Abstract The attempt to explain atomic structure was one of the main motives for the genesis of quantum mechanics (Bohr’s atomic model, spectral lines, etc.); at the same time it represented the first true testing ground for the new theory. The explanation of the existence of discrete atomic levels, their quantitative prediction, understanding of the periodic structure of the elements, the correct interpretation of chemical bonds, etc., can be regarded without exaggeration as one of the greatest achievements of the scientific thought of our time. A door to an entirely new world— that of the atomic and subatomic phenomena—opened, triggering an impressive advance in basic scientific knowledge and modern technology, unprecedented in human history. The very concepts and many of the mathematical tools of quantum mechanics have, in many cases, sprung from the efforts to solve these problems. In this chapter, we study atomic structure, and try to illustrate some of these features.
Abstract The attempt to explain atomic structure was one of the main motives for the genesis of quantum mechanics (Bohr’s atomic model, spectral lines, etc.); at the same time it represented the first true testing ground for the new theory. The explanation of the existence of discrete atomic levels, their quantitative prediction, understanding of the periodic structure of the elements, the correct interpretation of chemical bonds, etc., can be regarded without exaggeration as one of the greatest achievements of the scientific thought of our time. A door to an entirely new world— that of the atomic and subatomic phenomena—opened, triggering an impressive advance in basic scientific knowledge and modern technology, unprecedented in human history. The very concepts and many of the mathematical tools of quantum mechanics have, in many cases, sprung from the efforts to solve these problems. In this chapter, we study atomic structure, and try to illustrate some of these features.
The early twentieth century was a revolutionary time in the history of physics. People often think of Einstein's special theory of relativity of 1905, which changed our conceptions of time and space. But among physicists, quantum mechanics is usually regarded as an even more radical change in our thinking about the physical world. Quantum mechanics, which was developed between 1900 and 1926, began as a theory of atoms and light but has now become the framework in terms of which all basic physical theories are expected to be cast. We need quantum ideas not only to understand atoms, molecules, and elementary particles, but also to understand the electronic properties of solids and even certain astronomical phenomena such as the stability of white dwarf stars. The theory was radical in part because it introduced probabilistic behavior as a fundamental aspect of the world, but even more because it seems to allow mutually exclusive situations to exist simultaneously in a “quantum superposition.” We will see later that the possibility of quantum superposition is largely responsible for a quantum computer's distinctive advantage over an ordinary computer. The present chapter is devoted to introducing the basic principles of quantum mechanics. There are essentially four components of the mathematical structure of quantum mechanics. We need to know how to represent (i) states, (ii) measurements, (iii) reversible transformations, and (iv) composite systems. We will develop the first three in stages, starting with a very simple case – linear polarization of photons – and working toward the most general case.
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.
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.
The first objective of this chapter is to briefly review some of the basic ideas about the structure of matter, in particular the concepts of microscopic physics, in order to recall the knowledge gained in previous physics (and chemistry) courses and make it more precise. Our review will be very concise, and most statements will be made without any proof or detailed discussion. A second objective is to give a brief description of some of the crucial stages in the early development of quantum physics. We shall not follow the strict historical order of this development or present the arguments used at the beginning of the last century by the founding fathers of quantum mechanics; rather, we shall stress the concepts which we shall find useful later on. Our last objective is to give an elementary introduction to some of the basic ideas, like those of a quantum particle or energy level, that will reappear throughout this text. We shall base our review on the Bohr theory, which provides a simple, though far from convincing, explanation of how energy levels are quantized and how the spectrum of the hydrogen atom arises. This chapter should be reread later on, once the basic ideas of quantum mechanics have been made explicit and illustrated by examples. From the practical point of view, it is possible to skip the general considerations of Sections 1.1 and 1.2 at the first reading and begin with Section 1.3, returning to those two sections later on as needed.
Modern scientists describe the universe in terms of two basic partial theories: firstly, the general theory of relativity proposed by Albert Einstein, and secondly, the theory of quantum mechanics. This theory introduces an immense reality and simultaneously creates complex questions for the world of physics. Quantum mechanics is an incompletable knowledge however scientists are finding the basic theories from this theory. They propose complex mathematical expressions from quantum mechanics to prove the basic law of atoms. Quantum mechanics is the entire knowledge of subatomic particles such as electrons, protons, and neutrons. Quantum mechanics reveals that what we see is just a part of a deeper reality. Then we can grasp from this theory that the material world is not the completed reality; there are other things beyond human physical senses. The concept of quantum mechanics describes particles that can exist in multiple states. The concept of quantum mechanics is directly proportional to contemporary world technologies. The super specialty of this theory is that scientists cannot be explained universe without quantum mechanics. The uncertainty principle of Heisenberg in quantum mechanics suggests that there are limits to what we can know about a particle state of an atom. Humans cannot yet find out what it looks like and how we can imagine the existence of subatomic particles in an atom. That was an unseen dynamite for humans' perspective observation of the material world.
When physicists began to explore the world of atoms more precisely, as they endeavoured to understand its structure and the laws governing its behaviour, they soon encountered serious difficulties. Our intuitive concepts, based on our daily experience of the macroscopic world around us, proved to be completely erroneous on the atomic scale; the atom was incomprehensible within the framework of classical physics. In order to uncover these new mysteries, after a great deal of trial and error, entirely new concepts therefore had to be elaborated: the concepts of quantum physics. What are the main stages that have led us to this modern understanding of the atom? What is the present state of atomic physics? How has it contributed to the development of our knowledge and where is it heading?