Kunming Xu
Environmental Science Research Center Xiamen University, Fujian Province 361005, China
kunmingx@xmu.edu.cn
Following a previous proposition of quaternity spacetime for electronic orbitals in neon shell, this paper describes the geometrical course each electron takes as it oscillates harmonically within a certain quaternity space dimension and provides the concrete connections between geometries and trigonometric wavefunctions that observe Pythagorean theorem. By integrating four quaternity space dimensions with conventional Cartesian coordinate systems in calculus, we explain electronic motions by Maxwell’s equation and general Stokes’ theorem from the principles of rotation operation and space/time symmetry. Altogether with the previous reports, we have effectively established quaternity spacetime as a successful theory in elucidating the orbital shapes and motions of electrons within inert atoms such as helium and neon. We point out once again that 2px, 2py, and 2pz orbitals have different geometrical shapes as well as orthogonal orientations, contrary to the traditional 2p orbital model.
Keywords: electronic orbital; harmonic oscillation; spacetime; Pythagorean theorem; Stokes’ theorem；quaternity
1. Introduction
In a previous report [1], we introduced a fresh spacetime concept to account for a four-dimensional spherical layer such as neon shell where eight electrons are oscillating harmonically in four various space and time dimensions, obeying quaternity equation individually and simultaneously:
4
∂4Φ4∂Φ=v ∂t4∂l4
(1)
where Φ is electronic wavefunction, l is a generalized space dimension, and t is a generalized time dimension in neon shell. As typical solutions to quaternity equation, we have also characterized 2s2p electrons by eight trigonometric wavefunctions:
1
&sinΩcosΩ⎛cosΩ0cosΩ2−Ω⎞⎛Φ0⎞002⎜⎟⎟⎜
&Φ⎜−v(sinΩ0sinΩ2+Ω0cosΩ0sinΩ2)⎟⎜1⎟
⎜2⎜Φ⎟&sinΩcosΩ)⎟ΩΩ−Ω(coscosv202002⎟⎜⎜⎟
3&cosΩsinΩ)⎟⎜−v(sinΩ0sinΩ2+Ω⎜Φ3⎟002
⎟ ⎜⎟=C⎜4
&ΦΩΩ+ΩΩΩ(coscossincos)v02002⎟⎜⎜4⎟
3&cosΩsinΩ)⎟⎜−v(sinΩsinΩ−Ω⎜Φ5⎟02002⎜2⎟⎜⎟&Φ⎜v(cosΩ0cosΩ2+Ω0sinΩ0cosΩ2)⎟⎜6⎟
⎜⎟⎜−v(sinΩsinΩ−Ω&cosΩsinΩ)⎟⎝Φ7⎠02002⎠⎝&=−∂Ω0 Ω0
∂t
(2)
(3)
& is a complex number where C is a constant, v denotes a generalized velocity dimension, Ω0
notation and a gateway function connecting both 1s and 2s2p layers, and Ω0 and Ω2 are time and space radian angles respectively, relating to two curvilinear vectors of 1s electrons that observe rotation relation:
−
∂Ω0∂Ω2
= ∂t∂l
(4)
Between every two adjacent electrons in neon shell, there is always a rotation relation distance, i.e., a differential operation with respect to a minus time dimension accompanied by an integral operation over a space dimension. For instance, the relationship between wavefunctions Φ0 and Φ1 can be characterized by or
Φ1=∫−
∂Φ0
dl ∂t
(5)
−
∂Φ0∂Φ1
= ∂t∂l
(6)
which can be derived by applying relationship (4) to wavefunctions in equation (2). For brevity, we shall only consider the relationship of their first terms because the first term can be treated as a dynamic pointer pointing towards the head of the whole electronic curvilinear vector.
