  Recent lattice QCD results for the low-lying odd-parity excitations of the
nucleon near the $N^{*}(1535)$ and $N^{*}(1650)$ resonance positions have
revealed that the lattice QCD states have magnetic moments consistent with
predictions from a constituent-quark-model. Using Hamiltonian Effective Field
Theory (HEFT) to describe pion-nucleon scattering in the $I(J^{P}) =
\frac{1}{2}(\frac{1}{2}^{-})$ channel, we represent these two quark-model like
states as two single-particle bare basis states, dressed and mixed by
meson-baryon scattering channels. By constraining the free parameters of the
Hamiltonian with $S_{11}$ pion-nucleon scattering data, we perform the first
calculation of the finite-volume spectrum using two bare-baryon basis states.
By comparing this spectrum to contemporary lattice QCD results at three lattice
volumes, we analyse the eigenvectors of the Hamiltonian to gain insight into
the structure and composition of these two low-lying resonances. We find that
an interpretation of the two low-lying nucleon resonances as quark-model like
states dressed by meson-baryon interactions is consistent with both the
$S_{11}$ scattering data and lattice QCD. We introduce a novel HEFT formalism
for estimating scattering-state contaminations in lattice QCD correlation
functions constructed with standard three-quark operators. Not only are
historical lattice QCD results described with excellent accuracy, but
correlation functions with large scattering-state contaminations are
identified.
