The wave properties of particles are expressed in terms of a complex wave function, described by Arthur Beiser as follows:
The variable quantity characterizing de Brogiie waves is called the wave function, denoted by the symbol Ψ (the Greek letter psi). The value of the wave function associated with a moving body at the particular point x, y, z in space at the time ( is related to the likelihood of finding the body there at the time. Ψ itself, however, has no direct physical significance. There is a simple reason why Ψ cannot he interpreted in terms of an experiment. The probability P that something be somewhere at a given time can have any value between two limits: 0, corresponding to the certainty of its absence, and 1, corresponding to the certainty of its presence. (A probability of 0.2, for instance, signifies a 20 percent chance of finding the body.) But the amplitude of any wave may be negative as well as positive, and a negative probability is meaningless. Hence Ψ itself cannot be an observable quantity.
This objection does not apply to |Ψ|^2 the square of the absolute value of the wave function. For this and other reasons |Ψ|^2 is known as probability density. The probability of experimentally finding the laxly described by the wave function Ψ at the point x, y, z at the time t is proportional to the value of |Ψ |^2 there at t. A large value of |Ψ|^2 means the strong possibility of the body's presence, while a small value of |Ψ|^2 means the slight possibility of its presence. As long as |Ψ|^2 is not actually 0 somewhere, however, there is a definite chance, however small, of delecting it there. This interpretation was first made by Max Born in 1926.
There is a big difference between the probability of an event and the event itself, Although we shall speak of the wave function Ψ that describes a particle as being spread out in space, this does not mean that the particle itself is thus spread out. When an experiment is performed to detect electrons, for instance, a whole electron is either found at a certain time and place or it is not; there is no such thing as 20 percent of an electron. However, it is entirely possible for there to be a 20 percent chance that the electron be found at that time and place, and it is this likelihood that is specified by |Ψ|^2 .
Alternatively, if an experiment involves a great many identical bodies all described by the same wave function Ψ , the actual density of bodies at x, y, z at the time t is proportional to the corresponding value of |Ψ|^2 .
While the wavelength of the de Brogiie waves associated with a moving body is given by the simple formula
λ= h/(mv)
determining their amplitude Ψ as a function of position and time usually presents a formidable problem. We shall discuss the calculation of Ψ in Chap. 5 and then go on to apply the ideas developed there to the structure of the atom in Chap. 6. Until then we shall assume that we have whatever knowledge of Ψ is required by the situation at hand.
In the event that a wave function Ψ
is complex, with both real and i in aginary
parts, the probability density is given by the product Ψ* Ψ of Ψ
and its complex
conjugate Ψ*
. The complex conjugate of any function is obtained by replacing
i (√-1) by -i wherever it appears in the function. Every complex function Ψ
can be written in the form
Ψ = A + iB
where A and B are real functions. The complex conjugate Ψ*
of Ψ is
Ψ* = A - iB
and so
Ψ* Ψ = A^2 - i^2 B^2 = A^2 + B^2
since i^2 = -1. Hence Ψ* Ψ is always a positive real quantity.
end of quote
As I have shown here:
eijms.com/index.php/ms/article/view/51/75
the probability density need not be calculated from complex numbers at all