4.Wave particle duality of substance waves
4.1.From classical wave to De Broglie wave
De Broglie wave, also named as the material wave, is a kind of wave expressing spatial probability in microscopic particle occurrence, which refers to the probability that may occur in a specific spacial point at transient time, and this particle occurrence probability is controlled by the fluctuation law. Mechanical wave is the propagation of periodic vibration in the medium, and electromagnetic wave is the propagation of periodic electromagnetic field, whereas material wave is neither mechanical wave nor electromagnetic wave [21]. Thus the material wave is proposed to explain the light wave at quantum level, which can be hardly deduced by the classical mechanical wave or electromagnetic wave theories.
Davidson and Dermot emitted single energy electron onto the polished plane of nickel single crystal, aiming to observe the quantitative relationship between the intensity of scattered electron beam and scattering angle. Both emitting source of electron beam and the scattered electron detector were symmetrically placed on the normal of the crystal surface, and the experiment showed that the intensity of the scattered electron beam varied with the scattering angle (θ). When the scattering angle was adjusted to be a critical value, the scattering intensity reached the maximum value, which was the same as the phenomenon of X-ray diffraction, fully proving that the electron had wave particle duality. Because the electrons with low energy (in Davidson’s experiment, the electron energy was set to be about 30~400 ev), which could not penetrate deeply into the crystal, most of the electrons were scattered on the crystal surface [19].
According to the diffraction theory, the position of the maximum diffraction value is determined by the following formula [19]:
λ = 2d×cos(0.5θ) n = 1, 2, ... equation 21
Where n is the order of diffraction maximum, λ is the wavelength of the diffractive ray, and d is the crystal plane spacing of crystal Bragg scattering. Setting up parameter of d (the crystal plane spacing of nickel single crystal) = 0.091nm, when the electron beam energy is 54 ev and the scattering angle (θ) is 50°, respectively, a maximum diffraction peak is observed. According to the Bragg equation, the electron wavelength can be calculated to be 0.165 nm [19].
There is the parallel calculation according to the de Broglie relation, the wavelength of the electron can be also calculated in another way [19]:
(See PDF document) equation 22
In 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, me is the electron mass, and E is the kinetic energy of electron. Many experiments have shown that, in addition to electrons, all the other microscopic particles at various sizes, such as neutrons, protons, mesons, atoms, molecules and even C60 molecules, also display as 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 duality of various microscopic particles [19].
4.2.The wavelength of electron de Broglie wave
Under 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 Broglie wavelength of electron’s material wave is calculated as [19]:
(See PDF document) equation 23
Where (See PDF document) is the electron mass. If an electron is accelerated under the electric field with the potential difference of U in a transmission electron microscope, and usually the acceleration voltage of an electron microscope is 200~300 kV, then the electron kinetic energy is calculated as: E = eU, among which e is the electron charges, so the de Broglie wavelength of electron’s material wave is further estimated as [19]:
(See PDF document) equation 24
After determining the specific values of parameter h, μ, and e, the de Broglie wavelength 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 de Broglie wave of electrons is very short, whose order of magnitude is equivalent to the atomic spacing in crystals, much shorter than macroscopic linearity of mechanical wave.
Therefore, the wave particle duality of physical particles, which is derived empirically under general macroscopic conditions, will not be applicable on the microscale any more (particle nature is the main contradiction aspect), so a new mechanics - quantum wave dynamics - must be established to cope with the quantum level material wave [19].
4.3.The substance plane wave
In classical mechanics, if the angular frequency is ω and the wavelength is λ, the propagation motion of substance plane wave along the x-axis direction can be expressed as [19]:
ψ(x,t) = A×cos(k·x - ω·t) equation 25
Where the wave vector of k = 2π/λ; the constant A is the wave amplitude; t is the propagation time [19].
For the convenience of calculation, it is to transform the above wave function by integrating the Euler formula: eix = cos(x) + i×sin(x), among which parameter i is the imaginary unit. Then the substance plane wave function can be expressed as [19]:
ψ(x,t) = A×exp[i×(k·x-ωt)] equation 26
This wave function is used to describe the X-rays motion that displays as Bragg diffraction phenomenon. Since microscopic particles such as free electrons are also capable of showing diffraction stripes similar to X-rays, the de Broglie wave of free microscopic particles can also be described by this wave function. By inputting the above equations, E = hν and λ = h/p, into this classical mechanics function, the free particle plane wave function is expressed as [19]:
(See PDF document) equation 27
This is the function describing the substance plane wave associated with free microscopic particles, the transformed equation of de Broglie wave, which is derived from the classical macroscopic wave function. It describes the motion state of free particle by integrating momentum p and energy E, which is the de Broglie wave specifically describing the free microscopic particles. If the particle motion trajectory varies both spatially and temporally under the potential field, its momentum and energy are no longer constant (or both physical variables are not constant concurrently), and then the particle is no longer in the form of free particle, so its motion model cannot be described by the plane wave derived from the classical macroscopic wave function. Instead, as the particle still possesses the attribute of wave particle duality, it must be described by more complex wave function (e.g., Bloch wave function describing the electron wave motion in solids) [19].
