Retrieval from: www.crossref.comAuthor: Liu Huan (1983-), Master of Science (First Class Honours, 2009), TheUniversity of Auckland; Chief Editor, registered Editorial Board Memberships in Webof Science (Researcher ID: KCY-0721-2024).All rights reserved. Formally published on 15/09/2025.This research project is formally registered via Open Science Framework (OSF)Registry, with registration DOI: https://doi.org/10.17605/OSF.IO/KRPC3Abstract/Study objectives: it is first to conduct systematic review of mechanicstheories, from Newton and his classical motion Laws, Einstein and his quantummechanics, Black Body theory, wave-particle duality of De Broglie wave,photoelectric effect and Bohr atomic quantum model, finally to Schrodinger equation,based on which the new theories are proposed and discussed in detail in the first part;Then it is to select representative experiment research with regards to the abovemechanics theories as case studies in this observation research project, and to furthercollect research data for re-editing into adapted version, which are re-analyzed on thebasis of my newly proposed theories; Finally, it is to finish the artificial intelligenceprograms by C++ language, the 3D modeling of quantum chemistry proposed in myprevious article.Key Words: Classical Mechanics; Quantum Mechanics; De Broglie wave; BohrAtomic Quantum Model; Schrodinger Equation; AI and Quantum Chemistry.Hypotheses1.Why can these elementary particles (such as electron or proton) carry stable electriccharges, whereas other elementary particles (such as neutron) carry little electriccharges? If it can be explained, how they generate electric charges?2.Does photon possess the attribute of mass? If it is, how to characterize the mass ofphoton? And how to input this variable of mass into the Mass-Energy formula?3.Is the De Broglie wave of elementary particles significantly different from theclassical materials wave? If it is, what is the difference between them in detail? AndHow to re-calculate the wave function of De Broglie wave for elementary particles?1.Introduction of classical mechanicsQuantum Physics/量子力学2First of all, it is to briefly introduce the milestones of classical mechanics in thescience history that is outstandingly contributed by Newton. Then the applicableconditions and limitations of classical mechanics Laws are discussed to initiate thequantum mechanics chapter in modern science history.1.1.Newton's First law of MotionAny object's state remains in the forms of uniform motion or stationary state, until itis forced to change its state of motion, which is expressed as the equation:= = 0 equation 1Fi is the composition of external forces, and is the accelerated velocity,representing that when the composition of external forces is zero, the acceleratedvelocity is zero, so the object's state remains in the forms of uniform motion orstationary state [24].1.2.Newton's Second Law of Motion-Force and AccelerationThe acceleration of an object is positively relevant to the external force but negativelycorrelated to the mass of the object, with the same direction of acceleration as that ofthe resultant force, which is expressed as equation [25]:F = m×a equation 2In this equation, a is the accelerated velocity.1.3.Newton's Third Law of Motion-Force and AccelerationTwo interacting forces include both external force on object and the correspondingreaction force against the external force, so both are constantly equal in magnitude butopposite in direction acting on the same straight line, which is expressed as theequation [26]:F = -F equation 31.4.Law of Universal GravitationIn the universe, there is the force between any two objects to attract each other, calledthe universal gravitation, whose magnitude is proportional to the product of twoobjects' mass and inversely proportional to the square of their distance between them,with the expression equation [27]:F = (G×M₁×M₂)/R² equation 4Where F represents the universal gravitation, G is the Gravitational constant(6.67×10⁻¹¹ N·m²/kg²), M₁ and M₂ are the mass of two objects, and R is the distancebetween both of them [27].Quantum Physics/量子力学31.5.Limitations and pre-conditions of classical mechanics LawsThe classical mechanics Laws take effect that is subject to two hypotheses: firstly, it isassumed that the variables of both time and space are isolated (absolute) and thetransmission of interaction between two objects is instantaneous, so the measurementof both spatial length and time interval is independent of the motion of the observer,which means that classical mechanics Laws actually apply only to situations underwhich the speed of object is much lower than the light. At high speeds, variables ofboth time and length can no longer be considered independent of the observer'smotion; Secondly, all observable physics quantities can in principle be measured withinfinite precision, which follows that classical mechanics Laws apply only tomacroscopic objects, whereas in microscopic systems, it is in principle impossible tomeasure all physical quantities simultaneously. Therefore, the Laws of classicalmechanics are the generally approximate equations suiting for the macroscopicobjects of low-speed motion [28].1.6.The mathematical application on vector of mechanicsBecause the physical quantities of mechanics are usually the vectors, there are twocommon mathematical methods used to calculate the vectors' parameters inmechanics study, Lagrange method and Euler's method, both of which rely on partialderivatives and differential equations to resolve the component at each dimensionalvector.1.6.1.Lagrange methodThe spatial vector of r of a particle in space can be expressed as:r = r(a,b,c,t) equation 5In the Cartesian coordinate system (x, y, z), this vector formula is expressed as:x = X(a,b,c,t) y = Y(a,b,c,t) z = Z(a,b,c,t) equation 6When variables of a, b and c are fixed, the above equation represents the motionmodel in a specific spatial point; when t is fixed, it indicates the spatial distribution ofall particles at the same moment. Therefore, the above equation describes themovement of all particles in three-dimensional space [29].For a given spatial point, variables of a, b and c are constants, so if it is to find the rateof a physical quantity variation with respect to time, only the partial derivative ofabove equation need to be taken against 't' as [29]:V = = equation 7Quantum Physics/量子力学4u = ,v= , w = equation 8Where V is the speed of a particle in the spatial point (a,b,c), and variables of u, v andw are the component of vector 'r' at x, y, z axis respectively [29].Similarly, the accelerated velocity can be derived by secondly taken the partialderivative against variable 't' as [29]:a=∂2r(a,b,c,t)/∂t2equation 9ax= ∂2X(a,b,c,t)/∂t2ay= ∂2Y(a,b,c,t)/∂t2az= ∂2Z(a,b,c,t)/∂t2Where variables of a, ax, ay, az, are the accelerated velocity, and its component at x, y,z axis, respectively [29].1.6.2. Euler's methodEuler's method is another common method used to resolve the vector of wavefunction in mechanics study, which will be in detail illustrated in the following wavefunction sections. In Euler's method, the Operator of formula is adopted to simplifythe calculation process.2.Einstein and his quantum mechanics2.1.Study objective of quantum mechanicsMatter in nature exists at multiple scales, spanning from the vast cosmic space,macroscopic celestial bodies, conventional objects, tiny particles, fibers, crystals, tothe microscopic scale of molecules, atoms, and elementary particles. Forms ofmechanics motion encompass movement, rotation, flow, deformation, vibration,fluctuation, diffusion, among which the balance and stillness status is the special stateof motion. Mechanical motion stands as the most fundamental form of materialmotion, while other forms of material motion include thermal motion, electromagneticmotion, atomic and internal motion, as well as chemical motion, but mechanicalmotion usually co-exists with other forms of motion [14].Photon that was also defined as light quantum referred to a kind of elementary particletransmitting electromagnetic interactions, which was a gauge boson proposed byEinstein in 1905 and was subsequently officially named by an American physicalchemist, Gilbert Lewis, in 1926. It was proposed that photons were the carriers ofelectromagnetic radiation and in theory of quantum physics, photons were consideredas mediators of electromagnetic interactions. The static mass of photons was deemedto be zero, and photons moved at the speed of light, which possessed the properties ofenergy, momentum and mass [15].Quantum Physics/量子力学5Particularly, modern quantum mechanics has introduced the conceptional model ofhigh-dimensional spaces into its theoretical demonstration, not limiting tothree-dimensional spaces, such as Time-Space proposed by Einstein who argues thatthe fourth dimensional space is defined as the inertial system along time axis, andM-theory proposed by Hawkin who classifies the high dimensional spaces into 11dimensions in the universe. However, my another article has newly modified andredefined the conceptions of high dimensions in quantum mechanics theory [8].2.2.Attributes of photonsWhen Einstein pointed out that in addition to energy ε, photons must also becharacterized by using the attribute of impulse, the direction of which corresponded tothe direction of light propagation, in order to improve the interpretation of photons.According to his theory of special relativity, compared with most elementary particles,photons possessed zero mass in static, which meant that their propagation speed invacuum was the speed of light. Like other quanta, photons showed wave particleduality: photons could exhibit wave properties, such as refraction, interference,diffraction and etc. like classical mechanics waves, while the particle nature ofphotons could be demonstrated by the photoelectric effect. Photons could onlytransmit quantized energy, so they were space-lattice particles [15].After the establishment of the theory in quantum electrodynamics, it was confirmedthat photons were the medium particles for transmitting electromagnetic interactions,so charged particles interacted each other by emitting or absorbing photons, and bothpositively and negatively charged particles could be annihilated, converting intophotons, which could be generated in electromagnetic fields. More specifically, a pairof both positive and negative particles could annihilate into a pair of high-energy γphotons, while a pair of high-energy γ photons could also react at high temperature togenerate a pair of positive and negative particles correspondingly. For example, theconversion of photons into baryons, such as protons and neutrons, could occur at hightemperature T = 1015k. The positron - negative electron scattering, which was alsonamed as Bha-Bha scattering, was interpreted by Feynman diagram, whose wave linerepresented the process of exchanging virtual photons [15].Defined by the special relativity, photons were the particles in light that carried energy,which was proportional to the frequency of the light wave, so the higher the frequency,the higher the energy. When a photon was absorbed by an atom, an electron gainedenough energy to transition from the inner orbit to the outer orbit, turning the atomfrom the ground state to the excited state [15].2.3.Mass-Energy EquationAccording to Einstein's special relativity, the physical attributes could be convertedinto each other among photons' energy, momentum and mass. On the basis of themass-energy equation,Quantum Physics/量子力学6E = mc2= hν equation 10Then the mass of photons could be calculated into: m = hν/c2, where parameters of h,νand c were the Planck constant, frequency of light wave, light speed constantrespectively, and particularly this mass was defined as the relative mass only withoutstatic nature [15].Due to the differentiation of static mass from motion mass, there is the conception of'deficit conservation' proposed: when a cluster of particles aggregates to form acomposite