Axiomatic Foundation of Quantum Measurements and Survival Effect

The axiomatic theory of quantum first-kind measurements is developed in a rigorous form based on five Postulates. The measurement theory for an observable with a continuous spectrum is given in a rigged Hilbert space. This approach also describes measurements with non-ideal initial conditions. It yields the survival effect in the position measurement of particles. It is also found that there is no such survival effect in the momentum measurement of particles. These Postulates of axiomatic theory yield the survival effect, which violates the Heisenberg uncertainty relation. This theoretical result is demonstrated by the wave function with a minimum of position and momentum uncertainty of the particle. The survival effect leads to essential corrections to the uncertainty relations. These modified uncertainty relations can also be used for the experimental verification of the survival measurement effect.