4.4.Photoelectric effect
It is observed that when ultraviolet light or visible light at short-wavelength hits the metal in vacuum, electrons will escape from the metal surface, so this observed phenomenon is called photoelectric effect, and the escaped electron is called photo-electron, whose experiment has resulted in the conclusions below [19]:
Firstly, in the same period, the number of photo-electrons released from the radiated metal surface is proportional to the intensity of the incident light; Secondly, the initial kinetic energy of the photo-electron increases linearly with the frequency of the incident light (ν), regardless of the intensity of the incident light, which means that the intensity 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 no photoelectric effect occurring, if the frequency of incident light is less than the critical frequency limit (ν0) of the specific metal, regardless of the intensity of the light, which means that there is the threshold of incident light frequency for a specific type of metal to generate photoelectric effect; Fourthly, when the incident light frequency exceeds the threshold frequency (ν0), the photo-electrons are capable of being observed almost immediately as long as the light shines on it (about 10-9s after the incident 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 to the classical electromagnetic theory of light as: when the light shines onto the metal surface, the electrons in the metal are forced to vibrate by the electric field of the incident light, which causes to absorb the energy from the incident light and hence escape from the surface, so the amount of the obtained energy should be related to the intensity of the incident light and the period of the light radiation, but is independent of the frequency. According to this classical mechanics wave theory, as long as there is sufficient light intensity or sufficient irradiation time, there is always a photoelectric effect for any incident light frequencies, whose conclusions are all in direct contradiction to the experimental results. Consequently, these conclusions of the photoelectric effect are incapable of being explained by the classical mechanics wave theory [19].
4.5. Bohr atomic quantum model
The wavelength of light source can be derived from hydrogen atomic spectroscopy:
(See PDF document) equation 28
where 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]:
(See PDF document) (where ᶄ is Rydberg constant, K = 2,3,4,...; n>k) equation 29
In this equation, ṽ is the wave number (ṽ = 2π/λ), and thus the conclusions of the hydrogen atomic spectrum are drawn as: the wavelength is determined by the difference between the two spectral item (See PDF document); if the K value of first spectral item (See PDF document) is fixed and the n value of the second item (See PDF document) is given by different values, the wave number of each spectral line in the same spectrum is obtained; if it is to change the k value of the first term, then different spectrums can be obtained[19].
Rydberg constant ᶄ = (See PDF document) equation 30
The specific values of parameter μ, e, ε0, c and h are input into the above formula, and the R value accurately agrees with the values measured by the experiments [19].
However, these conclusions derived from the hydrogen atomic spectrum experiment are unexplained by the classical electrodynamics theories. Firstly, the classical theory cannot establish a stable atomic model to explain the continuous rotation motion of electrons around the nucleus. According to the classical electrodynamics, the movement of electrons around the nucleus is accelerated, generating the continuous radiation of electromagnetic waves, which results in continuous loss of electron energy, so the motion orbit of electron rotation around the nucleus can not be stable and continuous, and finally the electrons should fall into the nucleus due to the energy lose. However, this demonstration on the basis of classical electrodynamics theories is obviously not consistent with the fact; Secondly, the radiation generated by the accelerated electrons should be continuously distributed, which is inconsistent with the atomic spectrums that are discrete spectral lines; Thirdly, according to the classical theory, if the electromagnetic wave source emits a stream of wave with the specific frequency ν, it may also emit various harmonics with different frequencies concurrently, whose frequencies are the integer multiples of ν, but the experiment results of hydrogen atomic spectrum are inconsistent with this, because the frequency of spectral line results follows the principle of convergence (the spectral lines show a combination of different wave frequencies)[19].
Under this background, Bohr developed Planck's quantum hypothesis on the basis of the conclusions of hydrogen atomic spectrum experiment, leading to atomic quantum theory in 1913. There are two important hypotheses in Bohr quantum theory as following: Firstly, an atom has the attribute of discontinuous state when an electron rotates around the nucleus, so only when an electron orbit with angular momentum p equals to the integer multiples of h/2π, it is stable, which means that electrons rotate in the stable orbit. Under this state, electron possesses constant energy of En, and electron in the stable state does not absorb or emit radiation [19].
