The proposed work presents the derivation of the wave function and the Schrödinger equation based on the dual tangential equation of time with an imaginary rate. The derivation uses de Broglie formulas. The wave and complex-conjugate functions are considered, which participate in the probabilistic description of the position of a particle in a potential field. Another interpretation of the product of these functions is presented. Based on it, the classical structure of the electron is derived. The derivation determines the quantum value of the imaginary incident vector. When deriving wave functions, simple solutions immediately arise for energies corresponding to radiation emitted by portions and a microscopic harmonic oscillator. In quantum mechanics, the first solution was used by Planck as a hypothesis when deriving his famous formula for describing equilibrium thermal radiation. The second energy solution was obtained from the Schrödinger equation by cumbersome calculations.
The proposed work presents the derivation of the wave function and the Schrödinger equation based on the dual tangential equation of time with an imaginary rate. The derivation uses de Broglie formulas. The wave and complex-conjugate functions are considered, which participate in the probabilistic description of the position of a particle in a potential field. Another interpretation of the product of these functions is presented. Based on it, the classical structure of the electron is derived. The derivation determines the quantum value of the imaginary incident vector. When deriving wave functions, simple solutions immediately arise for energies corresponding to radiation emitted by portions and a microscopic harmonic oscillator. In quantum mechanics, the first solution was used by Planck as a hypothesis when deriving his famous formula for describing equilibrium thermal radiation. The second energy solution was obtained from the Schrödinger equation by cumbersome calculations.
The Schrodinger equation based on the de Broglie wave is the most fundamental equation of the quantum mechanics. There can be no doubt about it's prediction validity. However, the probabilistic interpretation on the quantum mechanics has insoluble semantic interpretations like reduction of wave packet on observations of physical values. Especially, it is not clear that the wave function which is described by complex function, is whether formality or reality to express the state of particle motion. On this paper, we interpret the wave nature of particle as not the inherency of particle itself, but the motional property of particle in fluctuated space-time due to the kinetic energy and momentum the belief that the kinetic energy and momentum fluctuates the microscopic space-time, and the particle move through the fluctuated space-time adversely. Then, through the particle motion in the Euclidean space-time, the particle will be recognized as if it has the wave nature. We estimate the governing equation of fluctuations of microscopic space-time based on the macroscopic law of motion. On this paper, the equivalence between the governing equation and the Schrodinger equation is indicated.
Abstract This paper intends to provide instruction for a single lesson in an introductory course on quantum mechanics by tying together probabilistic methods from statistical mechanics, the use of complex numbers, and a more founded explanation for the derivation of Schrödinger’s time-independent equation. Here presented derivation strongly supports the Copenhagen interpretation of quantum mechanics. Niels Bohr suggested a hydrogen atom model (similar to Kepler’s sun-planet model), where the negatively-charged electron moves in orbit around the heavy positively charged nucleus, this electron having some kinetic and potential energy. Starting from Hamilton’s differential equation for Kepler like atomic model, Schrödinger introduced a new unknown real function ψ in his derivation of time-independent equation (initially, he did not recognize it as the wave function). We show that hypothetical statistical spread of action on many (indefinite) levels is mandatory for the derivation of Schrödinger’s time-independent equation. The result of this hypothesis, the statistical probability constraint formula, is used in this derivation as we look for a minimum of probability for the electron energy distribution E ψ in the whole space. The Euler–Lagrange equation, with normalization of ψ as a constraint limitation, then gives Schrödinger’s time-independent equation for real numbers. A closer inspection of the normalized function ψ shows that it could be complex and is related to the de Broglie–Bohr model of the hydrogen atom. It can be postulated that the classical solution of Schrödinger’s time-independent equation should be a normalized complex function ψ .
