Wave Function, Developed Gaussian Distribution

[Hamid]

Normal distribution law reduces in an indisputable manner the quantum behavior of nature to a mathematical formula and wave function represents the sub-quantum structure(s) of this behavior at all scales, from the smallest sub-atomic particles to the entire universe.

Probability wave function can be derived from Gaussian Normal Distribution provided that a perfect definition of uncertainty to be available and the hidden variables as the pillars of this function to be recognized in bell curve.

In general terms, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is to say, the probability of obtaining each of the possible results from measuring an observable. In other words, before measuring the characteristics of a natural phenomenon its characteristics do not have definite values. Instead, they are described by normal distribution function. The degree of uncertainty in measurement depends mostly on the accuracy of measuring device. Consequently, higher level of technology enables us to measure the characteristics more precisely. As a matter of fact, the order of measurement is important [Ref.1].

For a normal distribution 99.73% of random variables or observations fall within 3 standard deviation (σ) of the mean value (µ), that is, between µ-3σ and µ+3σ. The probability wave function should at least meet this requirement.

Taking into consideration that discreteness in quantum mechanics is an absolute principle, it would be more appropriate to rename the random variables to quantum variables which can only have some set of discrete values.

Standard uncertainty has been defined in one of my articles [Ref.2], as follows:

The standard uncertainty ux of a measurement result X is equal to six standard deviation of X, i.e. ux = 6σx .

If µ would be the mean value of the measurement result, then X=µ±3σx .

The concept above has been used for interpreting the Diffraction Phenomenon [Ref.3], Double Slit Experiment [Ref.4] and Planck Length [Ref.1]. Most likely the latter could have a key role in explanations concerning the quantum gravity.

The hidden variables in normal distribution (bell curve) are µ-3σ, µ-2σ, µ-1σ, µ, µ+1σ, µ+2σ and µ+3σ. Each component of wave function, namely M3, M2, M1, Ci, P1, P2 and P3, is formed around one of these variables respectively [Refs.3&4].

The summary of above mentioned topics and the general configuration of wave function has been shown in Figure 1:

Figure 1

The distribution of all seven groups shall be jointly normal or approximately normal. They, totally, arrange the probability wave function or the love formula that we are going to generate:

Ψ = M3+ M2+ M1+ Ci+ P1+ P2+ P3

Distribution function of each group can be deduced from probability density function of Normal Distribution with the following formula, in which µ is mean value (mathematical expectation) and σ (sigma) is standard deviation that determines the width of the "bell":

The total area under this curve, from minus infinity to plus infinity, is one. That is to say it embraces 100% of random variables (quantum variables). Therefore the area under f(x) between any two values, say, from x=µ-3σ to x=µ+3σ, is the proportion of cases which lie between the two values:

This regulation has been used for calculating the area under the probability density function of each particular group (component) in wave function [Ref.4]. Table1 indicates the results:

Table 1

The mean value or mathematical expectation of different groups is given in Table 2.

Table 2

Even though the probability of all seven groups implies that their distribution singly is not normal, but we can obtain distribution function of each group from the bell curve. For doing this, it is sufficient to change the standard deviation to σ'=1/6σ (u=6σ'=σ) and by a factor less than one, proportional to each case, to reduce the height of the "bell". This means that the total area under every one, from minus infinity to plus infinity, will be less than one. Therefore they are abnormal distributions individually.

The probability distribution functions of all seven groups can now be constructed, see Figure 2. To summarize the functions it is supposed that µ=0.

Figure 2

By the same rule it is possible to make sub-quantum structures of different groups or components.

If µ=0 and σ=1, the Gaussian Distribution is called standard normal distribution, with the following formula:

By applying the above-mentioned standard, hidden variables or mean values will change into integers -3, -2, -1, 0, +1, +2 and +3. In other words, for every specific value of µ and σ we have all required data for drawing the graph of wave function, by Excel for example. The curve of standard wave function looks like Figure 3, which can be regarded as the Universal Wave Function.

Figure 3

The wave function introduced here is applicable to the measurement results related to all natural phenomena. For a while imagine that the brain which controls our thought and behavior may be a type of quantum variable. Where is our place in wave function of the human mind?