(See PDF document) equation 31
In this equation, n is the quantum number under the quantization condition. Secondly, electron switches to different orbits in the way of quantum transition: when an electron jumps from an orbit of stable state at energy Em to another orbit of stable state at energy En, the frequency ν of the absorbed or emitted photons is calculated as [19]:
ν = (See PDF document) (frequency condition) equation 32
According to Bohr's hypothesis, it is to calculate the electron orbital radius an by applying classical mechanics, and then an and the corresponding energy En of the electrons in hydrogen-like atoms is expressed respectively as [19]:
an = (See PDF document) equation 33
En = (See PDF document) equation 34
a0 = (See PDF document) equation 35
In this equation, a0 is called the first Bohr orbital radius of the hydrogen atom; μ is the mass of electron; in the International System of Units (SI), es = e (4πε0) -12, and e is the value of the electron charges (electron charge is -e); ε0 = 8.854×10-12 C2/N·m2, in the 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 of hydrogen atoms can only be estimated as a series of discontinuous values, which are all quantized, and these discontinuous values of energy are usually called as energy levels of circular orbital. Experiments show that not only the energy of the hydrogen atoms, but also all other element atoms are quantized [19].
According to Bohr's frequency condition, the spectral frequency of hydrogen atom can be deduced into:
ν = (See PDF document) equation 36
Among which both n’ and n are the energy levels of hydrogen atomic orbitals [19]. This equation quantifies the spectral frequency of electromagnetic waves absorbed or emitted by electron, when the electron switches from lower energy orbitals to higher energy 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 frequency condition, then it is completely consistent with Balmer formula that is used to represent the wavelengths of hydrogen atomic spectral line [19] [22]. By measuring the values of μ, e, ε0, c and h in experiment, the calculated Bohr's frequency (ν) is well consistent with the experiment measurement results. After that, this Bohr’s atomic quantum model is further developed, suiting the elliptic shape orbitals of electron. By proposing the quantum model, the questions of microscopic particle motion can be quantitatively solved by adopting a combination of both classical mechanics and discontinuous quantized orbital equation [19].
4.6.Double-slit experiment between classical mechanics and electron
In the theory of classical mechanics, the state of a particle is described by the physical quantities of momentum and spatial position (vector), both of which can be measured independently. However, this calculation is not applicable on the microscopic particles due to the wave-particle duality of microscopic particles in quantum mechanics. As both the vector position and momentum of microscopic particles cannot be determined, how to quantify the wave-particle duality by using a physical quantity to describe the motion state of the microscopic particles? In 1926, Born attempted to explain the statistical nature of de Brogyi wave, who believed that de Brogyi wave did not represent the fluctuation of real physical quantity like classical waves, so it was a kind of probability wave that described the probability distribution of particles in space. To clarify this concept, the double-slit diffraction of electron experiment was conducted 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-slit diffraction, it is first to compare the results of double-slit experiments between classical 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 only the seam S1 is opened, the spatial distribution of particle density is described by the wave figure of I1 after passing through the S1; when only the seam S2 is opened, the spatial distribution of particle density is described by the wave figure of I2 after passing through the S2; when the double seams of both S1 and S2 are opened concurrently, the spatial distribution of particle density through the S1 and S2 seams are completely added as: I = I1 + I2. However, for the classical waves passing through the double seams, the wave interaction between both is not simply added: when only S1 seam is opened, the spatial distribution of the wave intensity is described by the wave 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 after the wave passes through the S2; when the double seams of both S1 and S2 are opened concurrently, the spatial distribution of wave intensity through both S1 and S2 seams are 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 waves independently 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 classical particle density distribution [19].
