  In this study, we reveal the difference between Woods-Saxon (WS) and
Generalized Symmetric Woods-Saxon (GSWS) potentials in order to describe the
physical properties of a nucleon, by means of solving Schr\"odinger eq. for the
two potentials. The additional term squeezes the WS potential well, which leads
an upward shift in the spectrum, resulting in a more realistic picture. The
resulting GSWS potential does not merely accommodate extra quasi bound states,
but also has modified bound state spectrum. As an application, we apply the
formalism to a real problem, an $\alpha$ particle confined in Bohrium-$270$
nucleus. The thermodynamic functions Helmholtz energy, entropy, internal
energy, specific heat of the system are calculated and compared for both wells.
The internal energy and the specific heat capacity increase as a result of
upward shift in the spectrum. The shift of the Helmholtz free energy is a
direct consequence of the shift of the spectrum. The entropy decreases because
of a decrement in the number of available states.
