Uncertainty principle and probability concept

Another important development in quantum mechanics is the Uncertainty Principle set forth by W. Heisenberg in 1927. In its simplest terms, this principle says, The position and momentum of a particle cannot be simultaneously and precisely determined. Quantitatively, the product of the uncertainty in the x component of the momentum vector (Apx) and the uncertainty in thex direction of the particle s position (Ax) is on the order of Planck s constant  

While h is quite small in the macroscopic world, it is not at all insignificant when the particle under consideration is of subatomic scale. Let us use an actual example to illustrate this point. Suppose the Ax of an electron is 10-14 m, or 0.01 pm. Then, with eq. (1.2.1), we get Apx = 5.27 x 10-21 kg m s-1. This uncertainty in momentum would be quite small in the macroscopic world. However, for subatomic particles such as an electron, with mass of 9.11 x 10-31 kg, such an uncertainty would not be negligible at all. Hence, on the basis of the Uncertainty Principle, we can no longer say that an electron is precisely located at this point with an exactly known velocity. It should be stressed that the uncertainties we are discussing here have nothing to do with the imperfection of the measuring instruments. Rather, they are inherent indeterminacies. If we recall the Bohr theory of the hydrogen atom, we find that both the radius of the orbit and the velocity of the electron can be precisely calculated. Hence the Bohr results violate the Uncertainty Principle. 

With the acceptance of uncertainty at the atomic level, we are forced to speak in terms of probability we say the probability of finding the electron within 

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