Next it is to analyze the electrons’ double-slit diffraction experiments on the basis of conclusions drawn from the classical particles and waves. When the electron beam passes through the double slits, if the incoming flow of electron beam is weak, the electrons pass through the double slit almost one by one and subsequently hit the photosensitive screen. When the photosensitive time is short, it seems that the distribution pattern of the screen light dots is random and irregular. Nevertheless, when photosensitive time is prolonged enough, the screen light dots become more and more, 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 almost disappearing, so the final electronic screen distribution forms regular interference pattern. The distribution pattern of electron beam intensity is similar to the classic waves, but is different from the classical particle distribution pattern. In the above experiments, the pattern of diffraction is independent of the intensity of the incident electron flow. When the electron flow intensity is weaken to the state that almost the electrons are emitted one by one, the initial distribution pattern of screen light dots appears to show some irregular spots. However, as long as prolonged enough time, the same interference stripe as classical waves is still obtained on the photosensitive screen, showing the fluctuation of the electron. Therefore, it can be concluded that each particle diffracts independently of other particles, which means that diffraction is not the result of the interaction between these different particles and fluctuation is possessed by each microscopic particle, so each particle shows the nature of both particle and fluctuation. From the particle theory aspect, the peak intensity value in the interference pattern means that the electron projection probability is high, while there are few or no particles density at the minimum value; From the perspective of fluctuation theory, the intensity of the wave at the maximum in the interference pattern is great, and the intensity of the wave at the minimum is extremely small or even zero. Based on the theory of both particles and fluctuation, Born proposed statistical interpretation of the wave function, describing that the intensity of the wave function (the square of the absolute value of amplitude) at a point in space is proportional to the probability of particle occurrence at that point [19].
4.7.Wave-particle duality of de Broglie wave
Overall, the wave nature of microscopic particles is based on the statistics only, so it is to clarify the difference between classical waves and microscopic quantum waves: classical waves are usually defined as the transmission of substances in vibration forms. For this reason, there are two vectors of physical movements that need to be clarified in waves: the first movement vector is the vibration motion, and the second movement vector is the transmission motion, but what substance gets transmitted is the key to classify these waves. Based on the definition of two movement vectors in waves, it is easily to compare and contrast the classical wave nature with the microscopic quantum particles: for example, sound waves involve the up-and-down vibrations 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’; for instance, electromagnetic waves would not need the medium and is capable of propagating directly through the vacuum. In comparison to the classical waves, the waves of microscopic quantum particles also do not require the medium, but they transmit ‘probability’ by itself, which means that probability of particle occurrence is vibrating and this vibration is not identical to a kind of vertical displacement of particles 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 microscopic particles are constantly vibrating in the spatial positions (or velocities), and what gets transmitted is the ‘probability’ itself, a mathematical value, other than the physical quantity [19].
In experiments, phenomena such as light and electrons sometimes behave like waves but in other times display like particles, so these phenomena exhibit wave-particle duality, but it is impossible to observe both wave and particle properties simultaneously. These phenomena are explained as: when an object's de Broglie wavelength is comparable to the particle size or exceeds its size, its wave nature can be detected, which thus cannot be ignored. However, if its de Broglie wavelength is much smaller compared to its particle size, then the particle object's wave nature cannot be detected at all. Consequently, it is proposed that the theories of both particles and waves nature can be applicable on the microscopic quantum particles as complementary explanations [19].
4.8.The mathematical expression of classical wave and quantum wave
The physical properties of the wave function are expressed mathematically below: it is described that the wave function of Φ(x, y, z, t) is defined as the state of a particle at a spatial point A with coordinate (x, y, z) and time t. The intensity of the wave is defined as the magnitude of complex number
│Φ│² = Φ* × Φ equation 37
where Φ* is the complex conjugate of Φ[19].
According to the statistical interpretation of the wave function, the probability of the particle occurrence at the spatial point A is defined as dW(x, y, z, t), which is proportional 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 the spatial point A (x, y, z) is located at the center of this volume unit, the coordinate of this 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 38
In the formula, C is the proportional constant, so the probability of a particle occurrence in a volume unit at the point A (x, y, z) at time t is defined as the density of probability, ω(x,y,z,t), which can be calculated as [19]:
ω(x,y,z,t) = dW (x, y, z, t)/dr = C ×│ΦA(x, y, z, t)│² equation 39
As particles must appear in somewhere over the whole coordinate (space point A can be any point in coordinate axes), the sum of the probabilities, representing that particles will appear at all the points in the whole spatial coordinate, is equal to 1, which can be consequently calculated as [19]:
(See PDF document) equation 40
The infinity symbol of ∞ under the integral signs means to integrate the probability over all spatial points, so the proportional constant C is deduced by above equation as [19]:
(See PDF document) equation 41
Next it is to define the normalization of wave function as Ψ(x,y,z,t), and then this wave function is derived from the magnitude of complex number and the proportional constant [19]:
Ψ(x,y,z,t) = C×Φ(x,y,z,t) = Φ(x,y,z,t) / (See PDF document) equation 42
The wave functions of both Ψ and Φ describe the same state, and thus the probability of a particle occurrence in the volume unit of dr near the point (x, y, z) at time t is further expressed by the normalization of wave function [19]:
dW (x,y,z,t) = │Ψ(x, y, z, t)│² ×dr equation 43
Then the density of probability in particle occurrence, ω(x,y,z,t), is further calculated by using normalization of wave function [19]:
ω(x,y,z,t) = │Ψ(x, y, z, t)│² equation 44
The sum of the probabilities, representing that particles will appear at all the points in the whole spatial coordinate, is also expressed by inputting normalization of wave function [19]:
(See PDF document) = 1 equation 45
The procedure of turning the wave function of Φ(x,y,z,t) into (See PDF document) is called as the normalized process of quantum wave, and the proportional constant C is correspondingly defined as the normalization constant [19]. With regards to the above normalization procedure of wave functions, there are additional methods in adopting this normalized wave function under different scenarios:
Firstly, it is worthwhile noting that the above normalization of wave functions is not unique. For example, if Ψ is a normalized wave function, then the transformed wave function of eiδ (See PDF document) (where δ is any real constant) is also normalized. Consequently, the equation of │Ψ│²=│eiδ×Ψ│2 means that eiδΨ describes the same probability wave with the phase factor of constant eiδ, and a normalized wave function can contain any phase factor, so it is feasible to use this property of wave functions to simplify the complex wave functions via multiplying or dividing by a phase factor in the next [19].
