An approximate definition of the states and would
make the states and only approximate
energy eigenstates. But they can be made exact energy eigenfunctions
by defining and to
be the exact symmetric ground state and the exact antisymmetric state
of second lowest energy. The precise
function and can then be reconstructed from that.
Note that and themselves are not energy eigenstates, though they might be so by approximation. The errors in this approximation, even if small, will produce the wrong result for the time evolution. (The small differences in energy drive the nontrivial part of the unsteady evolution.)