  Physical potentials are routinely approximated to harmonic potentials so as
to analytically solve the system dynamics. Often it is important to know when a
quantum harmonic oscillator (QHO) behaves quantum mechanically and when
classically. Recently Su et. al. [Phys. Rev. A {\bf 85}, 052126 (2012)] have
theoretically shown that QHO exhibits quantum contextuality (QC) for a certain
set of pseudospin observables. In this work, we encode the four eigenstates of
a QHO onto four Zeeman product states of a pair of spin-1/2 nuclei. Using the
techniques of NMR quantum information processing, we then demonstrate the
violation of a state-dependent inequality arising from the noncontextual hidden
variable model, under specific experimental arrangements. We also
experimentally demonstrate the violation of a state-independent inequality by
thermal equilibrium states of nuclear spins, thereby assessing their
quantumness.
