The uncertainty principle comes into force because of the wave like properties of “particles” like electrons and protons. The waves tell the dynamics of the particle – its momentum, its energy and even its angular momentum. Wave packets are made by adding together many simple waves that run forever. If electrons have the wave properties then there must be a wave equation and a wave function to describe the electron waves just as the waves of light, sound and strings are described. The success of wave mechanics was well demonstrated by Erwin Schrödinger before an acceptable interpretation of the wave function was known. Max Born utilized the probability concepts of the uncertainty principle to give us the presently accepted ideas of the wave function.
The uncertainty principle comes into force because of the wave like properties of “particles” like electrons and protons. The waves tell the dynamics of the particle – its momentum, its energy and even its angular momentum. Wave packets are made by adding together many simple waves that run forever. If electrons have the wave properties then there must be a wave equation and a wave function to describe the electron waves just as the waves of light, sound and strings are described. The success of wave mechanics was well demonstrated by Erwin Schrödinger before an acceptable interpretation of the wave function was known. Max Born utilized the probability concepts of the uncertainty principle to give us the presently accepted ideas of the wave function.
Abstract Introduces the Schrödinger wave equation as the key equation governing the behaviour of electrons, and presents simple solutions (known as wave functions) as electron waves, and packets of electron waves, which can represent localised, particle-like electrons. Considers the behaviour of electron waves on meeting a potential barrier and trapped in potential wells with finite and infinite walls, and introduces the idea of energy levels. Discusses the accepted interpretation of the electron wave function, namely that its modulus square leads to the probability of finding an electron in a given region of space. Considers further the uncertainty relationship and associated philosophical implications.
This paper is a small discussion about the uncertainty principle. With the great discoveries and researches made by predecessors, I try to answer some questions within the uncertainty principle. Using the definition of momentum, the momentum in de Broglie wave, the property of cosine function, the Euler’s formula, the basic property of plane wave, the law of derivative, the energy of quantum discovered by Max Planck, the photon energy discovered by Albert Einstein, the energy equation of a system and the Hamiltonian operator, I figure out the Schrodinger equation successfully. With the help of Schrodinger equation, the possibility density of a wave function, the concept of normalization, the average value of position and momentum, the Fourier transform, the integration by parts and the limit of normalized wave functions, I find the position operator and the momentum operator. Then using some knowledge of the standard deviation and the variance, the concept of inner product of vectors, some property of Dirac symbol, the law of calculation of complex number and the law of operator calculation, I obtain the general uncertainty relationship of abstract mathematical quantities. According to the general relationship, I get the inequality of uncertainty principle, and discuss the specific relationship between the position and the momentum. After getting the inequality, I use Heisenberg’s experiment of measuring an electron by using the γ ray to hit the electron to explain why we cannot measure the position and the momentum simultaneously. Based on Heisenberg’s experiment, I give my own explanation by discussing the earth and a spacecraft. Transforming the perspective, we can easily see how the accuracies of position and momentum change. In addition, I attempt to figure out why the uncertainty exists. However, I can only explain it by discussing the time and frequency in mathematical method and have no idea to work it out in a pure physics approach.
Abstract While the theory proved successful at describing a broad range experimental results, from the black body spectrum to the hydrogen atom, the question arose as to what the theory was telling us about the nature of physical reality. Schrödinger’s attempt to represent particles as wave packets failed because of the spreading of the wave packets. Max Born, investigating how the theory could be applied to scattering processes, was led to introduce the probability interpretation of the wave function. In an attempt to summarize the physical meaning of the theory, Heisenberg introduced the uncertainty principle. Niels Bohr and his Copenhagen school developed the philosophical position known as “complementarity.” There was a bitter dispute between Schrödinger and the “Copenhageners” over the necessity of quantum jumps in the theory.
Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.
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”.
Abstract The second chapter introduces the notion of wave functions and their equation of motion proposed by Erwin Schrödinger. The interpretation and normalization of the wave function are discussed, leading to the continuity equation. The chapter also introduces basic mathematical operations that can be carried out for wave functions and operators, describing e.g., how to calculate scalar products and commutators, as well as expectation values and variances of observables. With these tools at hand, Heisenberg’s uncertainty principle is re-examined, focusing on the example of a Gaussian wave packet, which is studied both in position space and momentum space. We also discuss the time evolution of general states and of the exemplary wave packet. Finally, we will see how classical physics is recovered as a limiting case of the quantum-mechanical description