−
∂Φ01∂Φ11
=∂l∂t
(7)
where Φ01 and Φ11 refer to the first term of Φ0 and Φ1in equation (2) respectively. Equation (7) indicates that the rate of time component reduction from a full time dimension to vanishing is equal to the rate of space component change from a full space dimension to vanishing, which can be equivalently expressed in trigonometry as:
cosΩ0=sinΩ2
(8)
Thus radians Ω0 and Ω2 constitute two acute angles of a right triangle so that equation (8)
2
is an alternative expression of Pythagorean theorem. By the way, since both complementary angles represent time and space radians, respectively, they must coexist as a product in the wavefunction, i.e., one cannot substitute the other.
The forgoing deduction has adopted the following new interpretation of calculus on trigonometric functions:
Definition 1. For a trigonometric function such asy=cosθ, a differential operation of y with
respect to radian θ means increasing θ variable a displacement of π2 in the function. Definition 2. For a trigonometric function such as z=cosϑ, an integral operation of z over
ϑ means reducing ϑ variable a displacement ofπ2 in the function.
The correctness of both definitions can be easily proved by the property of cosine and sine functions in calculus. The difference between conventional differentiation concept and the above radian angle rotation definitions is that the latter traverses the full range of π2 radian whereas the former only catches the terminal state at a certain radian value. The latter is actually the dynamic implementation of the former. Thus, in a more general manner, the process of a differential operation on a wavefunction can be expressed as a cosine or sine function with a changing radian variable, which leads to two theorems with regard to dynamic differentiation:
Theorem 1. For a trigonometric wavefunction y, the derivative term −∂y∂t can be
expressed as the changing rate of variable y from a full time dimension t to vanish following a cosine rule such as
−
∂y
=Acosθ ∂t
(9)
where A is an orthogonal quantity to cosθ, t is a fix time dimension, and the sinuously changing range of y (t, 0) value corresponds to radian range θ(0, π2) in the dynamic process.
Theorem 2. For a trigonometric wavefunction z, the derivative term ∂z∂l can be expressed
as the changing rate of variable z from a full space dimension l to vanish following a sine rule such as:
∂z
=Asinϑ ∂l
(10)
where A is an orthogonal quantity to sinϑ, l is a fix space dimension, and the sinuously changing range of z (l, 0) value corresponds to radian range ϑ(π2, 0) in the dynamic process.
By interpreting differential operations in terms of trigonometric functions, we have provided the implementation for dynamic differential processes. Conventional differential
3
operation is only concerned with the final result of the operation, ignoring the intermediary course or process. However, the result is only instantaneous locating at a discrete point of θ (such as θ=π2) along a continuous curve while the process is perennial covering the full range of θ(0, π2) variable in the smooth curve. This sheds light on the relationship between the discrete variable and the continuous variable, i.e., the discrete result is only a special point on the continuous curve. Thus the dynamic process of differential operation upon a wavefunction is richer in physical meaning than its mere final result. Given these considerations, we shall examine the process of wavefunction transformations from one state to another via their dynamic pathways in trigonometry and geometry in the following section.
2. Pythagorean theorem in quaternity space
The electron octet as were listed in equation (2) form four pairs of conjugated complex functions, each pair with four velocity dimensions apart: Φ0 and Φ4 are a pair of 2s electrons; Φ1 and Φ5 are a pair of 2px electrons; Φ2 and Φ6 are a pair of 2py electrons; and Φ3 and Φ7 are a pair of 2pz electrons (Figure 1). Because of their various orbital types, electronic rotations from one state to another make four distinctive spatial courses that obey Pythagorean theorem in four unique right triangles. In the following subsections, we shall consider the geometry of four electronic transformations Φ0aΦ1aΦ2aΦ3aΦ4 in details. On the opposite sides of quaternity coordinates, another four electronic rotation steps Φ4aΦ5aΦ6aΦ7aΦ0 observe similar transformation principle but with regard to space and time dimension switches.
Figure 1. Transformation pathways of four conjugated electronic pairs that form four closed circles on quaternity coordinates where each pair orbit along both semicircular arcs of the same shaded circle in the abstract.
