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The Nature of Schrödinger Equation on Quantum Physics - Part I The Nature of Schrödinger Equation On Quantum Physics - Part I Xue-Shu Zhao, Yu-Ru Ge, Xin Zhao and Hong Zhao Abstract. We propose that the Schrödinger equation results from applying the classical wave equation to describe the physical system in which subatomic particles play random motion, thereby leading to quantum mechanics. The physical reality described by the wave function is subatomic particle moving randomly. Therefore, the characteristics of quantum mechanics have a dual nature, one of them is the deterministic nature carried on from classical physics, and the other is the probabilistic nature coined by particle’s random motion. Based on this model, almost all of open questions in quantum mechanics can be explained consistently, which include the particle-wave duality, the principle of quantum superposition, the interference pattern of double-slit experiments, and the boundary between classical world and quantum world. Keywords: quantum physics, Schrödinger equation, double-slit experiments, Quantum superposition, the boundary between classical world and quantum world, the dual nature of particle-waves. I Introduction The discovery of Schrödinger equation is a significant milestone in the development of the quantum mechanics [1]. The Schrödinger equation has been proven very accurate in extensive experiments. It is generally accepted that the wave function representing the state of the system contains all the information about the system that can be known before observation. However, the explanations about Schrödinger equation do not say any information about the wave function, but a pure theoretical concept [2]. The Copenhagen interpretations believe that the wave function of the Schrödinger equation cannot directly describe objective reality at all. The wave equation only describes the probabilities of the physical quantity measured on the system, and the probabilistic nature of the wave function is caused by measurement behavior [2]. If a theory does not directly deal with reality, and only focuses on the probabilities of measurement, it seems unacceptable. Therefore, Einstein believed, “any serious consideration of a physical theory must take into account the objective reality. The wave function does not provide a complete description of the physical reality. The theory is incomplete, has to be replaced at some later date by a more complete and direct one’’ [3]. Although there has been a debate about how to interpret the wave function for nearly a century: does it represent the objective reality? Alternatively, it merely represents subjective knowledge of the observer. So far, there is still no consensus among physicists and philosophers. 1 Although, the Schrödinger equation was published in 1926, the origin of the equation is still unknown through the physical world [1]. As Feynman said, “Where did we get that equation from? Nowhere. It is not possible to derive it from anything that you know, it came out of the mind of Schrödinger” [4]. In other words, the path of Schrödinger to obtain his equation becomes an eternal mystery. In fact, even if we knew the path, perhaps we may still unable to understand the real physical meaning of this equation. Like Einstein, Schrödinger never accepted the probabilistic interpretation of the wave function of the equation. However, this probabilistic nature is precisely the core of the quantum mechanics. At present, the most important task is to find the exact knowledge on which the Schrödinger equation is based. There is no rigorous derivation of this equation, but based on the classical energy conservation, its formula can be constructed in a number of ways by physical and mathematical arguments [5-7]. Such as E. Nelson derived the Schrödinger equation from Newtonian mechanics in 1966 and Field derived the same equation from the Hamilton – Jacobi equation in 2011, until recent Yan derived the Schrodinger equation based on the principle of translation invariance of Hamiltonian mechanics in 2021. Although these derivative methods have provided some useful information for understanding the nature of the Schrodinger equation, however, none of them provides a satisfactory explanation for the Einstein’s criticism; does the Schrödinger equation directly describe objective reality? If we accept that the Einstein’s criticism on the Schrödinger equation of the quantum mechanics is definitely right, but we also know that the Schrödinger equation is indeed correct. What we can do to solve this conflict? If we are more willing to believe that anything happens in nature is determined by causality, then there must exist a causality related to this conflict, which has been ignored by physicists for almost a century. In the following section, we try to construct the Schrödinger equation through the missed causality, and further understand the connection between classical and quantum mechanics more