Random Vector Representation of Continuous Functions and Its Applications in Quantum Mechanics

The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors where every random vector is defined on one of these probability spaces, such that the sequence of conditional mathematical expectations formed by the random vectors uniformly converges to the continuous function. This is random vector representation of continuous functions, which is regarded as a bridge to be set up between real function theory and probability theory. By means of this conclusion, an interesting result about function approximation theory can be got. The random vectors representation of continuous functionsis is of important applications in physics. Based on the conclusion, if a large proportion of certainty phenomena can be described by continuous functions and random phenomenon can also be described by random variables or vectors, then any certainty phenomenon must be the limit state of a sequence of random phenomena. And then, in the approximation from a sequence of random vectors to a continuous function, the base functions are appropriately selected by us, an important conclusion for quantum mechanics is deduced: classical mechanics and quantum mechanics are unified. Particularly, an interesting and very important conclusion is introduced as the fact that the mass point motion of a macroscopical object possesses a kind of wave characteristic curve, which is called wave-mass-point duality.


Introduction
From a physical point of view, continuous functions can describe a large proportion of certainty phenomena.For example, the trajectory of a projectile motion can be expressed as a continuous function.However random vectors can describe a lot of random phenomena in the natural world.So, if we consider the connection between some certainty phenomena and some random phenomena, we should research the relation between continuous functions and random vectors.
In the paper, we obtain a conclusion: for any given continuous function () ∈ [, ], there must be a sequence of probability spaces {(, ℱ,   )} and a sequence of random vectors {(  ,   )} where every random vector (  ,   ) is defined on the probability space (, ℱ,   ) , such that the sequence of conditional mathematical expectations {(  |  = )} uniformly converges to the continuous function () in [𝑎, 𝑏].
This can be called the random vector representation of continuous functions, which is like a bridge to be set up between real function theory and probability theory.
Step 3. Construct a sequence of probability spaces {(, ℱ,   )}and a sequence of random vectors {  } = {(  ,   )} which every random vector   is defined on (, ℱ,   ).In fact, let Then {  (, )}is a sequence of distribution functions.Take  = ℝ 2 and ℱ = ℬ 2 , where ℬ 2 is a Borel  algebra on ℝ 2 ; and   is taken as the probability measure corresponding to   (, ).We all know a fact in probability theory that   must exist and be unique [2][3][4].In this way we get a sequence of probability spaces as the following: Then on every probability space (, ℱ,   ) we define a random vector as follows: For any (, ) ∈ ℝ 2 , by noticing the following expression We can know that   () is really a random variable defined on (, ℱ,   ); in the same way,   () is also really a random variable defined on (, ℱ,   ).So   () = (  (),   ()) = ( 1 ,  2 ) =  is just a random vector defined on(, ℱ,   ).
So, the sequence of conditional expectations of the sequence of random vectors Step 4. Prove the fact that the sequence of conditional expectations {(  |  = )} converges to () everywhere in [, ].Firstly, we prove a result as the following: In fact, for any  ∈ [, ], clearly (∃ ∈ {1,2, ⋯ , }) ( ∈ [ −1 () ,   () ]).Then for any  ∈ [, ], we have It is easy to learn the fact that For this fixed  ∈ [, ], since   (, ) is a continuous function with respect to , we have By means of mean value theorem of integrals, there exists a point  ∈ Then we can get the following expression: By means of the first mean value theorem of integrals, there exists a point   () ∈ [ −2 () ,  +1 =   () For () being continuous and by noticing that   () ∈ [ −2 () ,  +1 () ], based on the medium value theorem for continuous functions, we have For  ∈ [, ] being arbitrary, we get the fact as follows Step 5. Prove the fact that the sequence of conditional expectations {(  |  = )} uniformly converges to () in [, ].
So above expression can be written as the following: By first means of mean value theorem for integrals, there exists a point   () ∈ [  −1 () ,   +1 ()  And then by above result we have the following expression: ).