object, due to the existence of interaction energy between the particles andthe kinetic energy based on the relative motion, the aggregated object as a whole isstationary, whose total energy is generally not equal to the sum of the stationaryenergies of all particles. Consequently, the difference between these two is expressedas mass-energy inequation below [16]:E0≠ ∑Mi0×c2equation 11where Mi0is the stationary mass of each elementary particle and is the total energy ofaggregated object. According to the mass-energy equation, it is deduced that thedifference between the two is attributed to the binding energy among the elementaryparticles of the aggregated object, which can be calculated as [16]:∆E = ∑Mi0×c2- E0equation 12Where ∆E is represented as the binding energy of the aggregated object.Correspondingly, the static mass of an aggregated object is not equal to the sum of thestatic mass among the elementary particles that make up it, whose difference betweenthe two is called mass loss (∆M), calculated as [16]:∆M = ∑Mi0- M0equation 13Where M0is the static mass of aggregated object. It is further deduced that there is therelationship between static mass loss and binding energy [16]:∆E = ∆M × c2equation 142.4.Energy - Momentum equationIn vacuum, the velocity of photons was identical to the velocity of light, with therelationship between energy E and momentum P as: P = E/c, under the hypothesis ofzero static mass. It was further deduced that both energy and momentum of photonswere only related to the frequency ν of light wave (or it was only related to thewavelength λ of light wave), so the momentum of photon could be also calculated as[15]:Quantum Physics/量子力学7P = h/λ= hν/c equation 15in comparison, for other elementary particles with static mass above zero inrelativistic mechanics, the energy-momentum (E-P) relationship of a elementaryparticle with static mass (m0) was calculated as [15]:E2= (Pc)2+ (m0c2)2equation 162.5.Explanation and modification of Einstein's special relativityHowever, another viewpoint disagreed with the meaning of the mass-energy equationabove, which argued that the mass-energy equation did not reveal the conversionrelationship between mass and energy, but the mass - energy equation only reflectedthe relationship between mass and energy in quantity. For a closed system in whichmass was conserved and energy was also conserved, when the existing form of matterchanged, the form of energy also changed correspondingly, but the mass was notconverted into energy in the process of material reaction and conversion, so both massand energy represented the properties of matter: mass property described inertia andgravitation, while energy property described the state of the system [16].The mass-energy equation described the relationship between mass and energy, so itdid not violate the law of conservation of mass. The formula showed that matter couldbe transformed into radiant energy, at the same time radiant energy could also betransformed into matter, which did not mean that matter would be eliminated, but thestatic mass of matter would be converted into another motion form [16].Einstein firstly proposed that the formula showed that an object still possessed energywhen it was stationary relative to a reference frame, which was contrary to Newton'ssystem, because in Newton's system, a stationary object was deemed as zero-energy,which became the reason why the mass of an object was called a stationary mass. Thevariable of E in the formula could be regarded as the total energy of the object, whichwas proportional to the total mass of the object, including both static mass and massin motion. Only when the object was stationary, it became proportional to the staticmass of the object that was consistent with the 'mass' definition in Newton's system.He also argued that the total mass of the object was different from the static mass, byillustrating that a beam of photons propagating in a vacuum was deemed to possesszero static mass, but they showed motion energy, so they also had mass [16].It was further explained that the static energy of an object was identical to its totalinternal energy, including the kinetic energy of molecular motion, the potential energyof inter-molecular interaction, the chemical energy that combined atoms with atoms,the electromagnetic energy that combined nuclei with electrons in atoms, and thebinding energy connecting protons and neutrons in nuclei. The revelation of the staticenergy of matter was considered as one of the most important corollaries of relativity,which pointed out that there was still motion inside a static particle that had certainQuantum Physics/量子力学8internal motion energy with certain mass; in turn, a particle with certain internalmotion energy showed certain inertial mass correspondingly. In the process ofelementary particle transformation, it was possible to release all the static energycontained in the particle, turning it into usable kinetic energy. For example, when a πmeson decays into two photons, the static mass of the photons is zero and staticenergy disappears, so the π meson contains all the static energy of matter before itdecays [16].In summary, this mass-energy equation critically contributes to the development ofthe atomic bomb. By measuring the mass difference between the mass of nuclei andthe sum mass of both independent protons and neutrons at the corresponding quantity,it is to obtain the estimated value of the binding energy contained in the nucleus,which does not only show the phenomenon that the nuclear binding energy may bereleased through both nuclear fusion (for lighter nuclei) and nuclear fission (forheavier nuclei), but also can be used to quantitatively estimate the amount of bindingenergy that will be released. It is worthwhile noting that the masses of both protonsand neutrons do not disappear over the nuclear reaction process, whose mass alsorepresents the energy value. This equation stems from Albert Einstein's study withregards to the relationship between the inertia of an object and its own energy. Thefamous conclusion of his study is that the mass of an object is actually to measure itsinternal energy. In order to better understand the importance of this relationship, it isto compare electromagnetic force with gravity force. According to the theory ofelectromagnetism, energy exists in both electric field and magnetic field that is relatedto force but independent of electric charges, whereas in the theory of universalgravitation, energy is contained in matter itself. Therefore, it is concluded that themass of matter can distort space-time in his special relativity, but the other three basicinteractions among elementary particles, including electromagnetic interaction, stronginteraction and weak interaction, cannot achieve the space-time distortion [16].2.6.Critical discussion of quantum mechanicsIt is to further clarify some conceptions originally proposed in my previous articles:2.6.1.Elementary particles and electric charge generationIn my previous article [10], it has been proposed that the magnetic line on the fourthdimension axis is perpendicular to the three-dimensional space and these magneticelementary particles cut along the magnetic line on the fourth dimensional axis, thusgenerating electric charges. My another article has proposed that "in ourthree-dimension space, the protons are positively charged and the electrons arenegatively charged; the materials in correspondingly symmetric three-dimensionalspace along the fourth dimension axis is called antimatter, in which the protons arenegatively charged and the electrons are positively charged [3]." Consequently, themagnetic line that is perpendicular to our three-dimensional space is definitely themagnetic line connecting both symmetric three-dimensional spaces, and theelementary particle motion cutting along this perpendicular magnetic line is theQuantum Physics/量子力学9spinning motion of elementary particle by itself. It can be easily deduced that thedirections of spinning motion are opposite between protons and electrons along thisperpendicular magnetic line connecting both symmetric three-dimensional spaces, sothat their electric charges are positive and negative correspondingly.My quantum physics article has defined that "quantum photon is the mass particle thatis coated and surrounded by dark matter in the fourth dimension, so that its static masscan be hardly detected by particle collider instrument, but this particle must be themass carrier for quantum energy." It is further proposed that photons generateelectromagnetic waves by the wave-like transmitting movement inside the dark matter,and this wave-like transmitting movement cuts the magnetic line along the fourthdimension axis that is parallel to the sphere surface of our three-dimensional space(different from the perpendicular magnetic line connecting both symmetricthree-dimensional spaces). Electromagnetic waves are the energy form matter existingonly as electric field and magnetic field. However, if the electromagnetic wave ofenergy form leaves the the carrier of photons, it can be easily absorbed by othermaterials (including dark matter) in this universe.It is clarified that there are two kinds of the fourth dimensional axes defined in myjournal: one is the axis along the magnetic line that is perpendicular to ourthree-dimensional space and connects two symmetric three-dimensional spaces, andthe other one is the axis along the magnetic line that is parallel to the sphere surface ofour three-dimensional space, as my previous articles [17][18] have demonstrated thatthe three-dimensional space where we are living is the curved sphere. Myelectromagnetism article has discussed the parallel space that "the propagationdirections of the electric field and the magnetic field are parallel and opposite in thefourth dimension axis, but in the corresponding three-dimensional space, they turn tobe the mutually vertical propagation directions due to the refraction effect [2]," andthis magnetic line in the fourth dimension axis refers to the first magnetic line that isperpendicular to our three-dimensional space and connects two symmetricthree-dimensional spaces. In summary, the generation of electric charges by thespinning motion of elementary particles in atom, the generating of electromagneticwave by the wave-like transmitting movement of photons, and the change of magneticline propagation direction due to the refraction effect at the sphere surface of ourthree-dimensional space are further demonstrated in Figure 1, Figure 2 and Figure 3,respectively.Quantum Physics/量子力学10Figure 1 illustrates the spinning motion of elementary particles in atom, generatingelectric charges. In this conceptional model, it is hypothesized that protons spin fromleft to right direction, while electrons spin from right to left, and both proton andelectron's spinning motion cuts the perpendicular magnetic line connecting bothsymmetric three-dimensional spaces, thus generating positive and negative chargesrespectively. In comparison, neutrons spin from up to down, whose spinning motiondirection is parallel to the perpendicular magnetic line connecting both symmetricthree-dimensional spaces, consequently generating little electric charges.Figure 2 illustrates the generation of electromagnetic wave by the wave-liketransmitting movement of photons in dark matter. The wave-like movement of photoncan be divided into two sub-vectors: one is the transmitting movement (indicated byQuantum Physics/量子力学11the blue arrow) along the magnetic line that is parallel to the sphere surface of ourthree-dimensional space, releasing magnetic field energy, and the other one is thecutting movement (indicated by the red arrow) against this magnetic line, releasingelectric field energy. Consequently, this motion model also reveals both magnetic andelectric energy distribution characteristics of electromagnetic wave in three-dimensionspace.Further more, when the elementary particles decay, it displays as the spring-likeshaking