Fermionic interaction potentials

Question

Are there any examples of fermionic particles or quasiparticles for which the interaction potential is a globally smooth function? i.e. no singularities or branch points.

As an example, in Flügge's Practical Quantum Mechanics, problem 148 has two repulsive particles on a circle. This is supposed to model the two helium electrons in the ground state. The equation he gives is

$$ -\frac{\hbar^2}{2mr^2}\left(\frac{\partial^2 \psi}{\partial x_1^2}+\frac{\partial^2 \psi}{\partial x_2^2}\right)+V_0\cos(x_1-x_2)\psi=E\psi$$

I don't quite follow why this potential does not have a singularity when $x_2\rightarrow x_1$. Are there other such examples?

Answer 1

Well there's no particular reason for a textbook problem to actually model a physical system... But one can certainly write something like this as a completely valid approximation. Take Flugge's example, with He3 so its fermionic[fn.2]. Say the size of the atoms is very small, much smaller than the scales on which the ground state wavefunction varies, which is reasonable enough.

Now there should really be a term $V_{repulse}(x_1-x_2)\psi$ where $V_{repulse}$ gets really big when $|x_1-x_2|\rightarrow 0$ to capture the fact that you can't put the two atoms on top of each other [fn.2]. But this is going to be really short ranged, almost zero if $|x_1-x_2|$ is significantly bigger then the size of the atom. On the other hand we know that $\psi(x_1,x_2)\rightarrow 0$ when $x_1 \rightarrow x_2$. So in precisely the region where $V_{repulse}$ would matter, $\psi$ is basically zero. So we can basically ignore $V_{repulse}\psi$. More exactly, the term is proportional to (size of atom)/(size of circle) squared, which could be very small.

So you don't always have to include a repulsive term. It can actually be quite negligible, even though it seems like a fact you can't ignore.

[fn. 1] There are also times when you can ignore the repulsive interactions of bosons, although its not suppressed like the fermions.

[fn. 2] Its not really true that it should diverge as $x_1\rightarrow x_2$. If you really got the two atoms on top of each they would stop behaving like pointlike atoms, so your model would stop being applicable, rather than anything going to infinity.

Answer 2

Apparently the Gaussian Effective Potential is used a fair amount. This would be something like $$V(x_1,x_2)=V_0 \exp(-\alpha(x_1-x_2)))$$. Thanks for the responses.

Answer 3

In atomic physics effective 1D soft-core Coulomb potentials are routinely used for the interaction between particles, particularly when external fields are present. For example, for the interaction between an electron and a (space-fixed) proton at the center of coordinates: $$V(x) = - \frac{1}{\sqrt{x^2 + \epsilon^2}}$$ where atomic units are used and $\epsilon$ is usual fitted or taken as one. It is much simpler to integrate the time-dependent Schrödinger equation for this potential.