There Also Can Be Fuzziness in Quantum States Itself—Breaking through the Framework and the Principle of Quantum Mechanics

In this paper, an attempt is made to synthesize fuzzy mathematics and quantum mechanics. By using the method of fuzzy mathematics to blur the probability (wave) of quantum mechanics, the concept of fuzzy wave function is put forward to describe the fuzzy quantum probability. By applying the non-fuzzy formula of fuzzy quantity and Schrödinger wave equation of quantum mechanics, the membership function equation is established to describe the evolution of the fuzzy wave function. The concept of membership degree amplitude is introduced to calculate fuzzy probability amplitude. Some important concepts in fuzzy mathematics are also illustrated.


Introduction
The research on combining fuzzy mathematics and quantum mechanics has been done in the prior work [1]- [8]. These theories, which are using fuzzy mathematics to understand quantum mechanics, can be regarded as the equivalent theory of quantum mechanics. They do not break through quantum mechanics.
In order to be able to break through quantum mechanics and find a deeper principle of the universe, in this article a mathematical theory named "fuzzy quantum probability" is proposed. Different from the prior theories, the theory in this paper is not using fuzzy mathematics to understand quantum mechanics, but using fuzzy mathematics to fuzzify the state function of quantum mechanics so that a fuzzy (unsharp) probability can be obtained. This work is not

Fuzzy Wave Function
Usually, in quantum mechanics it is thought that the value of the probability for the event ( ) , r t  . The meaning of the fuzzy wave function depicted by the membership function is showed in Figure 3.
can be represented as Here C is the plural set. The symbol " ∫ " does not represent integration but a sign, whereas "/" does not represent division but a kind of corresponding symbol.
The normalization problem of ( ) , r t ψ   will be discussed in Section 4.
Now it should be considered how to calculate fuzzy probability from ( ) Figure 2. This figure shows that quantum mechanics breaks through the principle of classical mechanics by replacing certainty with probability, while fuzzy quantum probability breaks through the principle of quantum mechanics by replacing clarity with ambiguity (using the uncertainty of probability (wave) to replace the certainty of probability (wave)).
the credibility of ( ) This is a possible function form (the membership degree of this form is α ) of the fuzzy wave strength ( ) Thus, the fuzzy quantum probability (density) of the event ( ) In addition, the membership function of ( ) can be easily solved as follows: t . This is the membership function for ( ) can be rewrit-

The Membership Function Equation
Next, an equation will be established for the membership function The membership function can be normalized, i.e. it should satisfy The normalized membership function reads as Then Schrödinger equation turns into real form ( ) Therefore, the integrand in the bracket is a function whose integral is zero about R, I.

The Membership Degree Amplitude and Fuzzy Probability Amplitude
In order to discuss the normalization of fuzzy wave function and the calculation of fuzzy probability amplitude, a new concept is now introduced, named "membership degree amplitude" The membership degree amplitude of ( ) The membership degree of ( ) It should be pointed out that the sum rule in Equation (10) is the normalized membership function which satisfies Equation (1).
For simplicity, the real number form is adopted. Order R iI φ = + , then Putting (11)

Rg R I r t Ig R I r t f R I r t R I
Rg R I r t Ig R I r t f R I r t R I can be regarded as the fuzzy source.

R R u v r t
Similarly, the imaginary part of the non-fuzzification of ( ) 3) By differentiating the variable t at both sides of (20), and by using (2) These two results are completely consistent with (16), so the above design of fuzzy probability amplitude is rational.

Conclusions
The subject of this paper is to do the work of "the fuzzification of quantum Unlike the work on using fuzzy mathematics to understand quantum mechanics is equivalent to quantum probability in a certain sense, my work on bluring quantum probability with the method of fuzzy mathematics in this paper is not equivalent to quantum probability. Quantum probability theory only advocates the uncertainty of events, while probability and probability waves are determined. Fuzzy quantum probability theory advocates not only the uncertainty of events, but also the uncertainty (fuzziness, unclearness) of probability and probability wave. The core concepts of quantum probability theory are probability wave and wave equation, while the core concepts of fuzzy quantum probability theory are membership degree (membership degree amplitude) and membership degree equation (membership degree amplitude equation). Since famous American automatic control expert Zadeh, L.A. pioneered the theory of fuzzy sets. Fuzziness has caused widespread concern and research. Its application has become increasingly widespread and in-depth. Here several pairs of important concepts are introduced.

1) Fuzziness and clarity
This pair of concepts is used to describe the property and state of things.
"Clarity" refers to an unequivocal and certain property-state, whereas "Fuzziness" to an equivocal and uncertain one. For instance, "far larger than 5", "pretty girl", "tall man" etc. are all fuzzy conceptions and there is no categorical or unambiguous boundary about them. "Larger than 5", "Chinese people", "round table" etc. are all clear concepts with distinct or certain boundaries about them.
2) Fuzziness and chance These two concepts both refer to uncertainty. However, the former means uncertain property-state of things, whereas the latter means uncertain the results of the event. The most critical quantity to depict fuzziness is "membership degree" (look below), whereas the most critical quantity describing chance is "probability".
3) General set and fuzzy set There is a categorical boundary about classical set: an object either belongs to a classic set or does not belong to this classic set, whereas there is not a clear boundary about fuzzy set: the membership relation about an object belongs to a fuzzy set is uncertain, and we only can say how large is the degree of membership of an object for a fuzzy set? Order U represents a collection of some objects, called "discussion domain". For a classical set A U ⊆ , we may use a characteristic function