  For finite quantum many-particle systems modeled with say $m$ fermions in $N$
single particle states and interacting with $k$-body interactions ($k \leq m$),
the wavefunction structure is studied using random matrix theory. Hamiltonian
for the system is chosen to be $H=H_0(t) + \lambda V(k)$ with the unperturbed
$H_0(t)$ Hamiltonian being a $t$-body operator and $V(k)$ a $k$-body operator
with interaction strength $\lambda$. Representing $H_0(t)$ and $V(k)$ by
independent Gaussian orthogonal ensembles (GOE) of random matrices in $t$ and
$k$ fermion spaces respectively, first four moments, in $m$-fermion spaces, of
the strength functions $F_\kappa(E)$ are derived; strength functions contain
all the information about wavefunction structure. With $E$ denoting the $H$
energies or eigenvalues and $\kappa$ denoting unperturbed basis states with
energy $E_\kappa$, the $F_\kappa(E)$ give the spreading of the $\kappa$ states
over the eigenstates $E$. It is shown that the first four moments of
$F_\kappa(E)$ are essentially same as that of the conditional $q$-normal
distribution given in: P.J. Szabowski, Electronic Journal of Probability {\bf
15}, 1296 (2010). This naturally gives asymmetry in $F_\kappa(E)$ with respect
to $E$ as $E_\kappa$ increases and also the peak value changes with $E_\kappa$.
Thus, the wavefunction structure in quantum many-fermion systems with $k$-body
interactions follows in general the conditional $q$-normal distribution.
