  The value of quarkonium wave function at the origin is an important quantity
while studying many physical problems concerning a heavy quarkonium. This is
because that it is widely used to evaluate the production and decay amplitudes
of the heavy quarkonium within the effective filed theory framework, e.g., the
non-relativistic QCD (NRQCD). In this paper, the value of the Schr${\rm
\ddot{o}}$dinger radial wave function or its first nonvanishing derivative at
zero quark-antiquark separation, i.e., $|(|c\bar{c})[n]\rangle$-,
$|(|b\bar{c})[n]\rangle$-, and $|(b\bar{b})[n]\rangle$-quarkonium, have been
tabulated under five potential models with new parameters of the heavy
quarkonium. Moreover, the production of the lower-level Fock states
$|(b\bar{Q})[1S]\rangle$ and $|(b\bar{Q})[1P]\rangle$, together with the higher
excited Fock states $|(b\bar{Q})[nS]\rangle$ and $|(b\bar{Q})[nP]\rangle$ ($Q$
stands for $c$- or $b$-quark; $n=2,\cdots,6 $) through top quark decays have
been studied with the new values of heavy quarkonium wave functions at the
origin under the framework of NRQCD. At the LHC with the luminosity ${\cal
L}\propto 10^{34}cm^{-2}s^{-1}$ and the center-of-mass energy $\sqrt{S}=14$
TeV, sizable heavy quarkonium events can be produced through top quark decays,
i.e., $4 \times10^5$ $B_c$ and $B^*_c$, and $2 \times10^4$ $\eta_b$ and
$\Upsilon$ events per year can be obtained according to our calculation.