movement, whose direction is vertical to the sphere surface of ourthree-dimensional space (in Figure 1), thus generating longitudinal wave (such asrays); in comparison, the electromagnetic wave generated by the wave-liketransmitting movement of photons in dark matter is the transverse wave.In Figure 3, there are two symmetric three-dimensional spaces outlined (ourthree-dimensional space is on the left and the symmetric three-dimensional antimatterspace is on the right in Figure 3). Between both symmetric three-dimensional spaces,the space is defined as the 'parallel space', in which the propagation directions of theelectric field and the magnetic field are parallel and opposite (the red line indicateselectric field line and the blue line indicates magnetic field line). In ourthree-dimensional space, the propagation directions of magnetic line and electric linebecome mutually vertical, but the refraction effect at the sphere surface of ourthree-dimensional space leads both to change the propagation directions as shown inFigure 3.2.6.2.Mass-Energy equationMy article agrees with Einstein's special relativity which firstly argues that mass canbe converted into energy, disagreeing with the 'energy/mass conservation Law.' It isfurther proposed that nuclear fusion reaction converts the mass of elementary particlesinto energy, while nuclear fission reaction converts the mass of dark matter intoQuantum Physics/量子力学12energy that is also the indicator of binding energy among elementary particles.Consequently, the mass of matter, including dark matter and elementary particles, isregarded as a kind of the form for energy storage.2.6.3.The energy-momentum equation of elementary particleFigure 2 has divided the movement of photon into two sub-vectors, indicated by theblue arrow and red arrow respectively, so the momentum of photon is correspondinglydivided into two kinds of vector momentum as well. It can be deduced that the cycleof the wave-like mechanical movement of photon (shown in Figure 2) determines thefrequency (ν) of the correspondingly generated electromagnetic waves, so therelationship between momentum of photons and energy can be further modified into:the electric energy (Ee) is positively correlated with its corresponding momentum(indicated by the red arrow) of photon and is negatively correlated with thewavelength (λ) of the wave-like mechanical movement (or positively correlated withthe frequency of the wave-like mechanical movement). Consequently, the vectormomentum of photon (indicated by the red arrow) can be calculated into:P1 = a×Ee/λ equation 17where the variable P1 and parameter a is the vector momentum of photon (indicatedby the red arrow) and its corresponding constant, respectively; the magnetic energy(Em) is positively correlated with the light speed (c) and its corresponding momentum(indicated by the blue arrow), so the vector momentum of photon (indicated by theblue arrow) is calculated as:P2 = b×Em×c equation 18where the variable P2 and parameter b is the vector momentum of photon (indicatedby the blue arrow) and its corresponding constant, respectively. Obviously, thevariables of both Eeand Emof light waves can be feasibly measured in experiment, sothat variables of both P1 and P2, together with constant a and b, can becorrespondingly estimated according to the regression analysis.In my quantum mechanics article, it is argued that "when two beams of chargedparticle collide in the particle collider, under the condition that the law ofelectromagnetic induction can be ignored, the kinetic mechanics of a beam of chargedmicro-particles only conforms to the principle of fluid mechanics (such as pressurecalculations), and is not applicable on the mechanical energy law of solid collision(such as conservation of momentum).[1]" My article further proposes that the beamsof charged elementary particles in the particle collider and the emitted chargedelementary particles during the atom decaying process also show the wave-likemechanical movement over the transmitting process due to the nature of dark-matter,which is similar to photon's movement shown in Figure 2, but their static mass can bedetected in experiment, because their aggregated masses display in ourQuantum Physics/量子力学13three-dimensional space. However, the static mass of elementary particles is notconstant in our three-dimensional space, as my quantum mechanics article hasproposed that "These elementary micro-particles randomly fluctuate along the fourthdimension axis, so the mass of micro-particles in the three dimensions space isn'tconstant.[1]" Hence Figure 4 of this article has further illustrated this movementmodel of the beams of charged elementary particles in the particle collider in moredetails.As shown in Figure 4, the movement model of a beam of charged elementary particlesin the particle collider instrument displays as the wave-like transmitting motion due tothe nature of dark matter. The boundary between the dark matter of the fourthdimensional axis and our three-dimensional space becomes the line partitioning thestatic mass that displays in our three dimensional space and can be consequentlymeasurable by using equipment. At the wave peak, the static mass displaying in ourthree dimensional space is the highest, then decreasing with the transmitting direction,and shows the smallest at the wave bottom position. Consequently, the static mass ofelementary particles that display in our three-dimensional space varies with thewave-like transmitting motion, so the energy-momentum (E-P) equation forelementary particles with static mass is further modified into:E2= α×(P×ν)2+ (m0×c2)2equation 19In this equation, the part of α×(P×ν)2represents the energy generated by the vectormotion (indicated by the red arrow) that is vertical to the boundary between darkmatter and three-dimensional space, in which the parameter α is the constant; theequation part of (m0×c2)2represents the energy generated by the vector motion(indicated by the blue arrow) that is parallel to the particle acceleration motiondirection, in which the variable m0is the static mass of elementary particles, and thevariable c is the acceleration motion speed.3. Black body3.1.Blackbody radiation and absorptionQuantum Physics/量子力学14An 'absolute blackbody' is a theoretical model that refers to an object absorbing alllight at any wavelengths without any reflection. Once a beam of light wave penetratesinto a cavity through a slit hole, it is difficult for the light wave to pass through thesame slit hole in the opposite direction, and the opening of this cavity can beconsidered a blackbody (Figure 5 (left)). In theory all objects in the world areconstantly absorbing and emitting infrared radiation, and a blackbody is considered asthe ideal emitter and absorber of infrared radiation in this theory. For example, blackholes are the typically idealized absolute blackbodies, which absorb the infraredwaves of all wavelengths [19].Experiments have shown that under the condition that a blackbody reachesequilibrium with thermal radiation, the radiation energy density (Er) is only related tothe thermodynamic temperature (T) of the blackbody, which is independent of theshape and composition of the cavity[19]. In the radiation spectral line of blackbodies,the color of light varies with the temperature ascending, which exhibits the order oftransition from red → orange red → yellow → yellow white → white → blue white.When the color of a light source is adjusted to be the same as the color of light that isemitted by a blackbody at a certain temperature (T), then this critical temperature ofthe blackbody is called the color temperature of the light source. The higher thetemperature of a blackbody, the more blue light wave components contained in itsspectrum, while the less red light wave component is emitted from the light source.For example, the color of incandescent lamps is warm white, with a color temperatureof 2700K, while the color temperature of daylight fluorescent lamps is 6000K,emitting fluorescent light waves [20].Kirchhoff Radiation Law hypothesizes that under thermal equilibrium, the ratio of theenergy radiated by an object to the absorptivity is irrelevant with the physicalproperties of the object itself, but is only related with wavelength and temperature.According to Kirchhoff Radiation Law, at a certain temperature, a blackbody must bethe object with the greatest radiation capacity, consequently called as the completeradiator, which means that blackbody radiation reaches the maximum amount ofemission at a specific temperature and wavelength. At the same time, blackbody is anobject that can absorb all incident radiation and will not reflect any radiation, thecolour of which is not necessarily black. For example, the sun is a gaseous planet,which can be considered that the electromagnetic radiation emitted to the sun isdifficult to be reflected back, so the sun is considered that the sun is a blackbody,although the absolute blackbody does not really exist. Theoretically, blackbody doesnot only absorbs electromagnetic waves of all wavelengths in the spectrum, but alsoemits electromagnetic waves of all wavelengths in the spectrum. Wien's displacementlaw describes the relationship between the peak wavelength of the electromagneticradiation energy flux density and the temperature of blackbody, which is furtherdeveloped by Planck [20].Planck hypothesis: there are charged linear harmonic oscillators in the radiationQuantum Physics/量子力学15material, and these harmonic oscillators can only exist under certain States. Underthese States, the corresponding energy is the integral multiple of the minimum energyε that is defined as energy quantum, so the radiation energy is expressed as: ε, 2ε, 3ε,nε,..., among which n is a positive integer. According to this hypothesis, the famousPlanck blackbody radiation formula is derived:ρ (v,T) = ×11/kThCeequation 20Where ρ is the radiation energy density, v is the frequency of harmonic oscillators, Tis the temperature, h is the Planck constant, λ is the wavelength, c is the light speed,and k is the Boltzmann constant. In this Planck blackbody radiation formula, theradiation energy density is the function of variable v and T, which is the improvedformula of Stefan-Boltzmann formula.3.2. Original discussion of blackbody radiation and absorptionMy article has firstly proposed that "According to the Figure 1 of my article [11], it isto further discuss the argument of the shielding effect of the electric field inside anatom and its effects on the electron orbitals; Multiple equipotential lines are formedbetween the zones of constructive interference and the zones of destructiveinterference." Consequently, it is here to argue that the shielding effect of theequipotential field lines inside an atom or a molecule causes the phenomenon of'blackbody' radiation and absorption described above.As can be seen from Figure 5 (left), the structure of sphere 'shell' and the slit hole onthe shell explains the old 'blackbody' formation theory, but my new theory proposesthat the equipotential field lines inside an atom or a molecule play the role in theshielding effect instead of the 'shell' of 'blackbody'. After the radiation waves haveQuantum Physics/量子力学16penetrated into the equipotential field lines, this shielding effect does not only stop theradiation waves from releasing out of the atom, but also causes a small proportion ofradiation waves that reflect on the surface of the shielding lines, shown in Figure 5(middle). Consequently, an 'absolute blackbody' does not really exist. When thetemperature ascends, free electron shifts from inner atomic orbital to the outer orbital,so that the equipotential field line causing shielding effects is shifted correspondingly(shown in Figure 5 (right)), which results in the variation of 'blackbody' radiationwaves, expressed as different color temperature discussed above.It is further concluded that the shielding effects of the equipotential field line inside anatom or molecule plays the important role