Secondly, if the magnitude of complex number, (See PDF document) , diverges, meaning that the wave function is not square-integrable over all space, then this wave function cannot be normalized according to the above steps, as this wave function would result in a normalization factor C = 0, which is clearly meaningless. For example, if the wave function of a free particle is defined as (See PDF document) that has a modulus squared of (See PDF document) , then this constant C is independent of time and coordinates. This independent constant means that the probability of a particle occurrence in a unit volume near any spatial point is the same, which is expressed as equation: (See PDF document) .
Consequently, such wave functions are not square-integrable over all space points and cannot be normalized according to the above 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 defined as ω(r, t) [19]:
ω(r,t) ×dr= │Ψ(r, t)│² ×dr equation 46
Using the above formula, it is to calculate the average value of particle coordinates (See PDF document) at x axis according to the common averaging equation by probability [19]:
(See PDF document) equation 47
Fourthly, if there is any mechanical quantity f(r) of a particle that is known, its average can be expressed as [19]:
(See PDF document) equation 48
Finally, the complex number form of wave function cannot be directly measured experimentally in quantum mechanics, so its mathematical equations only refer to the probability of particle occurrence in space, which is discussed above. Then the philosophy of quantum mechanics maths is that the variable of any real mechanical quantity of f(r) can be incorporated into this mathematical equation of occurrence probability.
4.9.The superposition of wave functions
In the linear system of classical physics, the linear differential equations (groups) are usually applicable on the physical quantities (including functions, vectors or vector fields) that meets the requirements of the linear equations (groups) describing their physical processes. For all the classical fluctuation processes, in which the principle of superposition is applicable, any fluctuation process φ is the result of the linear superposition of two possible fluctuation processes, φ1 and φ2, expressed as [19]:
φ = a×φ1 + b×φ2 (a, b are both constant) equation 49
For 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 the same space is the sum of the amplitudes generated separately by each wave. When the physical quantities are measured, only the amplitude of the synthetic variable is measurable, rather than the physical quantities generated separately by each wave, which means that its individual states participating in the superposition do not have their independent characteristics [19].
One of the main ways to calculate the physical quantities of a wave function is to sum the wave function as a superposition of some independent wave functions that are derived under particularly simple state, which is also called quantum superposition. For instance, because the Schrodinger wave equation is linear, the overall physical quantities of the wave function can be calculated on the basis of the superposition principle. In optics, the light interference and diffraction phenomenon can be explained by using the superposition principle: one beam of the incident electrons passes through slit S1 and the other beam passes through slit S2, represented by wave functions of ψ1 and ψ2 respectively [19].
The experimental results show that: the state of the particle after passing through the double slit is represented by the wave function of ψ, which is the result of the linear superposition of ψ1 and ψ2, calculated as [19]:
ψ = C1×ψ1 + C2×ψ2 (C1 and C2 can be any complex constants) equation 50
Only in this superposition way, the interference phenomenon can be explained, because the superposition of the interference pattern on the screen is measured by the interference intensity as below [19]:
│C1×ψ1 + C2×ψ2│² = │C1×ψ1│² + │C2×ψ2│² + C1C2×ψ1*×ψ2 + C1C2*×ψ1×ψ2* equation 51
Among which C1C2×ψ1*×ψ2 + C1C2*×ψ1×ψ2* is the interference item, explaining the interference intensity variation of the superposition waves [19].