4
2.1 One-dimensional harmonic oscillation
Figure 2(a) indicates the state change of a 2s electron to a 2px type along straight line OB. The 2s electron, which initially permeates the entire sphere as dimensionless cloud, gradually contracts its activity sphere while its spherical center moves towards point B, which may represent a solid charge. The transformation corresponds to wavefunction evolution from
&sinΩcosΩ &sinΩcosΩ to cosΩ0cosΩ2 to −Ωin time and from −Ω002002
−vsinΩ0sinΩ2 in space as was indicated by equation (5). The initial term cosΩ0cosΩ2 can be regarded as a four-dimensional time with zero dimensional space. Time dimension reduction is undergoing with space dimension increment as the electron transforms from misty cloud to a real particle. With the increase of Ω2 angle from 0 to π/2, the temporal radius of sphere O reduces from OB to EB. At a specific moment, center O moves to center E, which is the vertical projection of point A along a smooth semicircular arc OAB. In other words, spherical center E is traveling along straight line OB but follows harmonic oscillation principle as if it travels uniformly along semicircular arc OAB.
Figure 2 Electronic transformations track along semicircular arcs of various space dimensions in four harmonic oscillations: (a) 2px-oribtal, (b) 2py-orbital, (c) 2pz-orbital, and (d) 2s-orbital.
5
Sphere O is the maximum four-dimensional time sphere that is ever attained by 2s electron with a wavefunction of cosΩ0cosΩ2 whereas point A is −vsinΩ0sinΩ2 term representing a 2px electronic state. In right triangle OAB, side AB is the time component of the electron, corresponding to the product of hypotenuse OB and cosΩ2, projecting into radius EB, and decreasing, whereas side AO is its space displacement from its initial state, i.e., particle size, corresponding to the product of hypotenuse OB and sinΩ2 factor, which is increasing. The semicircular orbit OAB is the pathway for the electron to cover the abstract distance of diameter OB, the full 1D space element, or the final spatial displacement of the electron from its initial 2s state. At point B, the electron has possessed one-dimensional space OB, and has three-dimensional time accordingly. As the electron traverses along this 1D space dimension, we shall call this dynamic process a 2px orbital.
2.2 Two-dimensional harmonic oscillation
As shown in Figure 5(b), a 2py electron E is traveling along a semicircular arc from points B to C as radian angle Ω2 increases from π/2 to π (i.e., decreases from π/2 to 0 in right triangle BEC). The electron may manifest as magnetic flux during the process. In right
&cosΩsinΩ term spatially and point C denotes triangle BEC, point B denotes −vΩ002&cosΩcosΩ term spatially. At a specific moment along the path, side BE indicates v2Ω002space component of 2py electron while side CE is time component of the electron. Space dimension is increasing while time dimension is decreasing. Diameter BC corresponds to a full 2D space element for the electron to attain, and semicircular arc BEC is the pathway to traverse that dimension. As the electron at B tracks along the semicircular arc to C, it draws the shape of a 2py orbital as a semicircular ring on the plane.
2.3 Three-dimensional harmonic oscillation
Figure 5(c) explains the motion of a 2pz-electron. It is the revolution of semicircular arc BEC around the axis BC. The position of whole arc BEC is determined by angle Ω2 in right triangle DEF. With the increase of Ω2 angle from π to 3π/2 (i.e., from 0 to π/2 in right triangle DEF), arc BEC sweeps a hemispherical surface, which gives the shape of a full 2pz
&sinΩcosΩ spatially whereas final orbital. This initial arc position BDC denotes −v2Ω002arc BFC denotes −v3sinΩ0sinΩ2 spatially. At a specific moment along pz-orbital, the interval between arcs BDC and BEC measured by side DE in right triangle DEF represents space component of the electron whereas chord FE denote its time component. Space increases while time dwindles. A full 2pz orbital has wrapped the 3D hemispherical surface
6
represented by diametrical chord DF. Its geometrical shape is a hollow hemispherical surface such as BDCFE as was shown in Figure 2(c).
