The Schrödinger equation is the fundamental equation of quantum mechanics, primarily discussing the evolution of quantum states over time and solving the wave function under specific conditions. Its position is equivalent to that of Newton's equation in classical mechanics. Based on the literature “Teaching Attempts to Introduce Wave-Particle Duality in Quantum Mechanics,” this paper further establishes the Schrödinger equation through the Maupertuis principle. By comparing several typical teaching methods for establishing the Schrödinger equation, the characteristics of the method proposed in this paper are further highlighted, which helps students to integrate new and old knowledge organically, more easily accept and understand the basic equations of quantum mechanics, enhance students' ability to transfer knowledge, and stimulate students' creative thinking.
The Schrödinger equation is the fundamental equation of quantum mechanics, primarily discussing the evolution of quantum states over time and solving the wave function under specific conditions. Its position is equivalent to that of Newton's equation in classical mechanics. Based on the literature “Teaching Attempts to Introduce Wave-Particle Duality in Quantum Mechanics,” this paper further establishes the Schrödinger equation through the Maupertuis principle. By comparing several typical teaching methods for establishing the Schrödinger equation, the characteristics of the method proposed in this paper are further highlighted, which helps students to integrate new and old knowledge organically, more easily accept and understand the basic equations of quantum mechanics, enhance students' ability to transfer knowledge, and stimulate students' creative thinking.
th , 2012; revised: Jan. 8 th , 2013; accepted: Jan. 16 th , 2013 Abstract: In order that students have a whole realization of the wave mechanics foundation, we review the two impor- tant enlightened steps in details, i.e. the analogues between mechanics and optics, de Broglie matter wave. Based on the duality of wave and particle, we report the whole process that Schrodinger founded his wave mechanics. We introduce Schrodinger's interesting and deep method of deriving wave equation to quantum mechanics teaching.
The wave-particle duality for light has been well established by various experiments, such as Youngâs double slit experiment and the photoelectric effect. This led de Broglie to propose that a particle also has wave characteristics. Schrödinger further established the wave equation for a moving particle. Many times, quantum mechanics textbooks do not show how Schrödingerâs equation was developed in an intuitive manner that is appropriate at the undergraduate level. This article presents a new approach for introducing Schrödingerâs equation. This new approach starts with Maxwellâs equations and then applies the quantized energy of a light wave and special relativity. This more intuitive approach should help undergraduate students understand the origin of Schrödingerâs equation in a more natural way.
We propose that the Schrodinger 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 the classical world and the quantum world. The current quantum mechanics is a mixture of matrix mechanics and wave mechanics, which are sharply conflicting in principle. Matrix mechanics treats quantum particles as classical particles with fixed relation between the particle's position and its momentum. The matrix mechanics, in fact, belongs to the old quantum theory. Both Born's non-commutative relation and Heisenberg uncertainty relation originate from matrix mechanics. However, in wave mechanics, there is no any fixed relation between the particle's position and its momentum, and the particle's position and its momentum belong to immeasurable physical quantities. Therefore, there is no need for non-commutative relation and uncertainty relation in wave mechanics.
Quantum mechanics is one of basics theories of modern physics, and it is also a theory established with some debates of assumptions. However, in the teaching process, if the assumption of wave-particle duality is directly given, it will be particularly difficult for students to understand. Therefore, based on the teaching practice and previous research, we assume that light corresponds to a kind of real particle starting from the classical electromagnetic theory, according to the description methods of wave optics and geometric optics, and the form of functional equations. Subsequently, employing Planck's quantum conditions, we discuss the connection between the wave nature and particle nature of light. And then the Planck-Einstein relation and the de Broglie relationship are derived, which reflect the wave-particle duality of light and real particle.
Quantum Mechanics is one of the most important scientific developments of the twentieth century. It serves as the basis for understanding how matter and energy behave on very small scales. Unlike classical mechanics, quantum mechanics explains physical phenomena through probabilistic models, wave functions, operators, and discrete quantized states. This paper offers a thorough study of the fundamentals of quantum mechanics, covering its historical background, postulates, mathematical framework, wave-particle duality, Schrödinger equation, uncertainty principle, and operator theory. The paper also looks at how quantum mechanics is used in modern technologies such as quantum computing, lasers, semiconductors, and medical imaging. The goal of this paper is to provide a clear and detailed foundation for students and researchers who want to grasp the main principles of quantum theory. General Terms Quantum Mechanics, Physics, Wave Function, Uncertainty, Quantum Theory, Measurement, Operators.
This paper focuses on the principle of time dependent Schrodinger equation (TDSE) which simplified so many reviewed ideas, class room teaching, research ideologies and personal study meant for advanced knowledge. In this research work, the enlightenment of the basic concept of Schrodinger wave equation to improve knowledge about simple ways for mathematical understanding in deriving TDSE using different technique in more comprehensive approach. The research shows clearly that TDSE can be derived using time independent equation, wave mechanics, classical & Hamilton-Jacobi’s equations. Different methods and ways by different researchers/scholars have been used in the past. In this paper, we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational. Surely, a particle is what a particle detector detects. This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation. However, the result provides a comprehensive well derived derivation, derived using various approaches, which would make this research a unique one from different areas of specializations.