It is further to deduce the principle of state superposition in quantum mechanics from the equations of classical substance wave: If ψ1 and ψ2 are two possible states of the system, then their linear superposition of ψ = C1×ψ1 + C2×ψ2 (C1 and C2 can be any complex constants) is also a possible state of the system [19]. Consequently, the key difference in linear state superposition between quantum mechanics and classical substance wave is that the linear state superposition of quantum mechanics only refers to the superposition of probability under the same state, but the classical substance wave superposition is the superposition of physical quantities.
5.Schrodinger wave equation
5.1.The transformation of free particle plane wave function
As the equation to be established describes how the wave function of Ψ(r,t) changes over time variable of t at space unit volume of r, there are the following conditions to be met: Firstly, the equation is a differential equation of the first derivative to the wave function Ψ(r,t), with respect to the time variable of t, as it allows to determine the 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 this equation, and then their linear combination of aΨ1 + bΨ2 is also a solution, so the linearity of the equation ensures that its solutions are applicable on the principle of superposition; Thirdly, the coefficients in the equation (such as constant a and b) should not contain any parameters describing physical state quantities such as energy or momentum [19].
By clarifying the above pre-conditions, it is to convert the free particle plane wave function of Ψ(r,t) into Schrodinger wave equation and the conversion steps are deduced below [19]:
The free particle plane wave function of ψ(r,t) is expressed as:
(See PDF document) equation 52
Where the space unit volume of r is near the spatial point with coordinates (x,y,z), so it can be re-expressed as:
(See PDF document) equation 53
Where px ,py , pz is the momentum vector at x, y, z axis, respectively [19].
It is first to take the partial derivative of time variable t on the basis of ψ(x,y,z,t):
(See PDF document) equation 54
Then it is to calculate the second partial derivatives at coordinates x, y, z respectively:
(See PDF document) equation 55
It is to further integrate three equations of second partial derivatives into a whole:
(∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2)×ψ = (See PDF document) equation 56
Where 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 equation
To 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 derived into [19]:
E×ψ = iћ× (See PDF document) equation 57
(p·p)×Ψ = (-iћ∇ )(-iћ∇)×Ψ equation 58
In this formula, the symbol of ∇ is called as Nabla operator in Euler’s method (Please note: as the imaginary unit, i2 = -1; 1/i = 1 in this deducing process) [19].
(See PDF document) equation 59
Where 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 are expressed as the following operators acting on the wave function [19]:
(See PDF document) equation 60
From this formula, it is to replace the energy and momentum variables by the Nabla operators of Euler’ method in the classic energy-momentum relationship, and then to apply them on the wave function Ψ, so the wave equation for a free particle can be obtained. For a particle in the given potential field of U, it is to define the potential energy of the particle under the force field to be U(r), and the total energy of the particle becomes the sum of both kinetic and potential energy [19]:
(See PDF document) equation 61
Replacing the physical properties of both E and p in the above formula by Nabla operators, then the wave function is re-expressed as [19]:
(See PDF document) equation 62
This equation is called the Schrodinger wave equation, which usually refers to the time-dependent Schrodinger equation, and it describes the variation of particle state with the time change under the potential field of U(r). If the force field acting on the particle wave varies with time variable of t, the general form of the above equation is calculated as [19]:
(See PDF document) equation 63
It 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 the wave function of Ψ(r,t) by dividing it into the sub-function (ψ(r)) of variable r and the sub-function (f(t)) of variable t separately [19]:
Ψ(r,t) = ψ(r)×f(t) equation 64
Then it is to derive the particular integral of the Schrodinger equation [19]:
(See PDF document) equation 65
The relationship between this wave function and time t is sinusoidal, with its angular frequency of ω = E/ћ. It can be seen from De Broglie's equation that the constant E is the energy of the system under the stationary state, so its corresponding micro-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 the energy E is given a constant value [19].
The probability density of a particle occurrence (See PDF document) is independent of time, indicating that the probability distribution of a particle does not change over time. The equation is expressed as [19]:
(See PDF document) equation 66
The average of any mechanical quantity does not change over time, which means that the time variable is not included in the functions of any mechanical quantity mean value [19].
To fully describe the motion state of an electron, there are four quantum variables that must be required, including the principal quantum number, angular quantum number, magnetic quantum number and spin magnetic quantum number, among which the first three variables are all the solutions of Schrodinger equation, except that spin magnetic quantum number is not a solution of Schrodinger equation. Both principal quantum number and angular quantum number are related with electron energy, while magnetic quantum number characterizes the electron angular momentum [23].