2.4 Four-dimensional harmonic oscillation
Figure 5(d) indicates electronic transformation from 2pz to 2s types, which involves the expansion of a spherical surface, in which the radius OE, the maximum radius OG, and tangential line GE form a right triangle. The radial expands via the course of arc OEG. Point
&cosΩsinΩ whereas point G denotes v4Ω&sinΩcosΩ function in O denotes −v3Ω002002spacetime. With the increase of Ω2 angle from 3π/3 to 2π (i.e., decrease from π/2 to 0 in right triangle OEG), side OE is space component of 2s electron, which is increasing gradually, whereas side GE is time component of the electron, which diminishes in the meanwhile. Arc OEG is the pathway to traverse the 4D space element represented by diametrical chord OG.
As the whole spherical surface expands along the radial direction, the electron disperses over the entire sphere so that the geometrical shape of 2s electron is a continuous solid sphere of four dimensions in space. A full 2s orbital, whose time dimension becomes zero rendering the whole spherical space instantaneous, proceeds to further oscillation cycles via
Φ4aΦ5aΦ6aΦ7aΦ0 steps and finally returns to original state Φ0.
Figure 3 Harmonic oscillations of electrons in one, two, three, and four space dimensions in correspondence with radian angle Ω2 revolution in a 2π cycle.
The forgoing description demonstrated the same principle of right triangles, i.e., Pythagorean theorem as was expressed by equation (8). A 2p-orbital is one-dimensionally oriented in space; a 2py-oribital moves on a two-dimensional planar ring; a 2pz-electron sweeps around a three-dimensional hemispherical surface; and a 2s-orbital permeates over the four-dimensional sphere. These provide the visual connections between geometries and trigonometric wavefunctions as were listed in equation (2). Four distinctive diametrical chords OB, BC, DF, and OG in Figure 2(a), 2(b), 2(c), and 2(d), respectively, represent four
7
space dimensions for an electron to traverse in Φ0aΦ1aΦ2aΦ3aΦ4 processes respectively via their subtending semicircular arcs. We shall label these quaternity space dimensions as l3, l2, l1, and l0 that correspond to radian Ω2 revolution in a full period (Figure 3), their spatial orientations being 1D, 2D, 3D, and 4D, respectively. We may generalize l3, l2, l1, or l0dimension as l, which refers to the outstanding space dimension of concern or the most immediate dimension in the wavefunction of concerned.
3. Quaternity space in Cartesian coordinates
So far we have identified four distinct space dimensions l3, l2, l1, and l0 in spherical layer of neon shell, but how these dimension elements are related to Cartesian coordinates in X, Y, and Z orientations? First of all, consider l3 dimension in Figure 2(a), if we set up a one-dimensional X axis along OB direction with the origin at point O, then for the electron to traverse l3 dimension is equivalent to moving along X-axis, which gives the conformity of l3 and X measures for 2px transformation in one-dimensional harmonic oscillation:
∂Φ1∂Φ1
=∂x∂l3
(11)
Secondly, in Figure 2(b) for 2py-orbital transformation, we may add Y-axis to X-axis to form a two-dimensional Cartesian plane as shown in Figure 4. Chord BE is the space component of the electron in quaternity, which form a hypotenuse in right triangle BEP. Following the rule of vector addition, we have
BE=BP+PE
(12)
where vector BE can be expressed as ∂Φ2∂l2 in quaternity space in the dynamic transformation, vector PE can be expressed as ∂Φ2y/∂x in the same dynamic sense, and BP is equal to (OB−OP), which can be expressed differentially as −∂Φ2x/∂y because OB is a constant radius. Thus, equation (12) becomes:
∂Φ2∂Φ2y∂Φ2x
− =∂x∂y∂l2
(13)
or in two-dimensional Curl operator expression as
∂Φ2
=∇×Φ2 ∂l2
(14)
8
Figure 4 Schematic diagram for establishing vector relationship between quaternity space
and Cartesian coordinates.