By noticing the fact that We immediately have the result: there exists  2 ∈ ℕ + , for any  ∈ ℕ + , such that Take  = { 1 ,  2 }, and when  > , we get the following expression: By means of Equation ( 8 i.e.,   () →∞ → ().So, we get the conclusion: And then, similar to above proof, we can obtain the conclusion that the sequence of conditional mathematical expectations The elements in the set should be screened firstly.In fact, write () = { 0 ,  1 , ⋯ ,   }, and an equivalent relation ∼ is defined as the following: We can get a quotient set of() as being where [  ] is the equivalence class what   belongs to.Suppose the elements of () ∼ ⁄ are as the following: where 0 ≤ () ≤ , and we stipulate that the representation element    is taken as the least element in [   ].Then we have By using the nodes    0 () ,    1 () , ⋯ ,    () (𝑛) in [, ], a group of continuous base functions     () ,  = 0,1, ⋯ , () are formed as the following: ,  ∈    −1 () ,     () ; ,  ∈     () ,    +1 () ; 0， otherwise; So at the nodes    0 () ,    1 () , ⋯ ,    () () with respect to the representation elements   0 ,   1 , ⋯ ,   () , the group of continuous base functions as follows has been defined.And then, we should stipulate: for every equivalence class [   ], all the elements in [   ] have beencorresponded to the same continuous base function     () () .
Therefore, all the nodes in [, ], have been defined their continuous base functions: Hence, we can get a sequence of continuous base functions with two variables{  (, )}as follows Similar to above method we have ever been used, we can have a sequence of probability density functions {  (, )}defined on ℝ 2 = (−∞, +∞) 2 and a sequence of random vectors {  } = {(  ,   )} ; then we immediately get a sequence of conditional mathematical expectations {(  |  = )}, where Now we should prove the fact that the sequence of conditional mathematical expectations {(  |  = )} uniformly converges to () in [, ].
Next, we should consider the following two situations.
i) When    () () ≡    () (), we have ii) When    () () ≢    () (), we should have However, in either case, we can use the method similar to 1) to prove the result: Clearly this is a kind of degrading situation.So, we should take a distribution function: And we construct a probability space (, ℱ, ), where  is a probability measure corresponding to   (),  = ℝ 1 and ℱ = ℬ 1 .Take the random variable as follows :  → ℝ 1 ,  ↦ () = .It is easy to know that the distribution function of  is just   ().By noticing the following expression

The significance of function approximation of Theorem 1
In above section, we have proved the conclusion: the sequence of conditional mathematical expectations {(  |  = )} uniformly converges to () ∈ [, ] in [, ].Now we reveal the significance of function approximation of {(  |  = )} about continuous function ().We only consider the continuous function space [, ].In [, ], addition operation "+" and scalar multiplication operation "⋅" are defined as the following: We all know that ([, ], +, ℝ,⋅) forms a linear space, which can be simply denoted by [, ].In [, ], we define a norm operation as follows Then ([, ], ‖⋅‖) is a normed linear space, which can be also denoted by Clearly [, ] is an infinite dimension normed linear space.Suppose () ∈ [, ] is a "complicated" function; for every  ∈ ℕ + , we try to find a group of  + 1 linearly independent "simple" functions as following: and  + 1 real numbers  0 () , For some fixed , when  > , (spanΦ(), ‖⋅‖)is a  + 1 dimension normed linear sunspace of ([, ], ‖⋅‖) , such that, there exist real numbers  0 () ,  1 () , ⋯ ,   () ∈ ℝ , such that ‖  − ‖ <  , where span Φ() means that a normed linear subspace of [, ] generated by ().In other words, on this , we can use a kind of linear combination of the base functions in span Φ() as follows to take the place of () approximately, or we say that   () can approximate () to the approximation accuracy  .This is one of the basic ideas of function approximation.
where   () () ∈ [, ],  = 0,1, ⋯ , , such that, by using the sequence of the groups of base functions {()}, the sequence of interpolation functions constructed as the following Proof.Case 1.Let () be a strictly monotone functions, and we may as well assume that () is a strictly monotonically increasing function, as when () is a strictly monotonically decreasing function, the proof is the same as the increasing status.