Since the interest in the quantum phenomena started, there have been two main approaches to the quantum theory, the first approach investigates the motion of a wave-particle object by assuming the wave has the characteristics of a moving particle, momentum and energy, this approach was developed by physicists Max Planck, Albert Einstein, and others. The second approach investigates the motion of a particle-wave object by assuming the particle has the characteristics of a moving wave, wavelength, wave vector, and frequency, this approach was developed by physicists De Broglie, Schrodinger and others which led to the presence of quantum mechanics. This article focuses on the second approach and merging it with classical mechanics. An early mathematical formulation for the quantum theory was proposed by De Broglie’s relations and later on, another formulation was given by Schrodinger’s equation. A successful attempt to bridge the gap between classical mechanics with quantum theory implemented via a mathematical derivation for De Broglie’s relations and Schrodinger formulation, The derivation process overcame a main obstacle was the convergence of the deterministic nature of classical mechanics with the probabilistic nature of the quantum theory by adopting an innovative method based upon replacing constant values in a deterministic relation with probabilistic variables in the same relation, and this led to switch the relation from deterministic representation to probabilistic representation. Both classical and quantum explanation are given for the deduced formulas. An energy approach is instrumented. Since classical mechanics formulas are not sufficient to explain the quantum phenomena, a new concept of complementary energy introduced in order to connect the parameters of a moving particle with the parameters of an associated wave, the value of the complementary energy for each particle adjusted the total energy equation of N moving particles without changing the magnitude of the total energy of a system of N particles. In special cases, the value of the complementary energy for some particles equals the value of the total energy for the particles. The derivation process indicates that complementary energy corresponds to the potential energy on the surface of the material which prevents the particles to escape from the boundary of the matter to the outer space and maintains its coherence, complementary energy concept also manifest the impact of the moving particles on a single particle due to the energy conveyed during direct contact in case of collision or by the remote influence through an existing field in the space, so it represents an energy of dual nature for both matter and radiation wave combined. The complementary energy is not introduced as mathematical assumption in order to manipulate the total energy equation, it’s actually exists in the total energy equation and its validity has not been applied before. A general format for any potential energy is obtained regardless the nature of field it represents (a universal format for potential energy) that is by using mathematical manipulation for the equation of total energy of N moving particles. The derivation process led to a particle-wave formula. The substitution into the particle-wave formula with the speed of light, De Broglie’s relations, revealed. Some additional applications for the particle-wave formula are suggested, also as a well known energy quantization formula obtained by applying all of De Broglie’s relation, a general energy, and speed formulas resulting during the derivation process. Matching time independent equation called Schrodinger’s amplitude equation with classical mechanics formulation demonstrated by deducing both the wave function and the magnitude of the wave vector from the classical formulation after converting it to probabilistic formulation. A dedicated section for wave function solutions for Schrodinger’s time dependent equation was given in terms of the wave functions and the magnitude of the wave vectors obtained from the matching process of Schrodinger’s amplitude equation with classical mechanics.
Quantum mechanics began in very early twentieth century when Max Planck found the expression for black body thermal radiation in which the emitted light was not a continuum as postulated by classical physics. Erwin Schrodinger applied his equation to solve the quantum harmonic oscillator, the quantum rigid rotor, and a hydrogen-like atom. Pertaining to the solution of the Schrodinger equation to hydrogen atoms are the quantum numbers. C. K. Ingold’s overview of Schrodinger wave mechanics: “What can be known about an electron, considered as a particle, is of a statistical nature, and is summarized in the behavior of a wave. When considering singlet substances, the normal Zeeman Effect occurs, providing the discrete values of the orbital angular momentum. In an attempt to rationalize the anomalous Zeeman effect, W. Pauli proposed the fourth degree of freedom of the electron which he called “two-valuedness not describable classically”.
Till 1925 the idea of what the quantum theory would be was unknown. During 1925 three different but equivalent versions of quantum theory were proposed – Schrödinger-proposed wave mechanics; Heisenberg-developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie’s matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner (hence called wave function and denoted it as ψ). This function is generally complex and assumed to contain information about a system. Schrödinger set up a linear and time-dependent wave-like equation, called Schrödinger wave equation, to describe the wave aspect of a particle taking account of de Broglie’s relation for wavelength. Physically, |ψ(X, t)|2, where ψ(X, t) is the solution of the Schrödinger equation, is interpreted as position probability density. That is, |ψ(X, t)|2 is the probability density of observing a particle at position X at time t. ψ does not give exact outcomes of observations but helps us to know all possible events and their probabilities . Further, the probability interpretation allows us to find the average or expected result of a set of measurements on a quantum system.
In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.