Suppose I give you $2^N$ functions that are eigenvectors of a fermionic $H$. How do I determine which function describes which spin configuration?

Question

Consider the hamiltonian
$$
H = - frac{1}{2} nabla^2 + V.
$$
The potential $V : (mathbb{R^3})^N to mathbb{R}$ is symmetric, so for each eigenvalue, there is an antisymmetric eigenvector. There is not any spin operator inside $V$. It can be assumed that the hamiltonian describes a fermionic many-body problem.
Suppose I give you $2^N$ functions $f_i : (mathbb{R}^{3})^N to mathbb{C}$ for $i = 1,2,3,cdots,2^N$ that are square-integrable and eigenvectors of $H$ with eigenvalue $E_i$. If $sigma_j in {1/2,-1/2}$ for each $j in {1,2,3,cdots, 2^N}$; how would you find out which $f_i$ corresponds to the spin configuration $sigma_1sigma_2cdotssigma_N$?

Prof. Legolasov · Answer

(I'm improvising here -- please treat what follows with a degree of healthy skepticism)
So lets say your wavefunction is a multiplet of $2^N$ components -- functions over $mathbb{R}^{3N}$, with suitable integration condition imposed.
Your Hamiltonian $H$ acts on each component individually, and its action on differnt components is the same, by construction.
The spin operators (read, the operators that correspond to projections of the physical spin on some fixed axis) for any of the $N$ particles only shuffle the components and don't change any of the functions.
Therefore, the spin operator commutes with the Hamiltonian.
The two operators must therefore have joint eigenvalues.
Which means that your problem is essentially unsolvable, unless you allow the spin operator to enter $H$ as e.g. in the Pauli equation.

Suppose I give you $2^N$ functions that are eigenvectors of a fermionic $H$. How do I determine which function describes which spin configuration?

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