Abstract In the opening chapter on relativity, C. V. Vishveshwara (Vishu) takes the reader through an excursion on how Einstein laid down the remarkable foundations of special relativity. The chapter begins with an early treatment of relativity, namely Galilean relativity, which while successful with classical mechanics was not tenable with electromagnetism. Working in the seclusion of a patent office, Einstein thought deeply and despairingly of relativity that would encompass electromagnetism. He realized that not only spatial measures but also time varied among inertial observers moving with constant relative velocities. Thus, Einstein’s special theory of relativity came to be, and along with it, his celebrated equation, E = mc2. The chapter then surveys bizarre, observable consequences of this special relativity and how they all beautifully fit into a mathematical framework created by Minkowski in which space and time become united. The chapter concludes by describing physicists’ reactions to this radical new theory and its role in different areas, such as elementary particle physics and astrophysics. In the opening chapter of quantum physics, Smitha describes the circuitous historic path taken by many a physicist explorer that led to the establishment of its mind-boggling fundamentals. This revolutionary branch of physics was seeded by puzzles posed in the late 1800s related to the behavior of light from radiating objects. The chapter begins by exploring the nature of light and the story of blackbody radiation that led to Planck introducing the notion of the quantum. It then surveys the photoelectric effect, which heralds the concept of the photon, and wave-particle duality, which applies both to light and matter. The excursion continues into Schrödinger’s description of matter acting as a wave and its consequences, including energy quantization and quantum tunneling. The chapter then explains Heisenberg’s uncertainty principle from multiple angles. Finally, it presents the quantum picture of the atom and how it resolved some of the deep puzzles, a crowning testament to quantum theory.
Abstract In the opening chapter on relativity, C. V. Vishveshwara (Vishu) takes the reader through an excursion on how Einstein laid down the remarkable foundations of special relativity. The chapter begins with an early treatment of relativity, namely Galilean relativity, which while successful with classical mechanics was not tenable with electromagnetism. Working in the seclusion of a patent office, Einstein thought deeply and despairingly of relativity that would encompass electromagnetism. He realized that not only spatial measures but also time varied among inertial observers moving with constant relative velocities. Thus, Einstein’s special theory of relativity came to be, and along with it, his celebrated equation, E = mc2. The chapter then surveys bizarre, observable consequences of this special relativity and how they all beautifully fit into a mathematical framework created by Minkowski in which space and time become united. The chapter concludes by describing physicists’ reactions to this radical new theory and its role in different areas, such as elementary particle physics and astrophysics. In the opening chapter of quantum physics, Smitha describes the circuitous historic path taken by many a physicist explorer that led to the establishment of its mind-boggling fundamentals. This revolutionary branch of physics was seeded by puzzles posed in the late 1800s related to the behavior of light from radiating objects. The chapter begins by exploring the nature of light and the story of blackbody radiation that led to Planck introducing the notion of the quantum. It then surveys the photoelectric effect, which heralds the concept of the photon, and wave-particle duality, which applies both to light and matter. The excursion continues into Schrödinger’s description of matter acting as a wave and its consequences, including energy quantization and quantum tunneling. The chapter then explains Heisenberg’s uncertainty principle from multiple angles. Finally, it presents the quantum picture of the atom and how it resolved some of the deep puzzles, a crowning testament to quantum theory.
