Probability and the Uncertainty Principle
The fact is that such quantities as the position and momentum of an electron do not exist apart from a particular uncertainty. The magnitude of this inherent uncertainty is described by the Heisenberg uncertainty principle.
Probability and the uncertainty principle:
In any measurement of the position and momentum of a particle, the uncertainties in the two measured quantities will be related by
Similarly, the uncertainties in an energy measurement will be related to the uncertainty in the time at which the measurement was made by
These limitations indicate that simultaneous measurement of position and momentum or of energy and time are inherently inaccurate to some degree.
Of course, Planck's constant h is a rather small number (6.63 X 10-34 J-s), and we are not concerned with this inaccuracy in the measurement of x and px for a truck, for example On the other hand, measurements of the position of an electron and its speed are seriously limited by the uncertainty principle.
One implication of the uncertainty principle is that we cannot properly speak of the position of an electron, for example, but must look for the "probability" of finding an electron at a certain position.
Thus one of the important results of quantum mechanics is that a probability density function can be obtained for a particle in a certain environment, and this function can be used to find the expectation value of important quantities such as position, momentum, and energy.
We are familiar with the methods for calculating discrete (single-valued) probabilities from common experience.
Given a probability density function P(x) for a one-dimensional problem, the probability of finding the particle in a range from x to x dx is P(x)dx.
Since the particle will be somewhere, this definition implies that
if the function P(x) is properly chosen. The above equation is implied by stating that the function P(x) is normalized.
To find the average value of a function of x, we need only multiply the value of that function in each increment dx by the probability of finding the particle in that dx and sum over all x.
Thus the average value of f(x) is
If the probability density function is not normalized, this equation should be written