introduction-weak-interaction-volume-one

'wave functio n
at ( 1.3 .9 )
P —4 grad 3 (1 .3 .10 )
We now write down the four so-called 'harmonic' or plane wave
solutions for the Schrddinger wave function '1 (22) :
cos (kx -- ft) , (1 .3 .11 )
sin (kx - ft) , (1 .3 .12 )
ej(kx-ft),
(1 .3 .13 )
e j(kx-ft) . (1 .3 .14 )
1 .4 Interpretation of the Wave Function . (23)
In classical physics, the value of a wave function represents an
associated physical parameter . For example, in the case of waves in air, i t
represents the displacement of the air particles . In 1926 Born suggested (24 )
that the value of
(x, t) for the de Broglie wave of a particle corresponds
to the probability that the particle will be at a point x on the x-axis at a
given time t . However, a probability must be real and positive, but values o f
4T(x, t) are, in general, complex . Hence Born suggested that the probability
should be the product of the value of the wave function and its complex conjugate ,
so that the probability of finding the particle between x and x + dx is given by
P(x, t) dx . ^1iI*(x, t) ' T(x, t) dx . (1 .4 .1 )
This so-called 'position probability density' may readily be extended int o
three dimensions :
P(x, y, z, t) dx dy dz = x, y, z, t) S'(x, y, z, t) dx dy dz . (1 .4 .2 )
Let E be the volume of the infinitesimal element of space dx dy dz .
Since i t
is a certainty that the particle must exist somewhere in space ,
P(G , t) de = 1, (1 .4 .3 )
henc e
+,
T *(E , t) 'Is(e , t) dE = 1 . (1 .4 .4 )
_ oo v
Thus, in order to normalize a given solution to the Schrddinger equation, we
multiply it by its complex conjugate and integrate over all space, obtaining a
real number N . By dividing both the wave function and its complex conjugate by