Furthermore, with regard to 2pz electronic transformation on spherical surface as was shown in Figure 2(c), we may add Z-axis to X-Y axes to form a three-dimensional coordinates system. By expressing the same spherical surface vector in terms of quaternity space and of Cartesian coordinates, we obtain
∂Φ3y∂Φ3x∂Φ3x∂Φ3z∂Φ3∂Φ3z∂Φ3y
−)i+()j+()k −=(−
∂x∂y∂z∂z∂x∂l1∂y
(15)
where i, j, and k are unit vectors along X, Y, and Z directions respectively. Because l1 denotes a spherical surface dimension, the addition of three vectors, i, j, and k, amounts to a vector of constant length that equals the spherical surface radius. With three-dimensional Curl operator, equation (15) can be expressed compactly by
∂Φ3
=∇×Φ3 ∂l1
∂Φ2∂Φ3
= ∂t∂l1
(16)
In light of this, we may express the principle of rotation operation as
−
∂Φ2
=∇×Φ3 ∂t
(18)
−
(17)
which is in the same form as Maxwell’s equation that relates electric field strength E to magnetic field strength B.
−
∂B
=∇×E ∂t
(19)
This means that the relationship between 2py and 2pz orbitals observes electromagnetism strictly. The relationship between 2px and 2py orbitals has the same property.
Finally, from Figure 2(d) for 2s-orbital transformation, we can find the divergence operator as a differential reflection of Pythagorean theorem for three perpendicular
9
orientations:
OE=Xi+Yj+Zk (20)
whence
∂Φ4∂∂∂
=(++)Φ4 ∂l0∂x∂y∂z
(21)
or in div operator expression as
∂Φ4
=∇⋅Φ4 ∂l0
∂Φ3∂Φ4
= ∂t∂l0
(22)
In light of this, we may express the principle of rotation operation as
− (23)
−
∂Φ3
=∇⋅Φ4 ∂t
(24)
which express the general continuity relation between current density J and the charge density
ρat a point:
−
∂ρ=∇⋅J ∂t
(25)
Thus we have developed the relationships between four dimensions in quaternity space and traditional Cartesian coordinates in differential form. These integrations are very important for accurately understanding the concept of quaternity space and lay the foundation for physical integration of electronic harmonic oscillations with electromagnetism. One may interpret the principle of rotation operation in diverse physical manners.
4. Equilibrium of electrons by Stokes’ theorem
We have described electronic motions by trigonometry and geometry as smooth and continuous spacetime evolution processes. Each pair of 2px2, 2py2, 2pz2, and 2s2 electrons oscillate harmonically along a circle of one, two, three, and four spatial dimensions respectively. But how electrons synchronize their motions in various orbital states? We have shown that electronic rotation complies with electromagnetic laws in differential form. Here we shall further investigate the principle of rotation operation in integral form.
4.1 Principle of rotation operation and spacetime symmetry
Every two adjacent electrons obey the principle of rotation operation such as equation (6) where space dimension l refers to l3 that is one-dimensional in correspondence with X-axis as was formulated by equation (11). Equation (6) describes the relationship between a 2s
10
electron and a 2px electron in differential form.
By the principle of space and time symmetry, differentiations upon a wavefunction respect with to space dimension and to time dimension must be equal after balancing their resultant dimensions:
∂Φ0∂Φ0
=v1 ∂t∂l0produces
(26)
where v1 is a velocity serving as a dimension compensator. Combining equations (6) and (26)
v1
∂Φ0∂Φ1
=− ∂l0∂l3
(27)
which means that the rate of a 2s spherical contraction is proportional to the rate of a 2px displacement from the center nucleus during their full cycle oscillations (Figure 2a). The same principles can be expressed in integral form as follows.
4.2 Green’s theorem
In integral form, equation (6) can be expressed as
∫−Φ0dl3=∫Φ1dt
(28)
By the symmetry of space and time components, integrations of Φ1 over time and over space must be equal after balancing their resultant dimensions.
v2∫Φ1dt=∫Φ1dl2
(29)
Combining equations (28) and (29) yields
−v2∫Φ0dl3=∫Φ1dl2
(30)
or in X-Y coordinates of
−v2∫Φ0dx=∫Φ1xdx+Φ1ydy
(31)
which means that the integral of Φ1 over semicircle BEC is counterbalanced by the integral of Φ0 over diameter COB (Figure 5).
According to Green’s theorem, line integral of Φ1 over the entire circle BECPB represents the circular orbital motion of both 2py electrons, which equals the area integral over the enclosed circular region:
C
∫Φ1xdx+Φ1ydy=∫∫(
R
∂Φ1y∂x
−
∂Φ1x
)dxdy ∂y
(32)
where C refers to the periphery of the circle and R refers to the enclosed area. Hence we obtain the integral expression of quaternity space in Cartesian coordinates with a curl operator:
11
C
∫Φ1dl2=∫∫(∇×Φ1)dxdy
R
(33)
Figure 5 Oscillation pathways of two 2py (arcs BEC and CPB) and two 2px (diameters
COB and BOC) where the dotted lines are parts of 2px orbitals though not in the spherical center track of the orbitals.
4.3 Stokes’ theorem
By the principle of rotation operation and space and time symmetry, we may obtain a similar relationship between 2py and 2pz electronic orbitals to equation (30).
−v3∫Φ1dl2=∫Φ2dl1
(34)
where v3 is a velocity dimension. The right-hand side of the equation refers to the area integral of Φ2 over the smooth spherical surface and the left-hand side is the line integral of
Φ1 over the spherical surface boundary (Figure 6). In Cartesian coordinates, the right-hand side of equation (34) can be expressed as
∂Φ2y∂Φ2x∂Φ2x∂Φ2z∂Φ2z∂Φ2y
)dydz+()dzdx+()dxdy (35) −−−∫Φ2dl1=∫∫(
∂yzzxxy∂∂∂∂∂H
or written compactly as
∫Φ2dl1=∫∫(∇×Φ2)dA
H
(36)
where H refers to the hemispherical surface of a 2pz orbital.
From equations (34) and (36) and taking both 2py and both 2pz electrons into considerations, we get
−v3∫Φ1xdx+Φ1ydy=∫∫(∇×Φ2)dA
C
H
(37)
This equation is in the form of Stokes’ theorem that the line integral along the boundary circle of both 2py orbitals counterbalances the area integral over spherical surface of both 2pz orbitals.
12
lower hemispherical surfaces.
Figure 6 Two semicircular arcs in X-Y plane form the line boundary of the upper and
4.4 Gauss’ theorem
Gauss’ theorem is commonly known as divergence theorem. Let V be the spherical region of 2s electron in space with spherical surface boundary H, then the volume integral of the
V and the surface integral of Φ3 over the boundary H of V divergence ∇⋅Φ3 of Φ3 over are related by
V
∫(∇⋅Φ3)dV=∫Φ3dH
H
(38)
On the other hand, in quaternity space we may derive a relationship between 2pz and 2s electrons:
−v4∫Φ2dl1=∫Φ3dl0
(39)
which parallels equations (30) and (34). We realize that the right-hand side of equation (39) is equivalent to the left-hand side of equation (38) concerning the volume integral over the sphere.
∫Φ3dl0=∫∫∫(∇⋅Φ3)dxdydz
(40)
whence or
−v4∫Φ2dl1=∫∫∫(∇⋅Φ3)dxdydz
V
(41)
−v4∫∫(∇×Φ2)dA=∫∫∫(∇⋅Φ3)dxdydz
H
V
(42)
which indicates that the density changes of 2s electrons within the region of space V is counterbalanced by the flux into or away from that region through its spherical surface boundary of 2pz electrons.