For any given  ∈ [, ] , there must exist  ∈ {1,2, ⋯ , } , such that  ∈ [ −1 () ,   () ].Then we have and in the same time, we should notice the expressions (see step 1 in Theorem 1) as following: And we also should notice that  and  are different, which  is a temporarily fixed subscript, but  will tend to infinite.
Case 2. Assume () be not strictly monotonic.Similar to the method in Theorem 1, we can have Based on them, we get a sequence of interpolation functions:

Quantum mechanics representation of classic mechanics
As we all know, classic mechanics is the scope of macroscopical physics in which Newtonian mechanics is its main part.Classic mechanics is very different with microphysics, especially with quantum mechanics [10,11].For example, the motions of microscopic particles have wave-particle duality; but the motion of mass points in macroscopical physics only has mass point characters and no wave natures; in other words, there is no wave-mass-point duality in macroscopical physics.However, there has ever existed a correspondence principle: considering a kind of motion state in quantum physics, when quantum number  → ∞, the limit situation of the motion state in quantum physics must become a kind of motion state in macroscopical physics.In other words, the limit situation of the motion law in quantum physics is just some motion law in macroscopical physics.
Generally, Bohr suggested a generalized correspondence principle: the limit situation of any new theory must be in line with some old theory.
It is worth noting that above correspondence principle or generalized correspondence principle is all of unipolarity: the limit situation of the motion law in quantum physics is just some motion law in macroscopical physics, but the converse principle is clearly meaningless.
However, we can consider an important problem: must any one of motion states in macroscopical physics be the limit situation of some the motion states in quantum physics?
Apparently, this problem has not been observed, and of course there is no any answer.For example, we consider the well-known projectile motion.As we all know, a projectile motion can be expressed by the equation of locus of the projectile motion as following: where  ∈ (0,  2 ) is a mass ejection angle,  0 ∈ (0, +∞) is the maximum range of fire, and the initial velocity is repressed by  0 ∈ (0, +∞); here the air friction is omitted.
Clearly () ∈ [0,  0 ], i.e., a projectile motion can be described by a unary continuous function.For this continuous function, can we find some microscopic particles such that the limit of the group behavior of these microscopic particles is just this continuous function () when quantum number  → ∞?
In this paper, we will give a positive answer for this problem.It is easy to understand that almost all laws of classic mechanics are described by continuous functions.So, we can generalize above problem as such problem: for any a continuous function  , unary continuous function or multivariate continuous function, which should describe some motion law of some mass point in microscopically physics, can we find some microscopic particles such that the limit of the group behavior of these microscopic particles is just this continuous function  when quantum number  → ∞?
Now we start to try to solve out the problem.Theorem 3. Given arbitrarily a non-constant function () ∈ [0,1], there must exist some microscopic particles such that the limit of the group behavior of these microscopic particles is just this continuous function () when the quantum number  → ∞. Proof.
Step 1.We consider the wave function of a microscopic particle in infinite deep square potential well.
As a matter of fact, we take a particle  with quality , and  moves along  axis and is of determined momentum  =   and determined energy  = 2 where   is the velocity of  moving along .We take a special infinite deep square potential well as follows (see Figure 2): The particle  is complete free inside the potential well; only at two endpoints  = 0,  = 1, there are infinite forces to impose restrictions on  not to escape.It is easy to know that () = 0; so the probability of finding the particle in the interval (−∞, 0) ∪ (1, +∞) is zero.However inside the potential well, i.e.,  ∈ [0,1], we have () = 0; then the Schrodinger Equation turn into the following form: Let  = √2 ℏ ; then we have the following form:  2 ()  2 +  2 () = 0, which is the equation of motion of a simple harmonic oscillation, and its general solution is as following: () =     +    , where ,  are two arbitrary constants that can be determined by some boundary conditions.
