Physical Meaning of the Wave Function

We are ultimately trying to use the wave function Ψ(x, t) to describe the behavior of an electron in a crystal. The function Ψ (x, t) is 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.

(1)

Since the total wave function Ψ(x, t) is a complex function. it cannot by itself represent a real physical quantity. Max Born postulated in 1926 that the function |Ψ(x, t)|² dx is the probability of finding the particle between x and x + dx at a given time, or that |Ψ(x, t)|² is a probability density function. We have that

(2)

where Ψ*(x, t) is the complex conjugate function. Therefore

Then the product of the total wave function and its complex conjugate is given by

(3)

Therefore, we have that

(4)

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 particle 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.