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EXPECTATION VALUES OF DYNAMICAL QUANTITIES

According to Born the wave function Ψ has probabilistic interpretation, therefore it is essential to calculate the average or expectation value of any dynamical quantity defined by the wave function. In physics such dynamical quantities are space-coordinates, momentum and energy of the system.

The average or expectation value of a dynamical quantity is the mathematical
expectation for the result of a single measurement.

Or

It may be defined as the average of the result of a large number of measurements on
independent systems.

The expectation value of any observable f, for normalised wave function may be
written as