Origin of Probability in Quantum Mechanics and the Physical Interpretation of the Wave Function

The theoretical calculation of quantum mechanics has been accurately verified by experiments, but Copenhagen interpretation with probability is still controversial. To find the source of the probability, we revised the definition of the energy quantum and reconstructed the wave function of the physical particle. Here, we found that the energy quantum ê is 6.62606896 ×10 -34 J instead of hν as proposed by Planck. Additionally, the value of the quality quantum ô is 7.372496 × 10 -51 kg. This discontinuity of energy leads to a periodic non-uniform spatial distribution of the particles that transmit energy. A quantum objective system (QOS) consists of many physical particles whose wave function is the superposition of the wave functions of all physical particles. The probability of quantum mechanics originates from the distribution rate of the particles in the QOS per unit volume at time t and near position r. Based on the revision of the energy quantum assumption and the origin of the probability, we proposed new certainty and uncertainty relationships, explained the physical mechanism of wave-function collapse and the quantum tunnelling effect, derived the quantum theoretical expression of double-slit and single-slit experiments.


Introduction
As one of the pillars of modern physics, quantum mechanics has many notable achievements, with its theoretical calculation verified by experiments.However, its physical nature is still controversial.Studies on the counterintuitive phenomena of quantum mechanics, such as the quantum superposition state, wave-function collapse, quantum tunnelling effect, uncertainty principle, double-slit interference, and single-slit diffraction, have not reached a consensus.From a practical point of view, if the calculation results are correct, then the physical mechanism does not need to be debated.However, to understand nature, revealing the essence of matter and its interactions is the fundamental task of physics.Therefore, quantum mechanics must be studied to explore the essence of the material world.
The quantum superposition state, wave-function collapse, the quantum tunnelling effect, the uncertainty principle, double-slit interference, single-slit diffraction, and other counterintuitive phenomena are seemingly related to the probability of quantum mechanics, which has been physically interpreted by the Copenhagen school.
Although Planck, Einstein, Schrödinger, and other physicists strongly opposed it, they did not propose a better interpretation.In quantum mechanics, the probability is based on Born's hypothesis, which lacks physical sources; thus, Steven Weinberg has repeatedly questioned that the integration of probability into physics has confused physicists, but the difficulty of quantum mechanics related to its source rather than probability since 2016.Therefore, to understand the counterintuitive phenomenon of quantum mechanics, one must trace the origin of probability in quantum mechanics.

Smallest unit of energy -the energy quantum
Planck 1 proposed an energy density Formula (1) for black-body radiation in 1900, consistent with experimental data.Based on an assumption that the resonator of the black-body radiation source discontinuously radiates energy according to the smallest unit of , he theoretically derived the following formula. ν where is the energy density of black-body radiation; is the frequency of the resonator; c is the speed of light; k is the Boltzmann constant; T is the thermodynamic temperature; and h is the Planck constant.
In the process of deriving Formula (1), Planck 2,3 obtained Formula (2), which shows the energy radiated by the resonator of the black-body radiation source.
, n=1, 2, 3… ( Based on Formula (2), Planck believed that the energy radiated by a black body is discontinuous and thus can be taken only as an integral multiple of .If n=1, the resulting energy is the smallest energy unit possible (i.e., it cannot be subdivided), defined as the energy quantum.According to the definition of this energy quantum, de Broglie assumption that the energy E of a physical particle is equal to the energy of an energy quantum hν is clearly illogical, and it is also difficult to logically establish the wave function of a physical particle.
In the assumption of , Planck sets as the minimum energy released by the resonator of the black-body radiation source.Moreover, the dimension of the coefficient h is Joule•sec (J.s) to maintain the dimension of energy of when the dimension of the vibration frequency of the resonator is s -1 .However, the assumption of shows that the resonator, which radiates energy externally, is actually doing work outside.The minimum amount of power of a single resonator of a black-body radiation source is , where the dimension of is J.s -1 and the dimension of the coefficient h is Joule (J).Based on the above analysis, if both sides of Formula (2) are divided by unit time T 0 (1 s), then Formula (2) becomes Formula (3). ( Let and ; then where E is the energy radiated externally by a single resonator of the black-body radiation source per unit time, which is the power with dimension J.s -1 .If n = 1 in Formula (4), then Formula (5) holds.
J (5) Formula (5) shows that ê is the smallest unit of energy radiated by a single resonator of a black-body radiation source.We define the minimum unit of energy ê as an energy quantum with the unit of Joule.Clearly, the energy of the material world is discontinuous, and the energy quantum is ê instead of hv, with a value of 6.62606896 ×10 -34 J.The energy of the energy quantum is a physical constant that cannot be subdivided and changed and is not related to time or any other factor.However, the period of the corresponding vibration can vary.
Based on Planck's hypothesis, A. Einstein 4,5 believed that light wave energy is discontinuous and that the smallest unit of each discontinuous and indivisible packet of energy is hν, named the photon.
Everyone knows that the radiation of a black-body radiation source takes the form of electromagnetic waves, namely, light waves.According to the definition of the energy quantum, the minimum energy of a period of an electromagnetic wave radiated by a vibration of a resonator in a black-body radiation source is equivalent to the energy of an energy quantum, with a value of 6.62606896×10 -34 J, which is transmitted outward at the speed of light.Therefore, we defined the minimum energy transmitted by a periodic electromagnetic wave as a photon, then, the energy of a photon is instead of of Einstein's hypothesis.As a physical constant that cannot be divided and changed, it is not related to time or any other factor.However, the period and wavelength of the corresponding electromagnetic wave can vary.