The forgoing discussion has used v1, v2, v3, and v4 for referring to various velocity dimensions corresponding to l3, l2, l1, and l0space dimensions with t1, t2, t3, and t4 time dimensions (which we did not distinguish above), respectively. The generalized v refers to the
13
proper velocity dimension of concerned. We may further write equations (31), (37) and (42) in a generalized form:
−v∫dΦi=∫∫dΦi+1;i=0,1,2,3
(43)
where Φi and Φi+1 are two adjacent electronic orbitals that are coupled together through general Stokes’ theorem. Thus each electronic orbital is in dynamic equilibria with its adjacent ones.
We have found that the principle of rotation operation is closely connected with general
Stokes’ theorem. Since Green’s theorem, Stokes’ theorem, and Gauss’ theorem have broad physical implications in dynamics, fluid dynamics, electromagnetism, heat conduction, thermodynamics, etc., one may explain electronic behavior in diverse physical manners. This discovery opens up prospects for unifying classic mechanics with quantum mechanics. For example, ∫Φ1dl2 is an integral about a closed circle and can be regarded as the circulation of the velocity field in fluid dynamics, which corresponds to Z-component of the angular momentum of the electron described by colatitude quantum number in quantum mechanics.
5. Summary
We have described electronic orbitals in neon shell in reasonable geometrical details beyond quantum mechanics. Electrons are oscillating simultaneously and individually in harmonic ways in four various dimensions: a 2px electron is one-dimensionally oriented; a 2py is tracking along two-dimensional semicircular arc; a 2p electron assembles on three-dimensional hemispherical surface; and a 2s electron permeates over the entire atomic sphere that is four-dimensional in quaternity space. These four space dimensions are four diameters that electrons are traversing via their corresponding semicircular arcs during harmonic oscillations. It is interesting that electrons track along the circular arcs, but their space and time components at any moment are measured by their subtended chords. The space and time components of the electron as two sides and the diametrical dimension as a hypotenuse always form a right triangle so that Pythagorean theorem indeed governs electronic motions in any cases.
We have identified four space dimensions in neon shell as l3, l2, l1, and l0 and
established their mathematical connections with conventional Cartesian coordinates in differential forms (see equations 11, 13, 15, and 22) as well as in integral forms. We summarize the integral definitions of four qauternity space dimensions under the context of electronic wavefunctions in neon shell.
14
⎧∫Φ0dl3=∫Φ0dx⎪
⎪∫Φ1dl2=∫Φ1xdx+Φ1ydy
∂Φ2y∂Φ2y∂Φ2x⎪∂Φ∂Φ∂Φ2z
⎨∫Φ2dl1=∫∫(2z−−)dydz+(2x−)dzdx+()dxdy (44)
∂y∂z∂z∂x∂x∂yH⎪
⎪∂∂∂Φ=++dl()Φ3dxdydz∫∫∫∫⎪30
∂∂∂xyz⎩
These equations effectively integrated quaternity space definition with Cartesian coordinate system and made it easy to understand from the conventional perspective. It was striking that Maxwell’s equation, general Stokes’ theorem, and Pythagorean theorem boiled down to the same law as the principle of rotation operation that we have proposed for electronic motions since our first report [2].
Each electronic orbital is a continuous process driven by a radian angle rotation under the new definition of dynamic differentiation and all electronic processes are contiguous in spacetime. Each electron is in equilibria with its adjacent ones. Electronic interaction obeys general Stokes’ theorem that connotes rich physical significances. One may interpret electronic behavior as many ways as general Stokes’ theorem in physical applications. This original article is a further exploration beyond the previous reports. Readers are encouraged to consult the previous article [1] and monograph [2] should any conceptual questions arise.
References
[1] K. Xu, Novel Spacetime Concept and Dimension Curling up Mechanism in Neon Shell, http://xxx.lanl.gov/abs/physics/0511020
[2] K. Xu, Discovering the spacetime towards grand unification, the theory of quaternity,
Xiamen University Press, 1-135, 2005.
15
因篇幅问题不能全部显示，请点此查看更多更全内容