in the conserving the energy of an atom ormolecule, and keeping the stable structure of elementary particles, whose mechanismis similar to the 'blackbody' theory.4.Wave particle duality of substance waves4.1.From classical wave to De Broglie waveDe Broglie wave, also named as the material wave, is a kind of wave expressingspatial probability in microscopic particle occurrence, which refers to the probabilitythat may occur in a specific spacial point at transient time, and this particle occurrenceprobability is controlled by the fluctuation law. Mechanical wave is the propagation ofperiodic vibration in the medium, and electromagnetic wave is the propagation ofperiodic electromagnetic field, whereas material wave is neither mechanical wave norelectromagnetic wave [21]. Thus the material wave is proposed to explain the lightwave at quantum level, which can be hardly deduced by the classical mechanicalwave or electromagnetic wave theories.Davidson and Dermot emitted single energy electron onto the polished plane of nickelsingle crystal, aiming to observe the quantitative relationship between the intensity ofscattered electron beam and scattering angle. Both emitting source of electron beamand the scattered electron detector were symmetrically placed on the normal of thecrystal surface, and the experiment showed that the intensity of the scattered electronbeam varied with the scattering angle (θ). When the scattering angle was adjusted tobe a critical value, the scattering intensity reached the maximum value, which was thesame as the phenomenon of X-ray diffraction, fully proving that the electron hadwave particle duality. Because the electrons with low energy (in Davidson'sexperiment, the electron energy was set to be about 30~400 ev), which could notpenetrate deeply into the crystal, most of the electrons were scattered on the crystalsurface [19].According to the diffraction theory, the position of the maximum diffraction value isdetermined by the following formula [19]:nλ = 2d×cos(0.5θ) n = 1, 2, ... equation 21Quantum Physics/量子力学
ere n is the order of diffraction maximum, λ is the wavelength of the diffractiveray, and d is the crystal plane spacing of crystal Bragg scattering. Setting up parameterof d (the crystal plane spacing of nickel single crystal) = 0.091nm, when the electronbeam energy is 54 ev and the scattering angle (θ) is 50°, respectively, a maximumdiffraction peak is observed. According to the Bragg equation, the electronwavelength can be calculated to be 0.165 nm [19].There is the parallel calculation according to the de Broglie relation, the wavelengthof the electron can be also calculated in another way [19]:λ = =equation 22In this equation, λ is the wavelength, c is the electron speed, h is the Planck constant,P is the electron momentum, ћ is the reduced Planck constant, ћ = h/2 π, meis theelectron mass, and E is the kinetic energy of electron. Many experiments have shownthat, in addition to electrons, all the other microscopic particles at various sizes, suchas neutrons, protons, mesons, atoms, molecules and even C60 molecules, also displayas the wave motion, so de Broglie formula is also applicable on these particles.Therefore, de Broglie formula is a basic formula to express the wave particle dualityof various microscopic particles [19].4.2.The wavelength of electron de Broglie waveUnder the conditions that the kinetic energy of a free particle is E, the momentum is p,and the particle velocity is much lower than the speed of light, then the de Brogliewavelength of electron's material wave is calculated as [19]:λ = = equation 23Where is the electron mass. If an electron is accelerated under the electric fieldwith the potential difference of U in a transmission electron microscope, and usuallythe acceleration voltage of an electron microscope is 200~300 kV, then the electronkinetic energy is calculated as: E = eU, among which e is the electron charges, so thede Broglie wavelength of electron's material wave is further estimated as [19]:λ ≈ (Å) equation 24After determining the specific values of parameter h, μ, and e, the de Brogliewavelength of electron's substance wave is estimated as: when U = 150 kV, λ = 1Å=10-10 m; when U = 10000V, λ = 0.122 Å. It can be seen that the wavelength of the deBroglie wave of electrons is very short, whose order of magnitude is equivalent to theatomic spacing in crystals, much shorter than macroscopic linearity of mechanicalQuantum Physics/量子力学18wave. Therefore, the wave particle duality of physical particles, which is derivedempirically under general macroscopic conditions, will not be applicable on themicroscale any more (particle nature is the main contradiction aspect), so a newmechanics - quantum wave dynamics - must be established to cope with the quantumlevel material wave [19].4.3.The substance plane waveIn classical mechanics, if the angular frequency is ω and the wavelength is λ, thepropagation motion of substance plane wave along the x-axis direction can beexpressed as [19]:ψ(x,t) = A×cos(k·x - ω·t) equation 25Where the wave vector of k = 2π/λ; the constant A is the wave amplitude; t is thepropagation time [19].For the convenience of calculation, it is to transform the above wave function byintegrating the Euler formula: eix = cos(x) + i×sin(x), among which parameter i is theimaginary unit. Then the substance plane wave function can be expressed as [19]:ψ(x,t) = A×exp[i×(k·x-ωt)] equation 26This wave function is used to describe the X-rays motion that displays as Braggdiffraction phenomenon. Since microscopic particles such as free electrons are alsocapable of showing diffraction stripes similar to X-rays, the de Broglie wave of freemicroscopic particles can also be described by this wave function. By inputting theabove equations, E = hν and λ = h/p, into this classical mechanics function, the freeparticle plane wave function is expressed as [19]:ψ(x,t) = A×exp[×(p·x-E·t)] equation 27This is the function describing the substance plane wave associated with freemicroscopic particles, the transformed equation of de Broglie wave, which is derivedfrom the classical macroscopic wave function. It describes the motion state of freeparticle by integrating momentum p and energy E, which is the de Broglie wavespecifically describing the free microscopic particles. If the particle motion trajectoryvaries both spatially and temporally under the potential field, its momentum andenergy are no longer constant (or both physical variables are not constantconcurrently), and then the particle is no longer in the form of free particle, so itsmotion model cannot be described by the plane wave derived from the classicalmacroscopic wave function. Instead, as the particle still possesses the attribute ofwave particle duality, it must be described by more complex wave function (e.g.,Bloch wave function describing the electron wave motion in solids) [19].Quantum Physics/量子力学194.4.Photoelectric effectIt is observed that when ultraviolet light or visible light at short-wavelength hits themetal in vacuum, electrons will escape from the metal surface, so this observedphenomenon is called photoelectric effect, and the escaped electron is calledphoto-electron, whose experiment has resulted in the conclusions below [19]:Firstly, in the same period, the number of photo-electrons released from the radiatedmetal surface is proportional to the intensity of the incident light; Secondly, the initialkinetic energy of the photo-electron increases linearly with the frequency of theincident light (ν), regardless of the intensity of the incident light, which means that theintensity of incident light does not influence the kinetic energy of the photo-electrons;Thirdly, when the incident light radiates onto a specific type of metal, there is nophotoelectric effect occurring, if the frequency of incident light is less than the criticalfrequency limit (ν0) of the specific metal, regardless of the intensity of the light, whichmeans that there is the threshold of incident light frequency for a specific type ofmetal to generate photoelectric effect; Fourthly, when the incident light frequencyexceeds the threshold frequency (ν0), the photo-electrons are capable of beingobserved almost immediately as long as the light shines on it (about 10-9s after theincident light hits the metal), no matter how weak the incident light intensity is [19].Firstly, it is to try to explain the photoelectric effect experiment results according tothe classical electromagnetic theory of light as: when the light shines onto the metalsurface, the electrons in the metal are forced to vibrate by the electric field of theincident light, which causes to absorb the energy from the incident light and henceescape from the surface, so the amount of the obtained energy should be related to theintensity of the incident light and the period of the light radiation, but is independentof the frequency. According to this classical mechanics wave theory, as long as thereis sufficient light intensity or sufficient irradiation time, there is always a photoelectriceffect for any incident light frequencies, whose conclusions are all in directcontradiction to the experimental results. Consequently, these conclusions of thephotoelectric effect are incapable of being explained by the classical mechanics wavetheory [19].4.5. Bohr atomic quantum modelThe wavelength of light source can be derived from hydrogen atomic spectroscopy:λ = B× , equation 28where B is a constant, and its estimated value is B = 3.645610-7m; n=3,4,5,... [19].This wavelength equation is extended to the general formula[19]:ṽ = ᶄ × ( -), (where ᶄ is Rydberg constant, K = 2,3,4,...; n>k) equation 29Quantum Physics/量子力学20In this equation, ṽ is the wave number (ṽ = 2π/λ), and thus the conclusions of thehydrogen atomic spectrum are drawn as: the wavelength is determined by thedifference between the two spectral items ( and ); if the K value of first spectralitem ( ) is fixed and the n value of the second item ( ) is given by different values,the wave number of each spectral line in the same spectrum is obtained; if it is tochange the k value of the first term, then different spectrums can be obtained [19].Rydberg constant ᶄ = equation 30The specific values of parameter μ, e, ε0, c and h are input into the above formula, andthe R value accurately agrees with the values measured by the experiments [19].However, these conclusions derived from the hydrogen atomic spectrum experimentare unexplained by the classical electrodynamics theories. Firstly, the classical theorycannot establish a stable atomic model to explain the continuous rotation motion ofelectrons around the nucleus. According to the classical electrodynamics, themovement of electrons around the nucleus is accelerated, generating the continuousradiation of electromagnetic waves, which results in continuous loss of electronenergy, so the motion orbit of electron rotation around the nucleus can not be stableand continuous, and finally the electrons should fall into the nucleus due to the energylose. However, this demonstration on the basis of classical electrodynamics theories isobviously not consistent with the fact; Secondly, the radiation generated by theaccelerated electrons should be continuously distributed, which is inconsistent withthe atomic spectrums that are discrete spectral lines; Thirdly, according to the classicaltheory, if the electromagnetic wave source emits a stream of wave with the specificfrequency ν, it may also emit various harmonics with different frequenciesconcurrently, whose frequencies are the integer multiples of ν, but the experimentresults of hydrogen atomic spectrum are inconsistent with this, because the frequencyof spectral line results follows the principle of convergence (the spectral lines show acombination of different wave frequencies)[19].Under this background, Bohr developed Planck's quantum hypothesis on the basis ofthe conclusions of hydrogen atomic spectrum experiment, leading to atomic quantumtheory in 1913. There are two important