deeply. II the Schrodinger Equation and Quantum Superposition Principle It is well known that temperature field exists everywhere in the universe even in deep space far from Galaxies. The tiny particles in a temperature field will move randomly in all directions with equal probability and various speeds. The average kinetic energy of a particle with a mass, m, in an environment with a temperature T is m = T (1) Where k is Boltzmann constant, the square root of , ( is called the root mean square (rms) speed of the particle, which is close to the most probable speed at the same temperature. From (1) we get ½ 1/2 υrms = ( = ( (2) 2 Here the υrms is rms speed of the particle with a mass m. The equation (2) shows that the rms speed of a particle is proportional to the square root of the temperature, and inversely proportional to square root of the mass. At a given temperature, the speed (υrms) of lighter particles is faster than the heavier particles. The Table 1 lists the rms speeds of some subatomic particles, and their corresponding de Broglie wavelengths at 20oC. We can see from the Table 1 that electrons at room temperature have an rms-speed, 1.15 x 105 m/s, approaching one thousandth of the speed of light. Under a given temperature, and without any disturbing field, an electron will move randomly around its initial position x0 with a root mean square speed (Figure.1). Since movement in all directions has the same probability, the trajectory distribution pattern generated by an electron moving randomly should have spherical symmetry about its initial position x0. Table 1, the υrms speed and L.de Broglie wavelength ===================================================================== o o Particle mass (u) υrms at 20 C (m/s) υrms at 1000 C (m/s) λdb (m) Electron 5.5x10-4 1.15x105 2.4x105 6.6x10-9 Proton 1 2.7x103 5.6x103 1.5x10-10 2 3 -11 N2 28 5.1x10 1.07x10 2.8x10 -12 C60 720 100 209 5.5x10 Si particle with 1.4x105 7.2 15 3.9x10-13 5x103atoms Si particle with 2.8x105 5.1 10.7 2.8x10-13 104atoms To simplify the situation, the Figure 1 shows a schematic diagram of the instantaneous trajectory distribution of the random movement of an electron confined in x – y plane around its initial position x0. 3 Now we use (x, t) to represent the one-dimension electron distribution function describing the electron moving randomly within its wave packet. At a particular time, when the coordinate is beyond the interval x1 to x2, the distribution function (x, t) should equal to zero. In this situation, the electron distribution function (x, t) looks like standing wave fixed at the both ends of the string. The important feature of the standing wave function is that the complete wave function ( , t) can be written as the product of two terms, one of them depends only on , and another depends only on t. So that the complete electron wave function has the form ( , t) = ( ) (3) In classical physics, for waves travelling along axis, the general form of the wave equation is = (4) Where υ is the wave speed, if we substitute above wave function (3) into equation (4), we get = - ( ) Divides on both sides of above equation, we get 2 = - ( ) ( ) (5) The equation (5) only contains variable . Recall Einstein hypothesis that the energy and momentum of a photon are E = , respectively. Since the energy and momentum of a photon are originated from the transition of an electron from its higher energy level to a lower energy level, the transition process must satisfy both the law of conservation of energy and the law of conservation of momentum at the same time. Therefore, it is reasonable to 4 apply the equations of the photon energy and of its momentum to the corresponding electron physical quantities. Using formula of ω = and = we get = Where ω is angular frequency, p is momentum of electrons, is the reduced Plank’s constant. In classical physics, the total energy E of a system is the sum of the kinetic energy, , and the potential energy U, which can be expressed by 2 E = + U or p = 2m (E - U) Therefore, we get = = (E-U) Substituting above equation into equation (5) gives + (E- U) = 0 (6) The equation (6) is the famous one –dimension time- independent Schrödinger equation for electrons, which has enormous successes for precisely determining the structure of atoms, molecular, and solids. From the path for deriving time-independent Schrödinger equation, we can see clearly that the wave function of Schrödinger equation is no longer explained like Copenhagen’s interpretation; it is just an abstract mathematical structure, having nothing to do with the reality of the described system. The above derivation process gives a straightforward connection between classical and quantum worlds. The physical reality described by wave function is the randomly moving electron hidden in its wave function (shown as Figure 1). Now we can come to conclusions naturally that the Schrödinger equation is rooted in the solid foundation of classical physics.
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