Physical Meaning of the Schrodinger's Wave Function

We are ultimately trying to use the wave function to describe the behavior of
an electron in a crystal. The function a wave function. so it is reasonable to
ask what the relation is between the function and the electron. The total wave function
is the product of the position-dependent, or time-independent, function and the
time-dependent function. We have from Equation (2.7) that
Since the total wave function is a complex function. it cannot by itself represent a real physical quantity. Max Born postulated in 1926 that the function| |²dx is the probability of finding the particle between .x and x + dx at a given time, or that| |² is a probability density function. We have that

is the probability density function and is independent of time. One major difference
between classical and quantum mechanics is that in classical mechanics, the position
of a particle or body can be determined precisely, whereas in quantum mechanics,
the position of a partical is found in terms of a probability. We will determine the
probability density function for several examples, and, since this property is independent
of time. we will, in general, only be concerned with the time-independent
wave function.

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