Foundation for the statistical treatment of Matter
Kevin Gibson
January 10, 2011
Motivation
Quantum Theory has proven a useful predictive tool for working with matter. However
an issue that most often remains unattended is the core meaning behind the theory.
Instead of having a foundational theory to guide the development, various philosophical
points of view are put forth such as wave-particle duality or the many worlds
interpretation. In this piece the notion of matter's wave-particle duality will be rejected
in favor of a more statistical viewpoint. This piece will begin with a brief introduction
of probabilities before leading into the behavior of matter in general. The results will be
familiar to anyone who studies Quantum Theory.
Basic statistics with probabilities
To begin with consider that a series of N measurements. Let A represents a potential
discrete outcome. The outcomes resulting in A can be expressed in a the proportion of
all measurements. As the size of the measurement sample goes to โ then this proportion
would approach a number that can be called the
probability
of outcome A.
P
๎
A
๎=
lim
N
๎ โ
N
A
N
(1)
This same line of thought can be extended to cases where one wants both conditions A
and B to be satisfied.
P
๎
A
and
B
๎=
lim
N
๎โ
N
AB
N
(2)
Likewise one can speak of the probability that either A or B will be realized. However
one must use caution in using the definition set up in (1) to not over-count the cases
where both A and B are realized.
P
๎
A
or
B
๎=
lim
N
๎โ
N
A
๎
N
B
โ
N
AB
N
P
๎
A
or
B
๎=
P
๎
A
๎๎
P
๎
B
๎โ
P
๎
A
and
B
๎
(3)
The two-slit experiment
Consider a beam consisting of bits of some form of matter passing through a barrier with
two narrow slits placed close by each other. The beam that passes through the slits is
then projected onto a screen as shown in the figure on the following page.
What would one expect the observed pattern to be with regards to where the bits of
1.
matter will end up on the screen? We'll draw upon (3) to answer this question because
we presume that the matter will hit a point on the screen after having passed through one
slit or another but not both. Moreover the positions that the matter can be found on the
screen will lie on a continuum of possibilities rather than a discrete set. Therefore
probability densities
ฯ
need to be employed rather than probabilities themselves.
๎๎
x
๎=๎
1
๎
x
๎๎๎
2
๎
x
๎
(4)
Equation (4) suggest a bimodal distribution on the screen, one distribution for each slit.
The trouble however is that it does not always work. Sometimes the distribution has
multiple regions of high probability, and this is something that (4) cannot deliver.
What went wrong
Without intending to, an assumption was made in obtaining (4) for application to the
two-slit experiment, and this is that the physics working on our matter is local. In other
words the measurement in one part of the experiment does not simultaneously alter the
rest of the experiment. Indeed when we stated earlier that the distribution of
measurements would be bimodal, we inherently assumed a sum of distributions that
would be the same independent of whether any measurement at the slits had been made.
In this vein we could take the probabilities that the matter would reach a portion of the
screen assumed that we knew which slit it passed through and use these even when such
a measurement had not in fact taken place. The task is now to reconstruct the statistics
without making this assumption. At the same time making sure that our modified
statistical treatment can account for the outcomes of the above experiment.
2.
Beam
s
c
r
e
e
n
In developing a new statistic a few criteria must be insisted upon.
โ
We still want to be able to use the probabilities via the two slits as before.
โ
The equation cannot produce negative probabilities.
โ
Equation (4) can still be a valid equation in a special case, so we must allow for it
to be a possible outcome.
โ
If the matter cannot reach a point via one slit then the final probability must equal
that of the other slit. For example if
๎
1
๎
x
๎=
0
then
๎๎
x
๎=๎
2
๎
x
๎
.
The first criteria can be satisfied by applying a correction term to (4).
๎๎
x
๎=๎
1
๎
x
๎๎๎
2
๎
x
๎๎๎ญ ๎ ๎
x
๎
(5)
To further aide in the task before us, consider three cases.
Equation (4) is valid
In this case the outcome must yield an earlier conclusion.
