# Simplest fermionic normalized quantum many-particle wavefunction in position representation

What is the simplest fermionic normalized quantum many-particle wavefunction, expressed in the first-quantized position representation, that you can think of? The normal single-particle examples don't seem to be that simple: Slater determinant of Gaussians times Hermite polynomials for harmonic oscillator, the free particle is not normalizable, and the infinite square well doesn't seem any better.

I understand simple in a fairly intuitive way: easy to manipulate, possible to differentiate and integrate, not too many symbols etc..

The best candidate I can think of (which is still not that good according to my example criteria above) is a single full Laundau level for a charged particle in a magnetic field, expressed as

$\displaystyle\Psi(z_1,z_2,\ldots,z_N)=\big(4\pi(N-1)\big)^{-N(N-1)/2}\prod_{i<j}^N(z_i-z_j)e^{-\frac{1}{4}\sum_{i=1}^N|z_i|^2}\ ,$

in terms of complex coordinates $z=x+iy$ and with physical constants set to 1 (normalization from Wikipedia).

Do you know a simpler one?

I'm asking because I would like a simple one to test out concepts and understanding.

• Define simple. $\lvert N \rangle = {a^\dagger}^N \lvert 0 \rangle$ for any bosonic creation operator $a^\dagger$ looks pretty simple to me ;) – ACuriousMind Jan 19 '15 at 14:43
• Good point, I'll edit the question. – jorgen Jan 19 '15 at 14:45
• Still, I think simple is left undefined (do you mean "has only computable terms"? Do you mean "has the fewest amount of symbols"?). Anyway, here's an algorithm for getting the simplest many-particle state: Take the simplest bosonic one-particle state you know, $\psi_1(x)$, and write $\psi_N(x_1,\dots,x_N) = \prod_{i = 1}^N \psi_1(x_i)$. – ACuriousMind Jan 19 '15 at 14:55
• Haha forgot about bosons, that solves it. I guess I mean simple in a fairly intuitive way, not completely well-defined. But I still think the question can be meaningful; although we don't have hard arguments for why one is simpler than another one usually have a feeling about it and that is as rigorous as required for this question I think (something like easy to manipulate, doesn't have too many symbols, possible to differentiate and integrate etc.. I realize my example is not too simple in the last two regards but that's the best I could think of; hence the question). – jorgen Jan 19 '15 at 15:10