Natural Science Forum / Physics / Particle Physics / September 2004

The general wavefunction for a free particle is:

Y (x,t) = A cos( kx - wt) + iA sin (kx - wt)

and the complex conjugate is:

Y*(x,t) = A cos( kx - wt) - iA sin (kx - wt)

These are multiplied together to give,along with a proportionality
constant,
the probability of finding a particle at a particular place at a
particular time.It is not known why this should be so.Here is a
suggestion:

If the wavefunction  represents the sum of the square roots of two
other wavefunctions before it is  squared,
then multiplying it by another wavefunction  - the complex conjugate -
gives a probability.Writing:

Y1^1/2 = [A cos (kx - wt)], Y2^1/2 = [iA sin (kx -wt)]
and

Y1 + Y2 =  A^2 cos (kx -wt)^2 - A^2 sin (kx - wt)^2 = A^2 [1 - 2 sin
(kx -wt)]^2

Y1 - Y2 =  A^2 cos (kx -wt)^2 + A^2 sin (kx - wt)^2 = A^2

These combinations of wavefunctions happen for two bonded particles of
different mass e.g a hydrogen chloride molecule.
The second of these wavefunction combinations is constant.
So the probability of finding a particle that "bonds" (like an
electron in
hydrogen chloride)at a given distance at a given time is constant -
the particle is equally likely to be everywhere.This is not the case
where the wavefunctions were added.The question arises:is the original
particle with
wavefunction Y really a combination of at least three other particles?

Actually, the squared amplitude of the wave function defines a
probability density function.  The probability of locating a particle
at any given point is not well defined, but the probability of
locating it in a particular finite region is obtained by integrating
it's squared amplitude over the volume of the region.  In general, a
wave function has two parts that differ by in phase by pi/2, hence the
sine and cosine parts.  But it's still only one function, having one
real squared amplitude at any given point, namely Y Y*.