An eigenstate is a state of a quantum system in which a physical quantity, such as position or momentum, has a definite value. It is represented by a wave function that is a solution to the Schrödinger equation.
Uncertainty is important for an eigenstate because it allows us to understand the limitations of our knowledge about the state of a quantum system. Even in an eigenstate, there is a degree of uncertainty in the measurement of physical quantities.
The uncertainty for an eigenstate is calculated using the Heisenberg uncertainty principle, which states that the product of the uncertainties in the measurement of position and momentum must be greater than or equal to a constant value (equal to Planck's constant divided by 4π).
No, the uncertainty of an eigenstate cannot be reduced. This is a fundamental limitation of quantum mechanics and is a consequence of the Heisenberg uncertainty principle. Even with perfect measurements, there will always be a degree of uncertainty in the measurement of physical quantities in an eigenstate.
The uncertainty of an eigenstate is different from that of a non-eigenstate because in an eigenstate, the measurement of a physical quantity will always yield a definite value. In a non-eigenstate, the measurement of a physical quantity will yield a range of possible values, and the uncertainty is the measure of this range.