=E\psi(x).$$ In case of a local potential, $$V(x,x')=V(x)\delta(x-x').$$ In what sense, one is 'local' and the other is 'non-local'? What does a non-local potential mean physically? When does such a potential arise?

## 2 Answers

In your nonlocal potential, the potential that your partical feels at each point in space depends not on the value of some single function, but the sum (Integral) of all the values of a function defined over all space. This comes up a lot in multi-particle QM. Consider trying to find the electron repulsion in the helium atom $$1/{r_{12}}$$. Both electrons are delocalized over all space. So if I wanted to know the average value of the electron repulsion on electron 1 at point r', I would need to calculate the sum of all the electron repulsion of electron 2 at each point in space, so it becomes a non-local interaction and depends on the shape of the wavefunction for the other electron.

Contrast that with the harmonic oscillator V=$$1/2kx^2$$. If I want to know the potential energy interaction of the partical at point x', I just need to evaluate V(x).

Does that help?

Let me make some complements. What James Johns said about Helium atom is a quite a good example.

Basically, if we are capable of solving the many body Schrodinger equation directly, then no non-local potential appears. We have no non-local interaction in the nature. However, if we want some effective models or make some approximations, sometimes we have non-local potential.

The key point is that, for usual local potential, no matter how the particle behaves, the interaction never changes. But in the case of non-local potential, the behavior of the particle itself have back effect on the interaction.

For instance, still the Helium atom, if we solve the three body alpha+2electrons directly, then no way to confront a non-local potential. But if we construct an equivalent two body model about one single electron, then we will face a non-local potential.