References

1. Planck Length and Quantum Geometry, March 2007,
   

   theories.toequest.com
   
2. Definition of Uncertainty, June 2008, theories.toequest.com
   
3. How Can the Photons Tolerate Each Other?, October 2004,
   
   theories.toequest.com
   
4. Double Slit Experiment and Quantum Mechanics, December 2005,
   
   theories.toequest.com

[Hamid]

Wave Function of the Universe(s)

"[Physicist Stephen] Hawking is one of the founders of a new scientific discipline called quantum cosmology. At first, this seems like a contradiction in terms. The word quantum applies to the infinitesimally small world of quarks and neutrinos, while cosmology signifies the almost limitless expanse of outer space. However, Hawking and others now believe that the ultimate questions of cosmology can be answered only by quantum theory. Hawking takes quantum cosmology to its ultimate conclusion, allowing the existence of infinite numbers of parallel universes."

-Michio Kaku (Professor of Theoretical Physics)

http://deoxy.org/h_kaku2.htm

[Hamid]

The observer becomes part of the observed system

"The notion of the observer becoming a part of the observed system is fundamentally new in physics. In quantum physics, the observer is no longer external and neutral, but through the act of measurement he becomes himself a part of observed reality. This marks the end of the neutrality of the experimenter. It also has huge implications on the epistemology of science: certain facts are no longer objectifiable in quantum theory. If in an exact science, such as physics, the outcome of an experiment depends on the view of the observer, then what does this imply for other fields of human knowledge? It would seem that in any faculty of science, there are different interpretations of the same phenomena. More often than occasionally, these interpretations are in conflict with each other. Does this mean that ultimate truth is unknowable?" – Thomas Knierim

http://www.thebigview.com/spacetime/uncertainty.html

[Graybeard]

A conversation between Einstein and Heisenberg.

Heisenberg: "One cannot observe the electron orbits inside the atom. [...] but since it is reasonable to consider only those quantities in a theory that can be measured, it seemed natural to me to introduce them only as entities, as representatives of electron orbits, so to speak."

Einstein: "But you don't seriously believe that only observable quantities should be considered in a physical theory?"

"I thought this was the very idea that your Relativity Theory is based on?" Heisenberg asked in surprise.

"Perhaps I used this kind of reasoning," replied Einstein, "but it is nonsense nevertheless. [...] In reality the opposite is true: only the theory decides what can be observed."

(translated from "Der Teil und das Ganze" by W. Heisenberg)


We can easily see the rift between Einstein's intuitive and Heisenberg's empirical approach. Although Einstein's argumentation appears tricky, it is clear that he believes in a reality independent of what we can observe, which is in essence the view of realism. Kant's "thing in itself" comes to mind. - In contrast, Heisenberg believes that reality is what can be observed.



If there are different observations, there must be different realities, which depend on the observer. Insofar Heisenberg can be regarded as an advocate of philosophical idealism, which states that the objects of perception are identical with the ideas we have about them. The idealist view denies that any particular thing has an independent real essence outside of consciousness.


Is the moon still there when nobody is looking at it?
The two philosophies seem incompatible at first. Heisenberg is in good company with famous contenders of idealistic positions, such Plato, Schopenhauer, and Husserl, but so is Albert Einstein. If we take Heisenberg's view for granted, strict causality is broken, or better: the past and future events of particles are indeterminate. One cannot calculate the precise future motion of a particle, but only a range of possibilities. Physics loses its grip. The dream of physicists, to be able to predict any future event in the universe based on its present state, meets its certain death.


If we regard reality as that which can be observed by all, we have to find that there is no objective movement of an electron around the nucleus. This viewpoint would imply that reality is created by the observer; in other words: if we take Heisenberg literally, the moon is not there when nobody is looking at it. However, we must consider the possibility that there is a subatomic reality independent of observation and that the electron may have an actual trajectory which cannot be measured. The moon may be there after all. This conflict is the philosophical essence of the Uncertainty Principle.

Hi Hamid .....

I read the article you posted and I have quoted some more of it here. I am not sure if you are the author or not.

However, I don't fully agree with it. It seems to me to be an attempt to establish a philosophical relationship between QM and Human perception. The tone of the article is that such a relationship is de rigueur, and not to be considered otherwise, not to be questioned, simply to be further elucidated. Yet it gives no reason why such a relationship should exist. Not even a philosophical reason.

My opinion only

cool bananas ... greg

[Hamid]

Double Slit Experiment and Quantum Mechanics

The existing interpretation of the most beautiful experiment in physics (Double- Slit Experiment) has been founded on false knowledge.

http://www.toequest.com/forum/physics-articles/437-how-can-the-photons-tolerate-each-other.html

http://www.youtube.com/watch?v=to2QM...eature=related

[austintorn@aol.com]

The quantum state set upon the slit by the previous photon plays a role in where the next photon will end up.