Abstract This chapter explores the quantum uncertainty and unknown of the nucleus and its role as the engine of the stars. It elaborates on the theory behind quantum mechanics to further understand the stability of atoms and matter. Among its many implications, quantum mechanics dictates that an electron cannot go wherever it pleases in an atom, so electrons in atoms follow a fundamental regularity. The chapter explains the golden rule of quantum physics concerning higher and lower momentum or energy and the phenomenon of beta decay concerned with the charge of a nucleus changing by one unit. It looks into the consequences of nature seeking stability to reduce the energy in an unstable configuration.
Abstract This chapter explores the quantum uncertainty and unknown of the nucleus and its role as the engine of the stars. It elaborates on the theory behind quantum mechanics to further understand the stability of atoms and matter. Among its many implications, quantum mechanics dictates that an electron cannot go wherever it pleases in an atom, so electrons in atoms follow a fundamental regularity. The chapter explains the golden rule of quantum physics concerning higher and lower momentum or energy and the phenomenon of beta decay concerned with the charge of a nucleus changing by one unit. It looks into the consequences of nature seeking stability to reduce the energy in an unstable configuration.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.
This chapter provides a foundational overview of the physics essential to understanding quantum computing. It begins with atomic structure and the stability challenges posed by classical models, leading to Planck's quantization and the resolution of blackbody radiation. Einstein's photoelectric effect and Bohr's quantized orbits introduced discrete energy levels, later refined through quantum probability, half-life, and spectral transitions. The chapter details the shift from deterministic to probabilistic physics, introducing quantum numbers, the Pauli Exclusion Principle, and electron spin. It distinguishes fermions and bosons by spin statistics and explores Heisenberg's uncertainty principle and Schrödinger's wave function. Dirac's theory of antimatter is introduced through energy equations. The role of the strong nuclear force, quantum tunneling, and the implications for modern chip design are discussed. The double slit experiment illustrates superposition, interference, and the collapse of the wave function—concepts central to quantum computing. The chapter bridges quantum mechanics to computing by connecting atomic behavior to qubit operations.
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.
Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state. Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.
1. Origins - Chapter One: Atoms Waves Quanta Before and after Democritus Titus Lucretius Carus Isaac Newton on atoms. Chapter Two: Spectra Ions Radiant matter Atoms, electrons, waves Discovery of spectral analysis The beginnings of television William Crookes Kinetic theory of gases Mikhail Vasilyevich Lomonosov. Chapter Three: The planetary atom Spectral series Photons Victory of atomistics The indivisible atom The diffraction grating Just what hath Rutherford wrought? Light pressure Chapter Four: Pre-Bohr times The Bohr atom Post-Bohr times Formal model of the atom Niels Henrik David Bohr Experimental proof of Bohr's postulates. Chapter Five: Teachings of the Ancients First attempts Elements and atoms Table of elements The Periodic law Atoms and people. 2. Ideas - Chapter Six: Contemporaries comment on Bohr's theory Phenomenon, image, concept, formula Heisenberg's matrix mechanics The foundation of physics. Chapter Seven: Louis de Broglie Matter waves Optical-mechanical analogy Schrodinger's wave mechanics The life of Boscovich ... ... and his atom Paul Ehrenfest (1880-1933). Chapter Eight: Schrodinger's equation The meaning of the psi function The image of the atom Quantum truth Compton's experiment Electron diffraction. Chapter Nine: Wave-particle duality Uncertainty relation Complementarity principle Duality and uncertainty Poets and the complementarity principle. Chapter Ten: Heads or tails and target shooting Electron diffraction Probability waves Electron waves The atom and probability Probability and atomic spectra Causality and chance, probability and certainty People, events, quanta. Chapter Eleven - What is an atom? What is quantum mechanics? Physical reality In search of the last concepts. 3. Results - Chapter Twelve: Wilhelm Konrad Rontgen Antoine Henri Becquerel Pierre and Marie Curie Ernest Rutherford and Frederick Soddy The energy of radium X-ray waves. Chapter Thirteen: The chemisty of radioelements Isotopes uranium family Stable isotopes Radioactive decay energy Nuclear binding energy Uranium Earth and radium Knights of the fifth decimal place. Chapter Fourteen: Probing into the nucleus The neutron Artificial radioactivity Slow neutrons Nuclear fission Letters about fission. Chapter Fifteen: Tunnel effect Effective cross-sections of reactions Neutron cross-sections Nuclear fission Labelled atoms Radiocarbon dating. Chapter Sixteen: Chain reaction Nuclear reactor Spontaneous fission of uranium The natural nuclear reactor at Oklo. Chapter Seventeen: Atomic Energy Plutonium The atomic bomb The atomic problem A chronology of the atomic era Soddy on atomic energy. Chapter Eighteen: Solar light Crucibles of elements The fate of the Sun The Sun, life and chlorophyll Life under the Sun A sun on earth Quanta around us. 4. Reflections - Chapter Nineteen: Inception of the scientific method Essence of the scientific method and its development. Truth and completeness of the scientific picture of the world. Science and humanity Boundaries of the scientific method Science and art Future of science Epilogue.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.