what physical significance of operator and eigen value in quantam mechanics.?

Answer 1

The operator replaces the simple variable in classical mechanics. For instance momentum in QM becomes the gradient operator instead of the product of mass and velocity. This happens because QM has discreet states represented by probability density functions (wave functions). Now the eignevalues are the observables of the system. In other words, the eigenvalues indicate teh qunatities you can measure, or observe.

Answer 2

An hermitian operator represents an observation that can potentially be made in the laboratory. The eigenvalues of that operator are the permitted results of the experiment.

There are nonhermitian operators used in QM as well (such as the creation/annihilation operators) but these do not represent observable quantities.