Then, what are the boundary conditions?In fact, in quantum mechanics, the solution of a Schrodinger Equation in three-dimension space, i.e., the wave function (, , , ) should satisfy the following established standard conditions: ii)  and its three partial derivatives   ,   ,   are continuous everywhere; iii)  is a single-valued function about coordinates.By means of above conditions, when the potential function approaches infinite, based on the continuity of (), we can get the result as being (0) = (1) = 0, which can make the solution be continuous at both inside and outside of the potential well.Because of the following expression: 0 = (0) =   0 +   0 = , we get  = 0; thus we have the following equation: () =   .And then we take notice of the equation: 0 = (1) =   , if  = 0, then () ≡ 0 which is a trivial solution and cannot be normalized.Thus we only get the result:   = 0, and we know the following fact:  = 0, ±, ±2, ±3, ⋯ Clearly that  = 0 is meaningless, for this can also make that () ≡ 0. Besides,  with negative values cannot generate any new solutions because of the fact that ( − ) = −   and we can make the minus signs enter into the coefficient .Therefore, we have the result:  =   = ,  = 1,2,3, ⋯ We should notice a fact that, the boundary condition at  = 1 is not used to determine the coefficient  , but to determine the energy  because of the expression: = √2 ℏ , i.e., It is well-known that 2 is ground state, and others are follows:  2 = 4 1 ,  3 = 9 1 ,  4 = 16 1 , ⋯ which means that the energy of a particle can only take discrete values; in other words, the energy of a particle is quantized.And positive integer  is called the quantum number of the energy of a particle.So we can learn that the quantization of the energy of a particle is very natural in quantum mechanics.
Thus the solution of the Schrodinger Equation can be expressed by the quantum number as the following: In order to determine the coefficient , we can use the normalization condition Step 2. Based on an important fact that will be described as follows, we should consider weakening three standard conditions about the wave function (, , , ) mentioned above.
As a matter of fact, we can see that the derived function of the wave function   () = √2 ( ) is not continuous at  = 0,1.For this we only notice the following implication is true: It is well known that the movement of a particle in the infinite potential well is a typical example in quantum mechanics.However, as we have learned above, its wave function  and its three partial derivatives We should not forget a fact that wave function  does not represent a physical wave but only a mathematical wave; in other words, || 2 is a probability density function where it should be normalized.
We also know such a fact that, in probability theory, any probability density function is not required to be continuous at everywhere but only required to be almost everywhere continuous.Thus, we have enough reason to revise the three standard conditions which the wave function (, , , ) should satisfy mentioned above to be as the following: (ii)  and its three partial derivatives   ,   ,   cannot be continuous only at finite points (clearly the requirement is a little stronger than almost everywhere continuous); (iii)  is a single-valued function about coordinates.Moreover, by the viewpoint of Von Neumann, wave function  is defined in a Hilbert space ℒ 2 (ℝ 3 ), where the operations in quantum mechanics (momentum, work, and so on) are inner product operations, which may be enlightened by ∫ || 2 ℝ 3  = 1 and form a mathematical formalization structure.We all know the fact that, in a Hilbert space ℒ 2 (ℝ 3 ), we have no need to require wave function  to be continuous at everywhere but almost everywhere continuous to be enough.
Step 3. We continue to consider the wave function of the particle in the onedimension infinite deep potential well.We have known its general solution being as () =     +    , where ,  are arbitrary constants which can be determined by the boundary conditions.This time, we suppose ()  be continuous at the boundary points  = 0,1.We take notice of the following implication:  =   = ,  = 1,2,3, ⋯ Very similar to the method in Step 1, we have the expression of  again as follows: So the solution of the Schrodinger Equation can be expressed by means of quantum numbers as the following: Again by using the normalization condition, we can get that  = √2.Thus another solution of the Schrodinger Equation in the potential well is as the following: () = √2 ( ),  ∈ [0,1],  = 1,2,3, ⋯ (21) Let   () = ( ),  ∈ [0,1],  = 1,2,3, ⋯, and we have The function   () is also called essence wave function of the wave function   ().
It is interesting to note that the wave function   () = √2 ( ) is not continuous at boundary points  = 0,1 this time.Besides, since when the quantum number  is very large, the two wave functions   () and   () are almost no different; in other words,   () is just the situation that   () translates a Step 4. Supplementary instruction for the revision of the three standard requirements on the wave function .
