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I was interested in making what I thought would be a simple simulation of an electron encountering a positron by numerically solving the Schrodinger equation over several time steps, but I've run into 1 problem: the potential used in most textbooks (to solve the hydrogen atom for example) are still of a classical point particle, rather than a wavefunction. How do I generate the potential field from two real, moving and interacting wavefunctions?

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The electron and positron system can be denoted by the full wavefunction $\psi(\vec{r}_1,\vec{r}_2,t)$ and its evolution is determined by the Hamiltonian (can be found from the classical energy):

$$\mathcal{H} = -\frac{\hbar^2}{2m}\nabla_1^2 -\frac{\hbar^2}{2m}\nabla_2^2 - \frac{e^2}{4\pi\epsilon_0}\frac{1}{|\vec{r}_1-\vec{r}_2|} \tag{1}$$

So, for two particles interacting in 1D, you only need to solve the following differential equation for wavefunction $\psi(x_1,x_2,t)$:

$$i\hbar\frac{\partial}{\partial t}\psi(x_1,x_2,t) = (-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_1^2} -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x_2^2} - \frac{e^2}{4\pi\epsilon_0}\frac{1}{|x_1-x_2|})\psi(x_1,x_2,t) \tag{2}$$

Then you can use your favourite numerical scheme to solve this "2D" partial differential equation.

You should notice that your need quadratic number of grid points $N^2$ for this two particles numerical solution. For the full wavefunction $\psi(\vec{r}_1,\vec{r}_2,t)$, you need $N^6$ grid points which is not practical at all. It simply means that the position based method to find the wavefunction is not a good appoarch.

Edit: Regarding your question of point particle/wavefunction, you can interpret the equation (2) in the following way: The particles are still "interacting classically" with the electric attraction. If one of the particle is like a point charge confined very locally, then this point charge is actually interacting with another wavefunction over the whole space with probability amplitude $\psi(x_1,x_2,t)$.

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If you look at hwlau answer, you see that you can change coordinates from x1,x2 to the center of mass and relative coordinate. Just like you do for the Hydrogen atom. If you work in the system of coordinates where the COM is stationary, the solution will be simple - identical to the Hydrogen atom, in fact.

My guess is, however, that this equation will not contain all the interesting physics, because there is no way for the two particles to annihilate and emit photons. To see that you will need to work in 2nd quantization, I guess.

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