Electronic wavefunction and probability density function

Since an electron has wave character, we can describe its motion with a wave equation, as we do in classical mechanics for the motions of a water wave or a stretched string or a drum. If the system is one-dimensional, the classical wave equation is 

A typical wavefunction, or a solution of the wave equation, is the familiar sine or cosine function. For example, we can have 

It can be easily verified that P(x, t) satisfies eq. (1.3.1). An important point to keep in mind is that, in classical mechanics, the wavefunction is an amplitude function. As we shall see later, in quantum mechanics, the electronic wavefunction has a different role to play. 

Combining the wave nature of matter and the probability concept of the Uncertainty Principle, M. Born proposed that the electronic wavefunction is no longer an amplitude function. Rather, it is a measure of the probability of an event when the function has a large (absolute) value, the probability for the event is large. An example of such an event is given below. 

From the Uncertainty Principle, we no longer speak of the exact position of an electron. Instead, the electron position is defined by a probability density function. If this function is called p (x,y,z), then the electron is most likely found in the region where p has the greatest value. In fact, p dr is the probability of finding the electron in the volume element dr (= dxdydz) surrounding the point (x,y,z). Note that p has the unit of volume-1, and pdr, being a probability, is dimensionless. If we call the electronic wavefunction f, Born asserted that the probability density function p is simply the absolute square of tjr. 

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