It is well known that, in physics, harmonic oscillation is often described by complex exponential form; for example, the two wave functions that we just get can be described as the following: () = √2 () = √2 ( ) + √2 ( ) =   () +   () (23) In classic physics, this kind of expression is said to be more convenient for operation but with no more physical significance.However, here we can find its physical significance of the complex variables function () = √2  coming from quantum mechanics.As its real part of the () = √2  , Eve wave function as being   () = √2 ( ) is determined by the second boundary condition; and its imaginary part, Adam wave function as being   () = √2 ( ) is determined by the first boundary condition.These mean that the two boundary conditions are all useful and we cannot give up any one of them.Therefore, the revision of the three standard requirements is quite reasonable.
Step 5.The extension of the domain of definition of the wave functions.is not essentially different; so they can be regarded the same.
It is worth noting that, for Adam wave function, in the infinite deep square potential well, it should be written as the following complete form: where we only write out the expression just as being  ∈ [0,1].And for Eve wave function, in the infinite deep square potential well, it should be written as the following complete form: where we also only write out the expression just as being  ∈ [0,1].
Based on the statistical interpretation of wave functions, |(, )| 2 should be a kind of probability density function.Then from Equations ( 24) and (25), we can learn that 2  2 ( ) is a probability density function and 2  2 ( ) is a probability density function too.We have enough reason to call  2 ( ) and  2 ( ) essence probability density function of the probability density functions.So we get the essence probability density function family of Adam wave functions and Eve functions as the following: ∞ is of two-phase normalization property:  2 ( ) +  2 ( ) = 1.Step 6.The construction of the sequence of two-dimension probability density functions.
For convenience, let  = 2; but be careful, here  means subscript but not the quality of some particle.We are going to discuss our problem from the following two cases.
Case 1. Suppose () is a strict monotonous function.It assumes that () be a strict monotonous rising function because its proof is not of essence difference when () is a strict monotonous declining function.Therefore, we have the following partition: Then we consider the particle wave functions defined in the following subintervals one by one: Firstly, we treat with it in the closed interval [ 0 () ,  1 () ].And we consider the movement of a particle in the infinite deep square potential well that the closed interval [0,2 ( 1 () −  0 () )] is just the bottom margin of the potential well.The particle is denoted by  1 () which can be regarded as a descendant particle generated by the Adam wave function and Eve wave function of the original particle  in the case of energy level being  .The descendant particle  1 () moves along  axis with determined quality  1 () and determined momentum  1 () =  1 ()   (,1) and determined energy where   (,1) is the velocity of movement of  1 () along  axis.By means of the continuity of the wave function, it is easy to get the solution of the wave function in [0,2 ( 1 () −  0 () )] as following: Then again, by means of the continuity of the derived function of the wave function, we can get another solution of the wave function in the closed interval [0,2 ( 1 () −  0 () )] as following: Now we care more for the ground state of   (,1) () and   (,1) (), i.e., the wave functions when  = 1 as follows: We can omit the amplitudes of the wave and keep the essence wave function and do squaring operation on the essence wave functions, and get the probability essence wave functions as the following:  The next, we make a coordinate translation:  =  +  0 () , and then we have the following expressions: ( −  0 () ).
Thus we transfer the probability essence wave functions defined in the closed interval [0,  1 () −  0 () ] into the probability essence wave functions in in closed interval [ 0 () ,  1 () ].And we rewrite the variable  back to , and then we get the following expressions: The graphs of the probability essence wave functions in [ 0 () ,  1 () ] are shown in  In that way, we can regard the following expression ( −  0 () ), as Adam probability essence wave function of the movement of the descendant particle  1 () in [ 0 () ,  1 () ], and regard the following expression as Eve probability essence wave function of the movement of the descendant particle  1 () in [ 0 () ,  1 () ].