Smallest unit of quality -the quality quantum
Einstein proposed that the energy and quality of matter are equivalent; Formula (6) shows their relationship. ( where E is the energy; m is the quality; and c is the speed of light in a vacuum. When the energy takes the energy quantum ê, an indivisible minimum unit of quality exists, namely, the quality quantum ô.Formula (7) shows the relationship between ê and ô. (7) Substituting the values of the energy quantum ê and the speed of light c into Formula (7), the quality quantum ô is 7.372496×10 -51 kg.In addition, the photon rest mass is zero, with a total mass of 7.372496×10 -51 kg.In the material world, the minimum value of the rest mass of physical particles other than photons is ô.Therefore, the mass of photons and physical particles can increase or decrease only following the integral multiples of the quality quantum.Mass and energy discontinuities are the essences of the material world.If the energy quantum, quality quantum, and light speed are regarded as natural constants of the material world, then Formula (8) shows their relationship. (8)

Wave function of photons
According to electromagnetic theory, the electromagnetic wave function is as shown in Formula (9). ( where ψ and ψ 0 are the amplitude and maximum amplitude of the electromagnetic wave, respectively; ν and are the frequency and wavelength of the wave, respectively; and and t are the spatial position and time, respectively.
According to its definition, a photon's energy is ê, which corresponds to one period and one wavelength of electromagnetic wave energy in a single electromagnetic wave.
n( ) photons with a frequency ν are superimposed to form a superimposed photon, the energy E and momentum p of which are defined in Formulas (10) and Formula (10) shows that the superimposed photon is a photon as defined by Einstein.
ℏ (14) Formula ( 14) is a function relation that characterizes the spatial propagation of superimposed photon streams after the electromagnetic wave function is quantized.In the wave function (14) of photon streams, ψ and ψ 0 represent the intensity amplitude and the maximum intensity amplitude of energy transmitted by the photon streams, respectively.The modular square of ψ is the energy density of the photon streams.
Since Formula (9) is the wave function of a monochromatic electromagnetic wave with frequency ν, Formula ( 14) is the wave function of a single energy superimposed photon stream with frequency ν.More complex electromagnetic waves are mixed with multiple frequencies in the objective world, the wave function of which is formed by the linear superposition of the monochromatic electromagnetic wave functions, expressed as where ψ 0νj represents the maximum intensity amplitude of electromagnetic waves with frequencies ν j in the composite electromagnetic waves and j is a positive integer from 1 to ∞.
Formula ( 16) is the photon stream wave function at the composite frequency quantized by the composite electromagnetic wave function (15).
where ψ 0pj represents the maximum intensity amplitude of energy transmitted by the photon stream with momentum p j in the composite photon stream and ψ represents the energy amplitude delivered by the composite photon stream.In addition, the modular square of the wave function ψ of the composite photon stream is the energy density transmitted by the composite photon stream.
The quantization of the electromagnetic wave function does not change the nature of the energy transmitted by photons and electromagnetic waves, while the energy density corresponds to the number of photons in space and time.In the whole space and time range of electromagnetic waves and photon motion, N is the total number of photons, and N j is the number of photons with momentum p j .Let |C Ppj | 2 =N j /N, which represents the photon distribution rate with momentum p j .Since the photons move throughout the entire space-time range with a constant total number, the sum of the distribution rates equals 1.Then, Formula (16) can be transformed into Formula (17).
ℏ ，j=1, 2, 3… (17 where ψ represents the total momentum photon distribution rate amplitude at time t in space r.Formula (17) expresses the function relation of the distribution rate amplitude, which periodically changes with time and the spatial position when photons move in space, i.e., the wave function of photon motion.Moreover, the modular square of the wave function equals the photon distribution density at time t in position r.

