  Specific heat per particle ($c_v$) of an ideal gas, in many occasions, is
interpreted as energy fluctuation per particle ($\triangle\epsilon^2$) of the
ideal gas through the relation: $\triangle\epsilon^2=kT^2c_v$, where $k$ is the
Boltzmann constant and $T$ is the temperature. This relationship is true only
in the classical limit, and deviates significantly in the quantum degenerate
regime. We have analytically explored quantum to classical crossover of this
relationship, in particular, for 3-D free Bose and Fermi gases. We also have
explored the same for harmonically trapped cases. We have obtained a hump of
$\triangle\epsilon^2/kT^2c_v^{(\text{cl})}$ around the condensation point for
3-D harmonically trapped Bose gas. We have discussed the possibility of
occurring phase transition with discontinuity of heat capacity from existence
of such a hump for other Bose and Fermi systems.