hypotheses in Bohr quantum theory asfollowing: Firstly, an atom has the attribute of discontinuous state when an electronrotates around the nucleus, so only when an electron orbit with angular momentum pequals to the integer multiples of h/2π, it is stable, which means that electrons rotatein the stable orbit. Under this state, electron possesses constant energy of En, andelectron in the stable state does not absorb or emit radiation [19].p = n × (n=1,2,3,..) equation 31Quantum Physics/量子力学21In this equation, n is the quantum number under the quantization condition.Secondly, electron switches to different orbits in the way of quantum transition: whenan electron jumps from an orbit of stable state at energy Emto another orbit of stablestate at energy En, the frequency ν of the absorbed or emitted photons is calculated as[19]:ν = (frequency condition) equation 32According to Bohr's hypothesis, it is to calculate the electron orbital radius anbyapplying classical mechanics, and then anand the corresponding energy Enof theelectrons in hydrogen-like atoms is expressed respectively as [19]:an=equation 33En= - equation 34a0=equation 35In this equation, a0is called the first Bohr orbital radius of the hydrogen atom; μ is themass of electron; in the International System of Units (SI), es= e (4πε0) -12, and e isthe value of the electron charges (electron charge is -e); ε0= 8.854×10-12 C2/N·m2, inthe Unit System of cm·gram·second (CGS), es= e; Z is the atomic number [19].Consequently, the circular orbital radius of hydrogen atoms and the energy ofhydrogen atoms can only be estimated as a series of discontinuous values, which areall quantized, and these discontinuous values of energy are usually called as energylevels of circular orbital. Experiments show that not only the energy of the hydrogenatoms, but also all other element atoms are quantized [19].According to Bohr's frequency condition, the spectral frequency of hydrogen atomcan be deduced into:ν = = equation 36Among which both n' and n are the energy levels of hydrogen atomic orbitals [19].This equation quantifies the spectral frequency of electromagnetic waves absorbed oremitted by electron, when the electron switches from lower energy orbitals to higherenergy orbitals or from higher energy orbitals to lower energy orbitals, respectively.It is to input the Rydberg constant (ᶄ) into the above equation of Bohr's frequencycondition, then it is completely consistent with Balmer formula that is used torepresent the wavelengths of hydrogen atomic spectral line [19] [22]. By measuringthe values of μ, e, ε0, c and h in experiment, the calculated Bohr's frequency (ν) is wellconsistent with the experiment measurement results. After that, this Bohr's atomicQuantum Physics/量子力学22quantum model is further developed, suiting the elliptic shape orbitals of electron. Byproposing the quantum model, the questions of microscopic particle motion can bequantitatively solved by adopting a combination of both classical mechanics anddiscontinuous quantized orbital equation [19].4.6.Double-slit experiment between classical mechanics and electronIn the theory of classical mechanics, the state of a particle is described by the physicalquantities of momentum and spatial position (vector), both of which can be measuredindependently. However, this calculation is not applicable on the microscopic particlesdue to the wave-particle duality of microscopic particles in quantum mechanics. Asboth the vector position and momentum of microscopic particles cannot bedetermined, how to quantify the wave-particle duality by using a physical quantity todescribe the motion state of the microscopic particles? In 1926, Born attempted toexplain the statistical nature of de Brogyi wave, who believed that de Brogyi wave didnot represent the fluctuation of real physical quantity like classical waves, so it was akind of probability wave that described the probability distribution of particles inspace. To clarify this concept, the double-slit diffraction of electron experiment wasconducted and the results was interpreted from the perspective of both 'particles' and'fluctuations' to find out the connection between the two [19].In order to better understand the the wave-particle duality of electrons in double-slitdiffraction, it is first to compare the results of double-slit experiments betweenclassical particles (e. g., sand) and classical waves (e. g., acoustic and water waves).For the classical particles passing through the double seams (S1 and S2): when onlythe seam S1 is opened, the spatial distribution of particle density is described by thewave figure of I1 after passing through the S1; when only the seam S2 is opened, thespatial distribution of particle density is described by the wave figure of I2 afterpassing through the S2; when the double seams of both S1 and S2 are openedconcurrently, the spatial distribution of particle density through the S1 and S2 seamsare completely added as: I = I1 + I2. However, for the classical waves passing throughthe double seams, the wave interaction between both is not simply added: when onlyS1 seam is opened, the spatial distribution of the wave intensity is described by thewave figure of I1 after the wave passes through the S1; when only S2 seam is opened,the spatial distribution of the wave intensity is described by the wave figure of I2 afterthe wave passes through the S2; when the double seams of both S1 and S2 are openedconcurrently, the spatial distribution of wave intensity through both S1 and S2 seamsare NOT completely added as I = I1 + I2, but are expressed as I =I1+I2+2×I1×I2×cosδ, among which δ is the phase difference between the two wavesindependently passing through S1 and S2. Due to the existence of interference terms(2×I1×I2×cosδ), the classical wave intensity distribution is different from the classicalparticle density distribution [19].Next it is to analyze the electrons' double-slit diffraction experiments on the basis ofconclusions drawn from the classical particles and waves. When the electron beamQuantum Physics/量子力学23passes through the double slits, if the incoming flow of electron beam is weak, theelectrons pass through the double slit almost one by one and subsequently hit thephotosensitive screen. When the photosensitive time is short, it seems that thedistribution pattern of the screen light dots is random and irregular. Nevertheless,when photosensitive time is prolonged enough, the screen light dots become more andmore, and the distribution of screen light dots in some places become very dense,while in other places they are very thin and even in some places they are almostdisappearing, so the final electronic screen distribution forms regular interferencepattern. The distribution pattern of electron beam intensity is similar to the classicwaves, but is different from the classical particle distribution pattern. In the aboveexperiments, the pattern of diffraction is independent of the intensity of the incidentelectron flow. When the electron flow intensity is weaken to the state that almost theelectrons are emitted one by one, the initial distribution pattern of screen light dotsappears to show some irregular spots. However, as long as prolonged enough time,the same interference stripe as classical waves is still obtained on the photosensitivescreen, showing the fluctuation of the electron. Therefore, it can be concluded thateach particle diffracts independently of other particles, which means that diffraction isnot the result of the interaction between these different particles and fluctuation ispossessed by each microscopic particle, so each particle shows the nature of bothparticle and fluctuation. From the particle theory aspect, the peak intensity value inthe interference pattern means that the electron projection probability is high, whilethere are few or no particles density at the minimum value; From the perspective offluctuation theory, the intensity of the wave at the maximum in the interferencepattern is great, and the intensity of the wave at the minimum is extremely small oreven zero. Based on the theory of both particles and fluctuation, Born proposedstatistical interpretation of the wave function, describing that the intensity of the wavefunction (the square of the absolute value of amplitude) at a point in space isproportional to the probability of particle occurrence at that point [19].4.7.Wave-particle duality of de Broglie waveOverall, the wave nature of microscopic particles is based on the statistics only, so itis to clarify the difference between classical waves and microscopic quantum waves:classical waves are usually defined as the transmission of substances in vibrationforms. For this reason, there are two vectors of physical movements that need to beclarified in waves: the first movement vector is the vibration motion, and the secondmovement vector is the transmission motion, but what substance gets transmitted isthe key to classify these waves. Based on the definition of two movement vectors inwaves, it is easily to compare and contrast the classical wave nature with themicroscopic quantum particles: for example, sound waves involve the up-and-downvibrations of air, but the air itself cannot move horizontally and only the'displacement' from these up-and-down vibrations is transmitted horizontally.However, not all waves require the medium to transmit their 'displacement'; forinstance, electromagnetic waves would not need the medium and is capable ofpropagating directly through the vacuum. In comparison to the classical waves, theQuantum Physics/量子力学24waves of microscopic quantum particles also do not require the medium, but theytransmit 'probability' by itself, which means that probability of particle occurrence isvibrating and this vibration is not identical to a kind of vertical displacement ofparticles in space, so it is just a type of mathematical 'vibration.' To put it further,probability waves can be understood as: the 'probability values' of microscopicparticles are constantly vibrating in the spatial positions (or velocities), and what getstransmitted is the 'probability' itself, a mathematical value, other than the physicalquantity [19].In experiments, phenomena such as light and electrons sometimes behave like wavesbut in other times display like particles, so these phenomena exhibit wave-particleduality, but it is impossible to observe both wave and particle propertiessimultaneously. These phenomena are explained as: when an object's de Brogliewavelength is comparable to the particle size or exceeds its size, its wave nature canbe detected, which thus cannot be ignored. However, if its de Broglie wavelength ismuch smaller compared to its particle size, then the particle object's wave naturecannot be detected at all. Consequently, it is proposed that the theories of bothparticles and waves nature can be applicable on the microscopic quantum particles ascomplementary explanations [19].4.8.The mathematical expression of classical wave and quantum waveThe physical properties of the wave function are expressed mathematically below: itis described that the wave function of Φ(x, y, z, t) is defined as the state of a particle ata spatial point A with coordinate (x, y, z) and time t. The intensity of the wave isdefined as the magnitude of complex number│Φ│² = Φ* × Φ equation 37where Φ* is the complex conjugate of Φ[19].According to the statistical interpretation of the wave function, the probability of theparticle occurrence at the spatial point A is defined as dW(x, y, z, t), which isproportional to the magnitude of complex number representing the spatial point A as│ΦA(x, y, z, t)│², so at time t in the volume unit of dr (dr = dx×dy×dz) where thespatial point A (x, y, z) is located at the center of this volume unit, the coordinate ofthis volume unit is expressed as x~x+dx, y~y+dy and z~z+dz, and the probability dW(x, y, z, t) of the particle occurrence at spatial point A is proportional to │ΦA(x, y, z,t)│², which is expressed as [19]:dW (x,y,z,t) = C ×│ΦA(x, y, z, t)│² ×dτ equation 38In the formula, C is the proportional constant, so the probability of a particleoccurrence in a volume unit at the point A (x, y, z) at time t is defined as the density ofprobability, ω(x,y,z,t), which can be calculated as [19]:Quantum Physics/量子力学25ω(x,y,z,t) = dW (x, y, z, t)/dr = C ×│ΦA(x, y, z, t)│² equation 39As particles must appear in somewhere over the whole coordinate (space point A canbe any point in coordinate axes), the sum of the probabilities, representing thatparticles will appear at all the points in the whole spatial coordinate, is equal to 1,which can be consequently calculated as [19]:C× = 1 equation 40The infinity symbol of ∞ under the integral signs means to integrate the probabilityover all spatial points, so the proportional constant C is deduced by above equation as[19]:C = 1/( ) equation 41Next it is to define the normalization of wave function as Ψ(x,y,z,t), and then thiswave function is derived from the magnitude of complex number and the proportionalconstant [19]:Ψ(x,y,z,t) = C×Φ(x,y,z,t) = Φ(x,y,z,t) /( ) equation 42The wave functions of both Ψ and Φ describe the same state, and thus the probabilityof a particle occurrence in the volume unit of dr near the point (x, y, z) at time t isfurther expressed by the normalization of wave function [19]:dW (x,y,z,t) = │Ψ(x, y, z, t)│² ×dr equation 43Then the density of probability in particle occurrence, ω(x,y,z,t), is further calculatedby using normalization of wave function [19]:ω(x,y,z,t) = │Ψ(x, y, z, t)│² equation 44The sum of the probabilities, representing that particles will appear at all the points inthe whole spatial coordinate, is also expressed by inputting normalization of wavefunction [19]: = 1 equation 45The procedure of turning the wave function of Φ(x,y,z,t) into is called asthe normalized process of quantum wave, and the proportional constant C iscorrespondingly defined as the normalization constant [19]. With regards to the abovenormalization procedure of wave functions, there are additional methods in adoptingthis normalized wave function under different scenarios:Quantum Physics/量子力学26Firstly, it is worthwhile noting that the above normalization of wave functions is notunique. For example, if Ψ is a normalized wave function, then the transformed wavefunction of eiδ (where δ is any real constant) is also normalized. Consequently, theequation of │Ψ│²=│eiδ×Ψ│2means that eiδΨ describes the same probability wavewith the phase factor of constant eiδ, and a normalized wave function can contain anyphase factor, so it is feasible to use this property of wave functions to simplify thecomplex wave functions via multiplying or dividing by a phase factor in the next [19].Secondly, if the magnitude of complex number, , diverges, meaningthat the wave function is not square-integrable over all space, then this wave functioncannot be normalized according to the above steps, as this wave function would resultin a normalization factor C = 0, which is clearly meaningless. For example, if thewave function of a free particle is defined as Ψp (r,t) = A× Exp[ × (p×r-E×t)] thathas a modulus squared of =×=A2, then this constant C is independent of timeand coordinates. This independent constant means that the probability of a particleoccurrence in a unit volume near any spatial point is the same, which is expressed asequation: = A2= ∞. Consequently, such wave functions are notsquare-integrable over all space points and cannot be normalized according to theabove steps [19].Thirdly, if the state of a particle is described by the normalized wave function ψ(r, t),then the probability distribution function at spatial unit volume of r at time t is definedas ω(r, t) [19]:ω(r,t) ×dr = │Ψ(r, t)│² ×dr equation 46Using the above formula, it is to calculate the average value of particle coordinates ()at x axis according to the common averaging equation by probability [19]:= ∫ x│Ψ(r, t)│² dr equation 47Fourthly, if there is any mechanical quantity f(r) of a particle that is known, itsaverage can be expressed as [19]: = ∫ ψ*f(r) ψ dr equation 48Finally, the complex number form of wave function cannot be directly measuredexperimentally in quantum mechanics, so its mathematical equations only refer to theprobability of particle occurrence in space, which is discussed above. Then thephilosophy of quantum mechanics maths is that the variable of any real mechanicalQuantum Physics/量子力学27quantity of f(r) can be incorporated into this mathematical equation of occurrenceprobability.4.9.The superposition of wave functionsIn the linear system of classical physics, the linear differential equations (groups) areusually applicable on the physical quantities (including functions, vectors or vectorfields) that meets the requirements of the linear equations (groups) describing theirphysical processes. For all the classical fluctuation processes, in which the principleof superposition is applicable, any fluctuation process φ is the result of the linearsuperposition of two possible fluctuation processes, φ1 and φ2, expressed as [19]:φ = a×φ1 + b×φ2 (a, b are both constant) equation 49For classical waves that are driven by the superposition principle, such as water waves,acoustic waves, the synthetic amplitude of two or more waves propagating in thesame space is the sum of the amplitudes generated separately by each wave. When thephysical quantities are measured, only the amplitude of the synthetic variable ismeasurable, rather than the physical quantities generated separately by each wave,which means that its individual states participating in the superposition do not havetheir independent characteristics [19].One of the main ways to calculate the physical quantities of a wave function is to sumthe wave function as a superposition of some independent wave functions that arederived under particularly simple state, which is also called quantum superposition.For instance, because the Schrodinger wave equation is linear, the overall physicalquantities of the wave function can be calculated on the basis of the superpositionprinciple. In optics, the light interference and diffraction phenomenon can beexplained by using the superposition principle: one beam of the incident electronspasses through slit S1 and the other beam passes through slit S2, represented by wavefunctions of ψ1 and ψ2 respectively [19].The experimental results show that: the state of the particle after passing through thedouble slit is represented by the wave function of ψ, which is the result of the linearsuperposition of ψ1 and ψ2, calculated as [19]:ψ = C1×ψ1+ C2×ψ2(C1and C2can be any complex constants) equation 50Only in this superposition way, the interference phenomenon can be explained,because the superposition of the interference pattern on the screen is measured by theinterference intensity as below [19]:│C1×ψ1+ C2×ψ2│² = │C1×ψ1│² + │C2×ψ2│² + C1C2×ψ1*×ψ2+ C1C2*×ψ1×ψ2*equation 51Quantum Physics/量子力学28Among which C1C2×ψ1*×ψ2+ C1C2*×ψ1×ψ2* is the interference item, explaining theinterference intensity variation of the superposition waves [19].It is further to deduce the principle of state superposition in quantum mechanics fromthe equations of classical substance wave: If ψ1 and ψ2 are two possible states of thesystem, then their linear superposition of ψ = C1×ψ1+ C2×ψ2(C1and C2can be anycomplex constants) is also a possible state of the system [19]. Consequently, the keydifference in linear state superposition between quantum mechanics and classicalsubstance wave is that the linear state superposition of quantum mechanics only refersto the superposition of probability under the same state, but the classical substancewave superposition is the superposition of physical quantities.5.Schrodinger wave equation5.1.The transformation of free particle plane wave functionAs the equation to be established describes how the wave function of Ψ(r,t) changesover time variable of t at space unit volume of r, there are the following conditions tobe met: Firstly, the equation is a differential equation of the first derivative to thewave function Ψ(r,t), with respect to the time variable of t, as it allows to determinethe state at any given moment from the initial state of the microscopic system;Secondly, the equation is linear, meaning that if Ψ1 and Ψ2 are both solutions to thisequation, and then their linear combination of aΨ1 + bΨ2 is also a solution, so thelinearity of the equation ensures that its solutions are applicable on the principle ofsuperposition; Thirdly, the coefficients in the equation (such as constant a and b)should not contain any parameters describing physical state quantities such as energyor momentum [19].By clarifying the above pre-conditions, it is to convert the free particle plane wavefunction of Ψ(r,t) into Schrodinger wave equation and the conversion steps arededuced below [19]:The free particle plane wave function of ψ(r,t) is expressed as:ψ(r,t) = A×exp[×(p×r-E×t)] equation 52Where the space unit volume of r is near the spatial point with coordinates (x,y,z), soit can be re-expressed as:ψ(x,y,z,t) = A×exp[×(px·x+py·y+pz·z-E·t)] equation 53Where px,py, pzis the momentum vector at x, y, z axis, respectively [19].Quantum Physics/量子力学29It is first to take the partial derivative of time variable t on the basis of ψ(x,y,z,t): = - × E × ψ equation 54Then it is to calculate the second partial derivatives at coordinates x, y, z respectively:∂2ψ/∂x2= - (/2) × ψ∂2ψ/∂y2= - (/2) × ψ equation 55∂2ψ/∂z2= - (/2) × ψIt is to further integrate three equations of second partial derivatives into a whole:(∂2/∂x2+ ∂2/∂y2+ ∂2/∂z2)×ψ = - (/2) × ψ equation 56Where the parameter of (∂2/∂x2+ ∂2/∂y2+ ∂2/∂z2) is replaced by the symbol of ∇2,which is re-named as Laplacian operator in Euler's method [19].5.2.The operator and Schrodinger equationTo integrate the relationship of a free particle between the energy E (kinetic energy)and momentum p, E=p2/2μ, where μ is the mass of the particle, the equation is derivedinto [19]:E×ψ = iћ× equation 57(p·p)×Ψ = (-iћ∇)(-iћ∇)×Ψ equation 58In this formula, the symbol of ∇is called as Nabla operator in Euler's method (Pleasenote: as the imaginary unit, i2= -1; 1/i = 1 in this deducing process) [19].∇= i× + j× + k× equation 59Where i, j, k is the imaginary unit at x, y, z axis respectively [19].By introducing the Nabla operator, both energy E and momentum p of a particle areexpressed as the following operators acting on the wave function [19]:E → iћ p → -iћ∇equation 60From this formula, it is to replace the energy and momentum variables by the Nablaoperators of Euler' method in the classic energy-momentum relationship, and then toapply them on the wave function Ψ, so the wave equation for a free particle can beobtained. For a particle in the given potential field of U, it is to define the potentialQuantum Physics/量子力学30energy of the particle under the force field to be U(r), and the total energy of theparticle becomes the sum of both kinetic and potential energy [19]:E = + U(r) equation 61Replacing the physical properties of both E and p in the above formula by Nablaoperators, then the wave function is re-expressed as [19]:iћ× = [- ×∇2+ U(r)] ×Ψ equation 62This equation is called the Schrodinger wave equation, which usually refers to thetime-dependent Schrodinger equation, and it describes the variation of particle statewith the time change under the potential field of U(r). If the force field acting on theparticle wave varies with time variable of t, the general form of the above equation iscalculated as [19]:iћ× = [- ×∇2+ U(r,t)] ×Ψ equation 63It is hypothesized that the force field acting on the particle does not change over time,and then the potential energy formula of U(r) does not include the time variable of t.Under this situation the wave function is called stationary state. It is to re-define thewave function of Ψ(r,t) by dividing it into the sub-function (ψ(r)) of variable r and thesub-function (f(t)) of variable t separately [19]:Ψ(r,t) = ψ(r)×f(t) equation 64Then it is to derive the particular integral of the Schrodinger