It is easy to know that () ⋅ ℬ() a linearly independent group in the continuous function space ([0,1] × [, ]).Now we take the diagonal elements of () ⋅ ℬ() to make a set as follows: Then this sequence of binary nonnegative continuous functions as being{  (, )} =1 ∞ are normalized as the following: where becomes a sequence of probability density functions defined on ℝ 2 , and   (, ) is called the probability density function when the quantum number is just .
And now by means of the sequence {  (, )} =1 ∞ , we can construct a sequence of functions of one variable as follows: Apparently, {  ()} =1 ∞ is just the sequence of conditional mathematical expectations formed by {  (, )} =1 ∞ .
Case 2. Suppose :  →  be not strict monotonous function and not constant function.
other words, this motion curve of a mass point  = () have been quantization, which is the limit state of these microscopic particles wave functions when  → ∞.Clearly this fact meets the Bohr's correspondence principle.
We finally end the proof of the theorem.
Example 1 Suppose we cast an object  with quality  0 , which is regarded as a mass point.So the movement of  can be described by its equation of locus as follows: where  ∈ (0,

Duality of mass point motion
We firstly review the projectile motion in Example 1.The property of mass point motion is shown as its momentum  =  0  0 and its energy as the following: Actually, more straightway, its property of mass point should be described by its equation of locus as the following: In other words, the property of mass point can be described by its momentum and energy or by its equation of locus; these two methods are equivalent.
Then we ask an interesting and important problem: is there wave nature on mass point motion in classic physics?Alternatively, we can ask the question: is there wave mass point duality in classic physics?
For answering this problem, we firstly review the particle nature and wave nature in quantum mechanics.As we all know, a microscopic particle has no determinate movement locus so that it has no an equation describing its movement locus.Thus, its nature of particle can only be described by its momentum  =  and its energy  = 1 2  2 .Based on the viewpoint of de Broglie, an object particle is of wave-particle duality, which means the particle also has its nature of wave.The nature of wave should be shown by its wave function , and the wave function  should be the solution of Schrodinger Equation.The wave as being the solution of Schrodinger Equation is called de Broglie wave.Then Born gave Schrodinger Equation the statistical interpretation of de Broglie wave, which means that || 2 should be a kind of probability density function.So || 2 is often called probability wave.In fact, in quantum mechanics, the probability wave || 2 is much more important than the wave function  itself.
It is worth noting that, the probability wave |  ()| 2 describes the probability density that the particle  appears at  in [0,1] when the quantum number is .Because the particle  does one-dimension motion along  axis, |  ()| 2 is a curve on two-dimension plane.It is well-known that the wave nature of simple harmonic wave is constructed by its frequency  and its wave length .When the quantum number is , its energy , and the wave frequency is as following: This just gives the result that   ⋅   = 1, which means that the relation between the wave nature and the particle nature can be established by using Planck number ℏ.
Now we return to continue to discuss the motion of projectile.Its mass point nature reflected in its equation of locus.
Especially, when  =  4 ,  0 = √, the equation of locus is as follows: Because this sequence of conditional mathematical expectations as being {  ()} =1 ∞ uniformly converges to  = () in [0,1], for arbitrarily given a  > 0, there must exist a natural number  ∈ ℕ + , such that where ‖⋅‖ is a kind of norm in the linear normed space ([0,1], ‖⋅‖) and defined as the following: For  > 0 is small enough, that ‖  − ‖ <  means that the difference between   and  is very small so that   can be replaced by  approximately.
We now take notice of the following important expression:     Above discussion reveals an important conclusion: the motion of mass point in classic mechanics is surely of waviness so that the motion of mass point in classic mechanics also has wave mass point duality, which is very same with wave-particle duality in quantum mechanics.
Furthermore, the relationship between the wave nature and particle nature is established by means of Schrodinger Equation and the energy of the particle  and the momentum of the particle  can be respectively expressed by the frequency  and the wavelength  of the particle as the following:  = 2ℏ,  = Because  = () is the equation of locus of motion of the mass point, it completely represents the mass point nature of motion of the mass point; while (, ) is the probability density function which is the probability wave of itself so that (, ) Given arbitrarily a non-constant function () ∈ [0,1], there must exist some microscopic particles such that the limit of the group behavior of these microscopic particles is just this continuous function () when the quantum number  → ∞.