6.Further development of mechanics models in this article
6.1.Mechanical movement refers to the vector variation in the displacement of the mass point in matter both temporally and spatially, which is different from the movement of matter existing as energy only. According to the new definition of photon in my another quantum physics article [13], this article proposes that photons are the most elementary research object for mechanical motion, which are also the smallest partitioning mass unit of mass matter, so the natural Law of electromagnetic wave particle duality is a basic attribute of mass matter, not limited to the basic properties of energy matter.
6.2.According to the Figure 1 of my article [11], it is to further discuss the argument of the shielding effect of the electric field inside an atom and its effects on the electron orbitals:
6.2.1.For the adjacent atoms of the same element, the frequency of the electromagnetic waves generated by adjacent atoms is the same, so interference waves of electromagnetic waves are easily formed between adjacent atoms;
6.2.2.Multiple equipotential lines are formed between the zones of constructive interference and the zones of destructive interference. The destructive interference zone is relatively neutral due to the offsetting between wave peaks and bottoms, and electrons tend to undergo rotation motion in the destructive interference zones, thus becoming an important factor affecting the electron rotation orbit. In the motion model shown in Figure 1, the shielding effect of the electric field inside the atom causes the electron orbitals to be relatively fixed rather than the randomly disordered orbitals;
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.
6.3.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 shielding effects inside them. However, unlike the shielding effect inside microscopic atoms, macroscopic celestial bodies mainly rely on the substance boundary layers to form field shielding effects. The rupture of the boundary layer leads to the destruction of the shielding effect, which is the main factor causing various natural disasters such as tornadoes, earthquakes, solar flares, etc [6][7][8]. Therefore, the stable boundary layer and the generated field shielding effect is the important influencing factor in the development motion of celestial bodies. Similar to the internal equipotential lines of microscopic atoms, the overall equipotential lines inside celestial bodies tend to form closed loops. Due to the shielding effect generated by equipotential lines, substances move parallelly to the equipotential lines in both sides [12]. This motion model is the main factor that enables celestial bodies in our three-dimensional space to evolve into regular spherical shapes.
6.4.It is to re-analyze the wave-particle duality of de Broglie wave below:
Figure is indexed in BASE: Link
My article re-defines the classical material wave as mass wave, and it is to divide the De Broglie wave generated by elementary particles into two components, including mass wave and energy wave (both electric and magnetic field energy) shown in Figure 6. Then the difference in physical quantities is critically compared between classical material wave and quantum wave in the Table 1.
Table 1. Comparison between classical material wave and quantum wave.
| Classical material wave | Quantum wave |
| | Mass wave and energy wave (both electric and magnetic field energy) |
Energy form transmitted by wave | | Kinetic energy and electromagnetic energy |
Interaction form between two waves | Collisions among particles of two mass waves | Through wave nature of dark matter driven by two waves |
The product of two waves’ interaction | Interference wave by two mass waves | Interference wave by two mass waves; Interaction between positive and negative poles of two energy waves |
Under the hypothesis that De Broglie wave is divided into mass wave and electromagnetic energy wave, the wave-particle duality of de Broglie wave can be easily understood: de Broglie wave does not only possess the same attributes of mass particles as the classical material wave, but also shows the physical quantities of electromagnetic energy wave that is generated and carried by the beam of elementary particles (photons, electrons, proton...) at quantum level. Consequently, the wave functions of mass wave and electromagnetic energy wave need to be calculated separately 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 two micro-particles undergo wave motion in the same phase, the poles of electromagnetic wave show the same nature, so the interaction product of two micro-particles is to generate the repelling force, expressed as the mathematical equation, i2 = -1. Under this hypothesis, the imaginary unit of ‘i’ is given the realistic nature, rather than just facilitating the mathematics calculation.
6.5. In summary, this paper firstly reviews the classical principles of mechanics, and classical mechanics can effectively solve physics cases under the common limitation conditions, that include macroscopic physical conditions and low-speed motion model. However, under the situations of quantum micro-scale, cross-galaxy motion models and material aging process, new physical models need to be established to solve physical problems. My previous papers have fully discussed the particle collision motion model at microscopic quantum field [1], the microscopic quantum mechanics model under the electric field shielding effects of the overall atomic structure [10], the force balance analysis at each mass point inside an atom[2][3], inter-molecule force generating sources [4], thermal motion model of micro particles in the process of materials aging [5][9], friction resistance model at quantum scale [4], charged particles 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 my previous articles as well as by this current article.