Wave function of physical particles
As a particle, photons have zero rest mass, while the photon velocity at all frequencies in a vacuum equals the speed of light c.Energy transmitted by photons has periodic changes and fluctuations in space and time directly related to the number of photons.
Additionally, the energy fluctuation causes the photon distribution rate amplitude to change periodically with time and space.The photon distribution density also fluctuates, as represented by the wave function (17).
A moving physical particle has energy; thus, it is a particle that transmits energy.
Regarding energy transfer, physical particles are similar to photons, and their rest mass is not zero, with a moving speed less than that of photons.According to Formula (8), the energy quanta transmitted by the physical particle are proportional to the number of the quality quanta contained Therefore, the distribution densities of the energy and quality quanta at time t and position r are equal.
When the energy transmitted by a particle equals that of a superimposed photon of the same frequency, the energy and momentum of the latter in Formula (17) can be transformed into those of the former.In addition, |C pj o | 2 represents the distribution rate of the particles' quality quanta with the momentum p j in a multi-particle system, where C pj o represents the distribution rate amplitudes of the quality quanta.
Regardless of the interaction between particles, the wave function of the superimposed photons becomes that of the multi-particle system.Therefore, the multi-particle system is a quantum objective system (QOS), and Formula (18) is its wave function.
where ψ represents the distribution amplitude of the quality quanta of the particles at time t in space r.Meanwhile, the modular square of the wave function ψ equals the distribution density of the particles' quality quanta at time t and position r.
In addition, if all particles have the same quality, ψ represents the amplitude of the distribution rate of the particles at time t in space r, which changes periodically with time and space.Moreover, the modular square of the wave function ψ equals the distribution density of the particles at time t and position r.Formula (18) is the QOS wave function.
Since the total number and energy of particles in space are constant, the particle

Origin of probability in quantum mechanics
Formula (18) is the wave function of a QOS composed of all state particles with the same quality.Regardless of the particle interaction, each particle has its energy, momentum, and state of motion.Considering a particle in the QOS, we do not know which one of the many in the system it is before measurement.Thus, it may be any one of the QOS.To facilitate a comprehensive study of this particle, we built a mathematical system artificially, called a quantum mechanics system (QMS).In the QMS, we study only one particle, which may be any particle in the QOS before measurement.The particle may correspond to all state particles in the QOS, meaning that the particle in the QMS has the possibility of all state particle distribution density at time t in space r in the QOS.We refer to the distribution rate of the particles in the QOS as the probability of the particle occurrence at time t in space r in the QMS, which is the origin of the Born probability hypothesis 6 in quantum mechanics.
Therefore, for the particle in the QMS, Formula (18) is the wave function.The particle wave function in the QMS is represented as Formula (20) to distinguish the QOS wave function.
If the wave function of the QMS is continuous, Formula (20) can be expressed in integral form (21). Since the particle always exists in the whole space, the particle probability in that space equals 1.
Since the QOS has no concept of probability, no randomness exists.However, the QMS introduces probability through artificial mathematical operations; thus, randomness is introduced.

Revision of the uncertainty principle and physical nature
In a QMS, Heisenberg's uncertainty relation, shown in Formula (25), is deduced based on the state superposition principle of the wave function, Born's probability hypothesis, and the non-commutation relation of operators.
p ℏ 2 (25) where and p are the uncertainties of the position and momentum of a particle, respectively.The physical meaning of the uncertainty relation is that a particle's position and momentum cannot be determined simultaneously.It indicates that the particle does not have a defined trajectory of motion.
Based on the essence of the state superposition principle in a QMS and the origin of the Born probability, and in Formula (25), where x and p are the measured values of the determined particle position and momentum, respectively; and are the average values of all possible positions and momenta in the QMS, which can be calculated by Formulas ( 26) and ( 27), respectively, and correspond to the average value of all particle positions and momenta in the QOS, rather than that of the measured particle.Therefore, and p indicate the deviation degree of the particle from the system (namely, the standard deviation) instead of being the measurement errors of the particle position and momentum.Use of the system's standard deviation to indicate the measurement error of the position and momentum of a particle is not appropriate. (26) where and are the position and momentum operators, respectively.
In a QMS, a particle can be measured by calculating its wave function with the operator.However, the wave function is expressed by the superposition state of particles with various states in the QOS; the position and momentum obtained by each measurement may be different; and the position and momentum of another particle are used to represent those of the particle to be measured.Therefore, in Formula (25), the position and momentum of a specified particle should not be expressed with those of another particle.
Based on the above analysis, the mathematical treatment of the state superposition the probability amplitude of a particle at time t in space r, which changes periodically with time and space.Meanwhile, the modular square of the wave functions represents the probability density of the particle appearing at position r at time t.The wave function is exactly the same as de Broglie7 wave function.From a mathematical point of view, Formula (20) and (21) indicate that the wave function of the particle in the QMS is always on a linear superposition state in all states of the particles in the QOS, and this is the physical nature of the state superposition principle of the wave function.Particles in different states in the QOS have different momenta and energy, indicating different moving velocities.A particle swarm composed of multiple particles expands in space over time.Therefore, the continuous expansion of the space volume occupied by the QOS is the physical nature of the continuous expansion of a particle's wave function in the QMS.