| |
 A schematic drawing
of our experiment
is shown in
Fig. (A).
An electron beam
comes from the distant left,
passes through
two slits,
and
 ,
and
exposes the photographic film
on the screen F.
A rough sketch
(sectioned drawing) of
the wave functions in the
above experiment
Fig. (A)
is shown in
Fig. (B).
The wave function would be
separated into two
spherical waves
(precisely speaking,
spherical-wave-like waves)
after passing the slits
and they are superposed
to interfere to each other.
Now let us execute
the double-slit experiment
as shown in
Fig. (A)
on a PC.
It is assumed
that an electron beam
(cathode rays)
coming from the distant
left to pass
through the two slits,
and
,
and exposes the photographic film F.
Suppose the intensity
of the beam, which is
the number of electrons
passing through a unit
cross section
par a unit time,
be enough weak to avoid
a meaningless interaction
between electrons.
The sectioned drawing
of the feature of
the wave functions
is shown in
Fig. (B).
Since the incident beam
is free particles,
the corresponding wave
is a (right-going) plane wave.
After passing through
the two slits,
it separates into two waves,
and
 ,
each of which is rather similar
to a spherical wave.
(Because the slits are
of rectangular form,
they might be similar
to a cylindrical wave.)
After passing through the slits,
the total wave function
would be
a superposition of these waves,
and
,
as
 ,
and this wave function
reaches
the film F on the screen.
These wave functions
are of a constant energy,
i.e. of a constant frequency.
Accordingly, they
are described as
stationary states.
The functional forms
of the wave functions
and
are approximately
plane wave
or
spherical wave
These functional forms
are explained
on the page,
2-2-A: Plane Wave and Spherical Wave.
You may understand
the forms of
the wave function
in the double-slit experiment
shown in
Fig. (A)
and
(B).
Using the result
we can approximately calculate
the probability
that an electron will be
found at any point on
the film F.
When an electron hits
some point on the film,
an image (a mark)
is made at that point.
If the wave function
is a plane wave,
the probability is uniformly
distributed everywhere.
When electrons are repeatedly
ejected many times,
the distribution
of their images would be
completely random.
Generating uniform
random numbers
with a computer program,
we can make such
a random distribution of images
on a PC.
Multiplying the probability
to the uniformly random distribution,
we can make a picture
of a distribution of images
or prints of electrons
hit on the film F.
The photograph-like picture
below
was made in this way.
Assuming the number
of the incident electrons
in a second (s)
to be about 10,
we made pictures of
the electron distribution
varying the "exposure time"
from 1 s
to 256 s.
While, for a very short
"exposure time",
the distribution of the images
is very sparse,
it would become dense
for a longer exposure time
and the interference fringes
would be seen
clearly.
This is an accurate simulation
of the interference pattern
of the electrons
and it conforms
to the time-dependent
behavior observed
in actual experiments.
Looking at these results,
you may understand that
the particle nature
and the wave nature
of electrons are beautifully
mixed and unified
within quantum mechanics.
Although the interference
fringes in the following
picture
have been made
by numerical calculations
using quantum mechanics,
a very similar photograph
was taken in an actual experiment
by A. Tonomura
(Hitachi, Japan)
(1989).
|
Top of Part 2
Last page
Next page
Top