Table 2. Summary of mechanics models originally proposed in our sponsored journals.
Scope level | Mechanics model | References |
| The particle collision motion model | [1]; Figure 4 of this article. |
| Mechanics model under the electric field shielding effects of the overall atomic structure | [10]; Section 6.2 of this article. |
| The force balance analysis at each mass point inside an atom | |
| The conduction of thermal motions among elementary particles (both electrons and protons) | |
| The resonant state of metastable composite nucleus and gravitational wave generation | |
Atomic or molecular level | Inter-molecule force generating sources | |
Atomic or molecular level | Thermal motion model of micro particles in the process of materials aging | |
Atomic or molecular level | Friction resistance model at quantum scale | |
Atomic or molecular level; Macro materials level | Charged particles motion model under free state at the substance boundary layers in nature | |
Macro planet or star level | Both parallel and vertical convection motions along the substance boundary layers forming field shielding effects of a planet or star. | |
| Dark matter principle and its application on inter-galactic motion model | |
| The magnetism along the fourth dimensional space and the driving force of celestial rotation | |
关键知识点译文：
1.机械运动亦称为力学运动，指代物质的质点在时间、空间中的位移矢量变化，区别于仅仅以能量物质存在的运动。本文根据本人在另一篇量子物理学论文中对于光子的新定义[13]，将光子作为机械运动的最基本的研究对象，也是质量物质中最微小的质点分割单元，因此电磁波的波粒二象性定理是质量物质的一种基本属性，并非局限于能量物质的基本属性。
2.根据本人一篇论文中的图1 [11]，进一步论述原子内部电场屏蔽效应与电子轨道的论点：对于同一种元素的相邻原子，产生的电磁波频率相同，因此相邻原子之间很容易形成电磁波的干涉波；相长干涉与相消干涉区域之间，形成多条等位线（等势线）。相消干涉区域由于波峰与波谷相抵，相对中性，电子会倾向于在相消干涉区域做自转运动，从而成为影响电子自转轨道的重要因子。在图1这种运动模型中，原子内部的电场屏蔽作用使得电子轨道一定相对固定，并非无序随机型；电子倾向于在某一条闭合环形等位线的外层空间做自转运动，符合电场屏蔽作用的形成条件。
3.与原子内部电场屏蔽效应进行对比与对照，宏观天体（比如恒星与行星）的星球内部也一定存在场量屏蔽效应，但是与微观原子内部的屏蔽效应不同，宏观天体主要依靠物质边界层形成场量屏蔽效应。边界层破裂导致屏蔽效应的破坏，这是导致各种自然灾害（比如龙卷风、地震、太阳耀斑等[6][7][8]）的主要因素，因此稳定的边界层以及产生的场量屏蔽效应是天体演化运动中重要影响因子。与微观原子内部等位线相似，天体内部的整体等位线一定倾向于闭合环形，由于等位线产生的屏蔽效应，物质沿着等位线两边做平行运动[12]。这种运动规律是我们所在的三维空间中的天体能够演变成为有规则球体形状的主要因素。
4.在假设德布罗意波分为质量波和电磁能波的前提下，德布罗意波的波粒二象性可以很容易被理解：德布罗意波不仅具备经典物质波的质量粒子属性，还在量子水平上表现出由基本粒子束（光子、电子、质子等）生成并携带的电磁能波的物理量。因此，接下来需要分别计算德布罗意波的质量波和电磁能波的波函数。在新的定义中，虚数概念指代在量子波中基本粒子的相位，具备现实、实际的物理性质，并非一种仅仅为了简化数学运算而引入的数学技巧。
5.总之，本文首先回顾经典力学原理，这些经典力学原理都在一个共同的局限条件下可以有效解决物理学上实际问题，即：宏观物理条件和低速运动模型。在量子微观尺度、跨星系运动模型、物质材料衰老等情境下，新的物理学模型需要建立起来才能解决实际问题。本人之前的论文已经充分论述了微观量子领域中粒子对撞运动模型[1]、原子整体结构的电场屏蔽作用下微观量子力学模型[10]、原子内各质点的受力平衡分析[2][3]、分子间作用力起源 [4]、材料在衰老过程中的热运动模型[5][9]、摩擦阻力的量子模型[4]、游离与自由态带电粒子在自然界物质边界层中的运动模型[6][7]、暗物质原理在跨星系间运动模型中的应用[8]等等。因此本文在表格2中全面总结了本人在之前论文以及本篇论文中论述的原创型力学模型。
发表于 2024-2-17 13:09:40
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