equation [19]:Ψ(r,t) = ψ(r) × Exp(- Et) equation 65The relationship between this wave function and time t is sinusoidal, with its angularfrequency of ω = E/ћ. It can be seen from De Broglie's equation that the constant E isthe energy of the system under the stationary state, so its correspondingmicro-particles' state is also stationary described by this wave function, including:The potential energy U(r) of the particle is independent of time variable t, and theenergy E is given a constant value [19].The probability density of a particle occurrence ( 2) is independent of time,indicating that the probability distribution of a particle does not change over time. Theequation is expressed as [19]:Quantum Physics/量子力学312= 2=2equation 66The average of any mechanical quantity does not change over time, which means thatthe time variable is not included in the functions of any mechanical quantity meanvalue [19].To fully describe the motion state of an electron, there are four quantum variables thatmust be required, including the principal quantum number, angular quantum number,magnetic quantum number and spin magnetic quantum number, among which the firstthree variables are all the solutions of Schrodinger equation, except that spin magneticquantum number is not a solution of Schrodinger equation. Both principal quantumnumber and angular quantum number are related with electron energy, while magneticquantum number characterizes the electron angular momentum [23].6.Further development of mechanics models in this article6.1.Mechanical movement refers to the vector variation in the displacement of themass point in matter both temporally and spatially, which is different from themovement of matter existing as energy only. According to the new definition ofphoton in my another quantum physics article [13], this article proposes that photonsare the most elementary research object for mechanical motion, which are also thesmallest partitioning mass unit of mass matter, so the natural Law of electromagneticwave particle duality is a basic attribute of mass matter, not limited to the basicproperties of energy matter.6.2.According to the Figure 1 of my article [11], it is to further discuss the argumentof the shielding effect of the electric field inside an atom and its effects on theelectron orbitals:6.2.1.For the adjacent atoms of the same element, the frequency of theelectromagnetic waves generated by adjacent atoms is the same, so interference wavesof electromagnetic waves are easily formed between adjacent atoms;6.2.2.Multiple equipotential lines are formed between the zones of constructiveinterference and the zones of destructive interference. The destructive interferencezone is relatively neutral due to the offsetting between wave peaks and bottoms, andelectrons tend to undergo rotation motion in the destructive interference zones, thusbecoming an important factor affecting the electron rotation orbit. In the motionmodel shown in Figure 1, the shielding effect of the electric field inside the atomcauses the electron orbitals to be relatively fixed rather than the randomly disorderedorbitals;6.2.3.Electrons tend to rotate in the outer space of a closed circular equipotential line,which meets the pre-conditions for the formation of electric field shielding.Quantum Physics/量子力学326.3.To compare and contrast with the shielding effect of electric field inside an atom,macroscopic celestial bodies (such as stars and planets) also have field shieldingeffects inside them. However, unlike the shielding effect inside microscopic atoms,macroscopic celestial bodies mainly rely on the substance boundary layers to formfield shielding effects. The rupture of the boundary layer leads to the destruction ofthe shielding effect, which is the main factor causing various natural disasters such astornadoes, earthquakes, solar flares, etc [6][7][8]. Therefore, the stable boundary layerand the generated field shielding effect is the important influencing factor in thedevelopment motion of celestial bodies. Similar to the internal equipotential lines ofmicroscopic atoms, the overall equipotential lines inside celestial bodies tend to formclosed loops. Due to the shielding effect generated by equipotential lines, substancesmove parallelly to the equipotential lines in both sides [12]. This motion model is themain factor that enables celestial bodies in our three-dimensional space to evolve intoregular spherical shapes.6.4.It is to re-analyze the wave-particle duality of de Broglie wave below:My article re-defines the classical material wave as mass wave, and it is to divide theDe Broglie wave generated by elementary particles into two components, includingmass wave and energy wave (both electric and magnetic field energy) shown inFigure 6. Then the difference in physical quantities is critically compared betweenclassical material wave and quantum wave in the Table 1.Quantum Physics/量子力学33Table 1. Comparison between classical material wave and quantum wave.Classical material waveQuantum waveWave typeMass waveMass wave and energy wave(both electric and magneticfield energy)Energy formtransmitted bywaveKinetic energyKinetic energy andelectromagnetic energyInteraction formbetween twowavesCollisions among particles oftwo mass wavesThrough wave nature of darkmatter driven by two wavesThe product oftwo waves'interactionInterference wave by two masswavesInterference wave by two masswaves; Interaction betweenpositive and negative poles oftwo energy wavesUnder the hypothesis that De Broglie wave is divided into mass wave andelectromagnetic energy wave, the wave-particle duality of de Broglie wave can beeasily understood: de Broglie wave does not only possess the same attributes of massparticles as the classical material wave, but also shows the physical quantities ofelectromagnetic energy wave that is generated and carried by the beam of elementaryparticles (photons, electrons, proton...) at quantum level. Consequently, the wavefunctions of mass wave and electromagnetic energy wave need to be calculatedseparately for de Broglie wave next.It is to give the imaginary unit of 'i' the realistic attribute: the imaginary unit of 'i'represents the phase of De Broglie wave (shown in Figure 6), and when twomicro-particles undergo wave motion in the same phase, the poles of electromagneticwave show the same nature, so the interaction product of two micro-particles is togenerate the repelling force, expressed as the mathematical equation, i2= -1. Underthis hypothesis, the imaginary unit of 'i' is given the realistic nature, rather than justfacilitating the mathematics calculation.6.5. In summary, this paper firstly reviews the classical principles of mechanics, andclassical mechanics can effectively solve physics cases under the common limitationconditions, that include macroscopic physical conditions and low-speed motion model.However, under the situations of quantum micro-scale, cross-galaxy motion modelsand material aging process, new physical models need to be established to solvephysical problems. My previous papers have fully discussed the particle collisionmotion model at microscopic quantum field [1], the microscopic quantum mechanicsmodel under the electric field shielding effects of the overall atomic structure [10], theforce balance analysis at each mass point inside an atom[2][3], inter-molecule forcegenerating sources [4], thermal motion model of micro particles in the process ofmaterials aging [5][9], friction resistance model at quantum scale [4], chargedQuantum Physics/量子力学34particles motion model under free state at the substance boundary layers in nature[6][7], dark matter principle and its application on inter-galactic motion model [8], etc.Therefore, Table 2 fully summarizes the original mechanics models proposed by myprevious articles as well as by this current article.Table 2. Summary of mechanics models originally proposed in our sponsoredjournals.Scope levelMechanics modelReferencesElemental particle levelThe particle collisionmotion model[1]; Figure 4 of this article.Elemental particle levelMechanics model underthe electric field shieldingeffects of the overallatomic structure[10]; Section 6.2 of thisarticle.Elemental particle levelThe force balance analysisat each mass point insidean atom[2];[3].Elemental particle levelThe conduction of thermalmotions among elementaryparticles (both electronsand protons)[32].Elemental particle levelThe resonant state ofmetastable compositenucleus and gravitationalwave generation[8].Atomic or molecular levelInter-molecule forcegenerating sources[4].Atomic or molecular levelThermal motion model ofmicro particles in theprocess of materials aging[5];[9].Atomic or molecular levelFriction resistance modelat quantum scale[4].Atomic or molecular level;Macro materials levelCharged particles motionmodel under free state atthe substance boundarylayers in nature[6];[7].Macro planet or star levelBoth parallel and verticalconvection motions alongthe substance boundarylayers forming fieldshielding effects of aplanet or star.[6];[7];[8];[12];[30].Astronomic levelDark matter principle andits application on[8].Quantum Physics/量子力学35inter-galactic motionmodelAstronomic levelThe magnetism along thefourth dimensional spaceand the driving force ofcelestial rotation[18].7.Experiment methods7.1.Particle collider dataThe basic structure and operation mechanism of particle collider instrument have beenintroduced in my another article [13]. This article systematically collects the particlecollision data from several research projects that are produced by Beijing ElectronPositron Collider (BEPC), which have been published in China. Consequently, theseparticle collision data are comparable due to the generation by the same instrument,although these data are analyzed by different research teams. The Beijing ElectronPositron Collider (BEPC) is a high-luminance collider of e-e+with multiple beambunches, designed to specifically study the physical processes in the r-charm energyregion. Its main parts consist of an injector, a beam transport system, and a storagering. To achieve its scientific goals and meet the technical requirements of the BEPC,the Beijing Spectrometer must satisfy specific conditions below: high-precisionrecording of the position, velocity and energy of both charged and neutral particles;high-resolution for the measurement of energy and angular velocity, especially forelectrons and photons; high-resolution for measuring momentum and time to improvethe analysis of charged particles; a front-end electronics system and data acquisitionsystem that can be adapted to multi-beam bunch modes; excellent particle recognitioncapabilities, particularly for T, K, and P particles; and high-precision software foroffline simulation, reconstruction, and calibration. The specific parts are mainlycomposed of main drift chamber (MDC), time of flight counter (TOF),electromagnetic calorimeter (EMC), muon discriminator (MUC) and othersub-detectors, hardware (such as the superconducting magnet) and software systemsused for particle and event identification [36].The main function of the MDC is to use gas amplifiers and wire detectors to measurethe trajectories and electric charges of particles, which are essential in particleidentification, including the measurement of both trajectory and momentum offinal-state charged particles. According to its function, this instrument can becategorized into time measurement and charge measurement; The TOF methodmeasures the duration of Δt from the particle's generation source to the detector, incombination with the measured flight path length (L) and momentum (P), todetermine the intrinsic mass (M0) of the particle; The EMC part primarily measuresthe energy of both charged particles such as electrons and neutral particles such asphotons, consisting of a barrel calorimeter and an end calorimeter; The Muondiscriminator (MUC) is designed to identify Muons and distinguish Muons from πmesons; The superconducting magnet provides the uniform magnetic field with
