  We consider the rationally extended harmonic oscillator potential which is
isospectral to the conventional one and whose solutions are associated with the
exceptional, $X_m$- Hermite polynomials and discuss its various important
properties for different even codimension of $m$. The uncertainty relations are
obtained for different $m$ and it is shown that for the ground state, the
uncertainity increases as $m$ increases. A one parameter $(\lambda)$ family of
exactly solvable isospectral potential corresponding to this extended harmonic
oscillator potential is obtained. Special cases corresponding to the
$\lambda=0$ and $\lambda = -1$, which give the Pursey and the Abhram-Moses
potentials respectively, are discussed. The uncertainty relations for the
entire isospectral family of potentials for different $m$ and $\lambda$ are
also calculated.