๎
0
๎
x
๎=๎
1
๎
x
๎๎๎
2
๎
x
๎
(6a)
Minimum probability
What is the minimal probability density possible for a given
ฯ
1
(x) and
ฯ
2
(x), other than
zero, that matches our criteria? The simplest candidate is
๎
min
๎
x
๎=
๎
๎
๎
1
๎
x
๎โ
๎
๎
2
๎
x
๎
๎
2
๎
min
๎
x
๎=๎
1
๎
x
๎๎๎
2
๎
x
๎โ
2
๎
๎
1
๎
x
๎โ ๎
2
๎
x
๎
(6b)
The subtraction serves to minimize the probability while the square eliminates any
negative results. Moreover the last term in (6b) can be said to take on the role of the
correction term in (5).
Maximum probability
So now what could be the maximum probability density? There is no definite upper
limit as to what a probability density can be, however one can safely deduce that in this
case the correction term in (5) would have to be positive and so.
๎
max
๎
x
๎โฅ๎
1
๎
x
๎๎๎
2
๎
x
๎
(6c)
Putting equations (5), (6a), (6b) and (6c) together suggests a methodology for combining
probabilities.
๎๎
x
๎=๎
1
๎
x
๎๎๎
2
๎
x
๎๎
2
๎
๎
1
๎
x
๎โ ๎
2
๎
x
๎โ
cos
๎ฎ
(7)
3.
A statistical view of matter
Equation (7) ties together all the previous pieces, and yet is still unsatisfactory. One
main reason is that it introduces a
ฮ
that is an unspecified quantity.
What is needed is an improved relation. To do so a lesson can be taken from (6b) that
allows for generalization. Define
ฮจ
to be a complex number such that
๎๎
x
๎=โฃ๎ต๎
x
๎โฃ
2
Let this new quantity be called the
probability amplitude
. Then let (7) be recreated by
simply adding these amplitudes.
๎ต=๎ต
1
๎๎ต
2
(8)
โฃ๎ตโฃ
2
=โฃ๎ต
1
๎๎ต
2
โฃ
2
๎๎
x
๎=โฃ๎ต
1
โฃ
2
๎โฃ๎ต
2
โฃ
2
๎
2
โฃ๎ต
1
โฃโ โฃ๎ต
2
โฃ
cos
๎
๎
1
โ๎
2
๎
๎๎
x
๎=๎
1
๎
x
๎๎๎
2
๎
x
๎๎
2
๎
๎
1
๎
x
๎โ ๎
2
๎
x
๎
cos
๎
๎
1
โ๎
2
๎
(9)
Here
ฯ
1
and
ฯ
2
are the complex phases. It can be seen by inspection that (9) is consistent
with (7). Moreover the unknown
ฮ
is now given significance as the difference in the
complex phase of the respective probability amplitudes.
Illustration: The toss of a die
Consider how the above can be applied to the real-world example of computing dice
probability. In particular, the probability that a die toss will produce either a 1 or a 6
roll. For simplicity let us assume that all amplitudes are either positive or negative but
not complex.
Consistent with our earlier definition our probability amplitudes will be taken to be
๎ต
1
=
ยฑ
1
๎
6
(10a)
๎ต
6
=
ยฑ
1
๎
6
(10b)
We have two possible outcomes, one where the signs are the same and one where they
are different.
P
same
=โฃ
1
๎
6
๎
1
๎
6
โฃ
2
=
2
3
(11a)
P
diff
=โฃ
1
๎
6
โ
1
๎
6
โฃ
2
=
0
(11b)
While these answers may not look right they in fact are. Consider that for a large
4.
number of rolls (a requirement if we are to speak of probabilities) both scenarios are
equally likely. So the actual probability will be the average of the two.
P
=
1
2
๎
P
same
๎
P
diff
๎=
1
3
(12)
This last result is indeed the expected outcome.
Conclusion
The behavior of matter is different then what on might expect based on common
experience. Specifically the statistical nature of matter behaves in such a manner that
probabilities can not, in general, be added together as previously supposed. Rather one
can work instead with probability amplitudes
ฮจ
defined such that:
โฃ๎ตโฃ
2
=
P
We add probability amplitudes largely the same as the probabilities earlier, except these
amplitudes need not be positive numbers at all.
A more detailed analysis of the consequences is beyond the scope of this piece.
Contact Information
5.
kevin.gibson@asu.edu
kevinlg@mesacc.edu
6.