Particularly, an interesting and very important conclusion is introduced as the fact that the mass point motion of a macroscopical object possesses a kind of wave characteristic curve, which is called wave-mass-point duality.
In this paper, we reveal an important problem: unified theory of classic mechanics and quantum mechanics.So-called unified theory here means almost every motion of a mass point in classic mechanics can be represented by the motions of an infinite sequence of particles in quantum mechanics, where limit operation plays an important role in the unified theory.Clearly this situation is just according with Bohr's Correspondence Principle.
It is worth noting that this kind of correspondence relation between classic mechanics and quantum mechanics cannot be expressed by the relationship between the mass point nature in classic mechanics and the particle nature in quantum mechanics because of Heisenberg's Uncertainty Principle (see Figure 14).As we all know, in classic mechanics, the motion of a mass point has no uncertainty so that we can use continuous functions to describe the movement locus of the mass point.However, in quantum mechanics, the motion of a particle has surely uncertainty so that we cannot use continuous functions to describe the movement locus of the particle.By now, we have known that the position and momentum of a particle are all random and they are related by Planck constant ℏ, i.e.,     ≥ ℏ 2 . Fortunately, we have pointed that the motion of a mass point in classic mechanics has also waviness in Section 5.The wave function of the motion of a mass point has surely no uncertainty.On the other hand, although the motion of a particle has surely uncertainty, the wave function of the particle must have no uncertainty.Thus, we can consider the relation between the wave function of a mass point in classic mechanics and the wave functions of some particles in quantum mechanics.As we discussed in Section 4, we have revealed the relation by means of Theorem 3. In other words, by using wave functions of both classic mechanics and quantum mechanics, classic mechanics and quantum mechanics are unified, which is the significance of our unified theory about the two kinds of mechanics.
We need to emphasize our new and important and interesting conclusion: The motion of a mass point has also so-called duality: wave-mass-point duality, which is very similar to the case of the motion of a particle in quantum mechanics and is an important support to our unified theory on classic mechanics and quantum mechanics.It is not difficult to understand that Theorem 3 should be the most important in physics.
Prigogine had ever pointed out his conclusion by many experiments: world is random not certain [12].In fact, Theorem 1 just prove his idea, because, as we all know, a large part of physical phenomenon can be described by some kind of continuous functions, and based on Theorem 1, any one of these continuous functions must be the limit of the sequence of conditional mathematical expectations of a sequence of random vectors.
At last, we should state the fact that, these results in this paper can be easily extended to the cases of multivariate continuous functions based on the methods in section 2.
Then by the same way we know that {  ()} =1 ∞ uniformly converges to () in  = [, ].The proof of the theorem has been finished.
continuous at everywhere (of course, in above case, there is only one partial derivative Then we get the following result: () =    .And then we pay attention to the equation we solve out the values of  as follows:

𝜋 2
phase position to the right side.For visualization, the function   () can be vividly called Adam wave function and   () be called Eve wave function.In fact, we care more about the function family of essence wave functions of Adam and Eve wave functions, denoted by {  (),   ()} =1 ∞ , and we can call   () to be Adam essence wave function and   () to be Eve essence wave function.Clearly   () and   () are defined on the unit interval  = [0,1].
definition of wave length, we know the wave length is   = motion of projectile where  = 0,  = 1, where   (, ) is a binary probability density function.When the quantum number  = 5, 10, 15, the graphs of the probability density function   (, ) are respectively shown in Figures 10-12.

Figure 13 .
Figure 13.Graph of the motion of projectile.

Figure 14 .
Figure 14.Unified frame of two kinds of mechanics.
So above result is correct.By this result, for any  ∈ ℕ + , the following expression Now we turn to prove the fact that the sequence of unary functions {(  |  = )} converges to () everywhere in [, ].In fact, for any  ∈ [, ], clearly we have Now we prove the fact that the sequence of conditional mathematical expectations {(  |  = )} uniformly converges to () in [, ].