particles [36].Finally, the collected particle collision data are adapted by the estimation ofobservation on the basis of published Figures, summarized in Table 3, Table 4 andTable 5.7.2. Quantum wave experiment and simulationThe electron wave diffraction experiment and double-slit experiment are introduced inabove review content to analyze the quantum wave theories. Correspondingly, tofurther explore the relevant scientific findings, this article selects the interferometerexperiment and empirical modeling of electron diffraction in transmission electronmicroscopy (TEM) to conduct the observation study project.Zeng Zi-Qi conducted spectral studies by using three types of interferometers,including the HOM interferometer, N00N interferometer and Franson interferometer.In addition to fundamental research, these three types of interferometers also play acrucial role in various industrial applications, which are capable of preciselymeasuring optical properties such as wavelength, phase, amplitude, coherence andpolarization. An improved interferometer of HOM, which are consisted of twopolarization beam splitters (PBS), two reflectors and two halfwave plates (HWP), hasbeen designed in this research. To gain the deeper understanding of the relationshipbetween HOM interference and N00N interference, controllable transformations weredemonstrated between HOM interference and N00N interference under the sameexperimental setup. In order to study the spectral-time characteristics and resolveFranson interference both theoretically and experimentally, it is to compare theinterference patterns of frequency-positive correlated, frequency-negative correlated,and uncorrelated two-photon states, which offers a new perspective on understandingthe spectral-time characteristics of the Franson interferometer [39].Electron diffraction is the crucial technique part in transmission electron microscopy(TEM). As the incident wavelength of electrons' beam wave in TEM is typicallymuch shorter than that of X-rays used for diffraction experiments, the Ewald sphereradius of electron diffraction becomes significantly larger, which allows morereciprocal lattice points to interact, generating additional diffraction beams, so itreveals more crystalline information. The most representative technique in TEM isnamed as selective area electron diffraction (SAED), which produces diffractionpatterns through parallel illumination of electron beams. There are both relativitydynamics model and relativity motion model compared in this research. It isworthwhile mentioning that the parameters used in dynamics model are based on theempirical measuring data and is widely applied in practice, while the motion model isthe theoretical model only [40].The spectral characteristics results are summarized in Table 6 by observationQuantum Physics/量子力学37estimation on the experimental figure results; the diffraction pattern data are drawn inFigure 7 ~ Figure 12 on the basis of observational estimation on the modeling graph.8. Particle mass, momentum and energy in the particle collision processIn this section, my article characterizes the physical quantities of particle mass,momentum and energy in the decaying chain of particle collision experiment, withemphasis of these parameters: mass/energy/momentum range along which the eventsdistribute, mass/energy/momentum level at which the peak events occur, amount ofpeak events.8.1. Case studiesBABAR collaboration team measured the produced cross section of e+e-→ Øηdecaying process by using the initial state radiation method, and observed the decayprocess of e+e-→η'Ø during the study of particle decaying process: e+e-→ K+K-π+π-π0,but due to the small amount of events in the e+e-→η'Ø decaying process, furtherresearch could not be carried out. Subsequently, the BEIII collaboration team studiedthe invariant mass spectrum of Øη' through the process of J/Ψ→ηØη', observing theexistence of resonant states with statistic significance of 5σ, as the complement to theBABAR team's findings [32].Indicated by both BABAR and BEIII research team, Sun Yan-kun (2019)subsequently conducted the experiment analysis to study the decaying process of bothK+K-and π+π-γ at energy level of 2125 MeV, which was able to clearly observe thesignals of decaying process: e+e-→ Øη', although there were massive backgroundevents appearing in the collision experiment [32]. However, Sun Yan-kun's study hasonly outlined the results by figures, without describing the data in more detail, so myarticle tries to further analyze the experiment data that are estimated according to thisstudy's figures and are summarized for comparison in Table 3.Ban Zhenglin (2020) studied the recoil invariant mass spectrum of particle state ofboth π+π-and π+π-π0in the decaying chain of Ψ(2S) → π0hc(hc→ 3(π+π-)), and theGaussian fitting optimization results were further analyzed according to thebackground data. Additionally, the invariant mass spectrum of double photons wasrecorded as the background data for analysis in this research. Another decaying chainstudied by this research was J/Ψ→γ3(π+π-), in which particle states of both γγ and3(π+π-) were recorded for the invariant mass spectrum by screening the 'noise' ofevent samples [33].Pan Xiang (2019) studied in total 9 decaying pathways, aiming to systematicallyanalyze the hadronic decaying characteristics of D meson, whose data were providedby BESIII collaboration team under the particle collision conditions of Ψ(3770).These nine decaying pathways include →K+π-,→K+π-π0,→K+π-π-π+, D-→K+π-π-,D-→π-, D-→K+π-π-π0, D-→π-π0, D-→π-π-π+, D-→K+K-π-[34].Quantum Physics/量子力学38Zhang HongHong (2019) studied the decaying pathway of Ψ(3686)→Λω, and Λbaryon was electrically neutral and composed of quark 'uds'. However, Λ baryon wasfurther decayed into the final particle states through four kinds of pathways: Λ→pπ-,→π+, ω→π+π-π0,π0→γγ, which included 6 charged particles (π+π+π-π-) and twophotons (γγ). Another decaying pathway of J/Ψ→Λη was also studied, whose finalparticle states included four charged particles (π+π-) and two photons (γγ) via threedecaying sub-pathways: Λ→pπ-,→π+, η→γγ[35]. The mass range, mass level atpeak events and peak events amount of these final state particles were selected by myarticle and summarized in Table 3 (24)~(30).Subsequently, Zhang HongHong (2024) further collected the particle collision dataproduced in 2021, aiming to analyze the excited state of Λ baryon in the decayingpathway of Ψ(3686)→Λω. The invariant mass spectrum of final particle states,including pπ-,π+, Recoil π+π-, π+π-π0, was selected and fitted in this study. Inaddition to the data of decaying pathway J/Ψ→Λη produced in 2009 and 2012, hisstudy also collected the data of the same decaying pathway generated from 2017 to2019 with approximately 8774 ×106events detected in total. Four charged particlesin forms of both pπ-and π+, as well as two photons, were analyzed in this decayingpathway [36].Wu XiongHao (2024) studied the decaying chain of τ+τ-→ K+K-K±µ±µµ(µ)TµTatenergy level of 3686 MeV. In total four types of final particle states were found in thisparticle collision research, including µ, K+K-K±, all of which were electrically chargedparticles. Particle's momentum during each particle decaying chain and total particleenergy were analyzed under different assumed limitations for calculation, accordingto the collision data collected from BEIII research team in 2009, 2012 and 2021. Thecriteria of χµwas introduced to screen the raw data, and χµreferred to the applicationof µ particles on theoretical calculations, expressed as [37]:χµ= [dE/dx (theoretical expectation) - dE/dx (actual measurement)]/(detectorresolution)equation 67Where variables of E and x represented the energy and position along the decayingchain, respectively [37].8.2. Results and descriptionTable 3. Summary of particle's mass during particle decaying process.Decaying processMass rangeMass at peakeventsPeak Events(1) J/Ψ→γØη'1.96 ~ 2.56(GeV/C2)2.11(GeV/C2)90(20 MeV/C2)(2) J/Ψ→ηØη'1.96 ~ 2.56(GeV/C2)2.38(GeV/C2)35(20 MeV/C2)Quantum Physics/量子力学39(3) K+K-1.0 ~ 1.8(2125 MeV)1.02(2125 MeV)710(0.004 GeV/C2)(4) π+π-γ0.3 ~ 1.1(2125 MeV)0.77(2125 MeV)540(0.004 GeV/C2)(5) K-π+0.62~1.20(2125 MeV)0.90(2125 MeV)580(0.003 GeV/C2)(6) K+π-0.62~1.20(2125 MeV)0.90(2125 MeV)590(0.003 GeV/C2)(7) π+π-0.28~1.05(2125 MeV)0.50(2125 MeV)820(0.004 GeV/C2)(8) Ø0.985~1.17(3097 MeV)1.02(3097 MeV)6400(0.001 GeV/C2)(9) π+π-γ0.9~1.1(3097 MeV)0.955(3097 MeV)2300(0.001 GeV/C2)(10) π+π-1.0~3.5(3686 MeV)3.10(3686 MeV)3000(0.0125 GeV/C2)(11) Double Photons0~0.16(3686 MeV)0.13(3686 MeV)8900(0.003 GeV/C2)(12) π+π-π00.4~2.8(3686 MeV)0.8(3686 MeV)4200(0.013 GeV/C2)(13) γγ(J/Ψ→π03(π+π-))0.02~0.20(GeV/C2)0.135(GeV/C2)1900(0.004 GeV/C2)(14) γγ(J/Ψ→γ3(π+π-))0.02~0.20(GeV/C2)0.18(GeV/C2)1100(0.004 GeV/C2)(15) →K+π-1.862~1.872(GeV/C2)1.866(GeV/C2)45000(0.00025GeV/C2)(16) →K+π-π01.84~1.88(GeV/C2)1.865(GeV/C2)66000(0.00025GeV/C2)(17) →K+π-π-π+1.84~1.875(GeV/C2)1.865(GeV/C2)58000(0.00025GeV/C2)(18)D-→K+π-π-1.865~1.875(GeV/C2)1.87(GeV/C2)70000(0.00025GeV/C2)(19)D-→π-1.865~1.875(GeV/C2)1.87(GeV/C2)8000(0.00025GeV/C2)(20) D-→K+π-π-π01.835~1.885(GeV/C2)1.87(GeV/C2)20000(0.00025GeV/C2)(21)D-→π-π01.835~1.885(GeV/C2)1.87(GeV/C2)14000(0.00025GeV/C2)(22)D-→π-π-π+1.835~1.885(GeV/C2)1.87(GeV/C2)11000(0.00025GeV/C2)(23)D-→K+K-π-1.835~1.885(GeV/C2)1.87(GeV/C2)6500(0.00025GeV/C2)(24)pπ-1.100~1.130(GeV/C2)1.116(GeV/C2)45(0.0002GeV/C2)Quantum Physics/量子力学40(25)π+1.100~1.130(GeV/C2)1.116(GeV/C2)48(0.0002GeV/C2)(26)Recoil π+π-2.95~3.25(GeV/C2)3.095(GeV/C2)60(0.0020GeV/C2)(27)π+π-π00.65~0.90(GeV/C2)0.78(GeV/C2)35(0.0020GeV/C2)(28)pπ-1.100~1.130(GeV/C2)1.116(GeV/C2)310(0.0002GeV/C2)(29)π+1.100~1.130(GeV/C2)1.116(GeV/C2)320(0.0002GeV/C2)(30)γγ0.50~0.60(GeV/C2)0.546(GeV/C2)300(0.0010GeV/C2)(31)pπ-1.105~1.13(GeV/C2)1.116(GeV/C2)500(0.00020GeV/C2)(32)π+1.105~1.13(GeV/C2)1.1165(GeV/C2)660(0.00020GeV/C2)(33)Recoil π+π-3.07~3.13(GeV/C2)3.098(GeV/C2)360(0.00120GeV/C2)(34)π+π-π00.65~0.90(GeV/C2)0.78(GeV/C2)210(0.00208GeV/C2)(35)pπ-1.06~1.16(GeV/C2)1.115(GeV/C2)6400(0.00050GeV/C2)(36)π+1.06~1.16(GeV/C2)1.115(GeV/C2)6400(0.00050GeV/C2)(37)γ (coupled with pπ-)1.2~1.6(GeV/C2)1.36(GeV/C2)950(0.00050GeV/C2)(38)γ (coupled with π+)1.2~1.6(GeV/C2)1.32(GeV/C2)950(0.00050GeV/C2)(1)(2) data is estimated according to the Fig 3.1 [32]; (3)(4) data is estimated according to the Fig 3.5(b)(c) [32]; (5)(6) data is estimated according to the Fig 3.6 (b)(c) [32];(7) data is estimated accordingto the Fig 3.8(a) [32]; (8)(9) data is estimated according to the Fig 3.9(a)(b)[32]; (10) data is estimatedaccording to the Fig 3-4(a) and Table 3-1 [33]; (11) data is estimated according to the Fig 3-3 [33]; (12)data is estimated according to the Fig 3.5 (a) [33]; (13) data is estimated according to the Fig 4.4(Green data) [33]; (14) data is estimated according to the Fig 4.4 (Blue data) [33];(15)(16)(17)(18)(19)(20)(21)(22)(23) data is estimated according to the Fig 3.3 [34]; (24)(25)(26)(27)data is estimated according to the Fig 3-3, 3-4, 3-8, 3-9, respectively [35]; (28)(29)(30) data isestimated according to the Fig 4-5, 4-6, 4-14, respectively [35];(31)(32)(33)(34) data is estimatedaccording to the Fig 4.30, 4.31, 4.35, 4.38 [36]; (35)(36)(37)(38) data is estimated according to the Fig5.6 (b)(c) and Fig 5.7 (a)(b) [36];As shown in Table 3 (1)(2), BABAR collaboration team's findings show that theparticle decaying events distribute along the mass range from 1.96 to 2.56 GeV/C2,with the peak value at mass level of 2.11 GeV/C2, whereas the BEIII collaborationteam's results report the peak values at mass level of 2.38 GeV/C2. However, theQuantum Physics/量子力学41events' amount at the peak observed by the BEIII collaboration team, 35 events (20MeV/C2), is
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Title: Fundamental Mechanics: From Classical Mechanics to Quantum Mechanics and Its Application in Quantum Chemistry
Authors: This paper is authored by researchers in the field of quantum mechanics and quantum chemistry.
Published in: The paper is available on ResearchGate.