By the end of the nineteenth century all the important elements of classical physics were fully mature. Newton's Laws had been applied with success to everything from industrial machines to planetary orbits. Classical thermodynamics had extended energy conservation arguments into thermal processes, and scientists including J. Willard Gibbs and Ludwig Boltzmann had developed the statistical interpretation of entropy. James Clerk Maxwell had explained all electromagnetic phenomena, from AC generators/motors to light, with the four Maxwell's equations and three material-dependent constitutive relations. There was serious discussion that physics was on the verge of being “finished” – becoming a closed system of knowledge like classical geometry with only essentially routine tasks left of determining how the known laws could be applied to engineering problems. Yet by 1930 the foundations of mechanics, electromagnetics, and atomic-scale systems had been shaken to the core by the new revolutions in the Theory of Relativity, the photon picture of light, and quantum physics. We are still reaping the benefits of the application of these new scientific paradigms in our technology today, including semiconductor electronic devices, lasers, and GPS systems. But the description of reality that these theories provide is still deeply anti-intuitive and disturbing. Nevertheless, quantum physics is one of the most extensively experimentally verified theories that we know. Even when quantum theory and common sense seem to diverge, quantum theory gives the right answers when tested by experiment. In the next chapter we will develop the framework of quantum theory that has been so successful when people “shut up and calculate,” as one scientist suggested as the way through the philosophical thicket of quantum physics. But in this chapter, we will spend a brief time considering the historical development and continuing paradoxical implications of quantum theory. The photoelectric effect and photons In 1905, Albert Einstein, who was a key figure in the revolution of modern physics, was working as a clerk in the patent office in Zurich, Switzerland. Einstein had finished his studies in physics, but had not obtained a coveted position as a university professor. He was intensely interested in the latest developments in classical physics and was aware of a few of the remaining loose ends that were resisting explanation in the classical model.
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.
This chapter will explore Einstein’s theory of special relativity. In addition, readers will find the basic concept of radiation as waves and particles including photoelectric and Compton scattering effects. Definition of X-ray and behaviors under conservation of momentum and energy will be provided and discussed. Various photon and particle wavelengths will be explored with various examples to help readers understand how to apply basic quantum theory dealing with the de Broglie wavelength, wave equation, wave–particle duality, and the uncertainty principle.
In the first two chapters we have examined the basic principles – superposition (p. 18), complementarity (p. 19), quantization (p. 44), statistical algorithm (p. 57), and correspondence (p. 72) (see also Subsec. 2.3.4) – and the basic entities, observables and states, of quantum mechanics, as well as the main differences with respect to classical mechanics. While what we have discussed so far is rather a static picture of observables and states, in this chapter we shall deal with quantum dynamics, i.e. with the time evolution of quantum-mechanical systems. Historically, after Bohr had provided a quantized description of the atom (see Subsec. 1.5.4), Einstein showed the quantized nature of photons (see Subsec. 1.2.1), and de Broglie hypothized the wave-like nature of matter (see Subsec. 1.5.5), the first building block of quantum mechanics was provided by the commutation relations, proposed by Heisenberg in 1925, whose consequence is represented by the uncertainty relation (see Subsec. 2.2.7 and Sec. 2.3). This was the subject of the previous chapters. The dynamical part of the theory was proposed by Schrödinger in 1926, and is known as the Schrödinger equation. It is also known as wave mechanics (see Subsec. 1.5.7). In this chapter we shall show that Heisenberg's and Schrödinger's formulations are only two different aspects of the same theory. We shall also come back to this point in Sec. 8.1.1. Here, first we shall derive the fundamental equation which rules quantum dynamics (Sec. 3.1), and, in Sec. 3.2, we shall summarize the main properties of the Schrödinger equation.
This chapter traces the sequence of discoveries which convinced physicists of the need to replace classical theories of light and matter. It is intended for readers who wish to learn how Einsteinian physics emerged through a sequence of discoveries that led the reader to the frontiers of current knowledge. The highlights the enormous struggle from which the revolutionary new understanding of the universe called Einsteinian physics emerged. Traditionally, quantum physics and general relativity have stood apart, separated by a fundamental mismatch between their language, their mathematics, and their structure. Einstein's conclusion made the conundrum of quantum physics quite clear. Photons behaved like particles with discrete energy. To provide some sense of a mechanism for understanding quantum behavior, Schrodinger and Bohr developed the Copenhagen interpretation of quantum physics. From a few small clues, Einstein and others realized that sound, heat, and light come in energy packets called phonons and photons.
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.