Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave-like behavior is an important part of the explanation of why atoms have discrete energy levels and make transitions in which electromagnetic radiation is absorbed or emitted at specific frequencies. Important aspects of how quantum systems may be observed and how they develop in time are presented. The Schrödinger equation is introduced and the importance of the wavefunction covered. Mathematical simplifications that arise from considering symmetry are covered. Several examples of simple quantum systems are presented in which the calculation of electronic energy structure based on boundary conditions is illustrated. Sections on wave–particle duality and quantum observables are followed by others on the dynamics of two-level systems, mathematically complete representations of arbitrary states, and standard approaches for handling the temporal development of wavefunctions.
Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave-like behavior is an important part of the explanation of why atoms have discrete energy levels and make transitions in which electromagnetic radiation is absorbed or emitted at specific frequencies. Important aspects of how quantum systems may be observed and how they develop in time are presented. The Schrödinger equation is introduced and the importance of the wavefunction covered. Mathematical simplifications that arise from considering symmetry are covered. Several examples of simple quantum systems are presented in which the calculation of electronic energy structure based on boundary conditions is illustrated. Sections on wave–particle duality and quantum observables are followed by others on the dynamics of two-level systems, mathematically complete representations of arbitrary states, and standard approaches for handling the temporal development of wavefunctions.
Introduction In this chapter we discuss the basic physical concepts and equations related to the behavior of particles in the nanoworld. We introduce the Schrödinger wave equation for particles and determine the ways to calculate observable physical quantities. We find that, in wave mechanics, the motion of a particle confined to a finite volume is always characterized by discrete values of the energy and standing-wave-like wavefunctions, i.e., such a motion is quantized. While motion in an infinite space (i.e., free motion) is not quantized and is described by propagating waves, the energy of the particle is characterized by a continuous range of values. Keeping in mind the diverse variants of nanostructures, by using wave mechanics we analyze some particular examples, which highlight important quantum properties of particles. Many of the examples analyzed can serve as the simplest models of nanostructures and will be exploited in following chapters to understand the fundamentals of processes in nanoelectronics. The Schrödinger wave equation From the previous chapter, we conclude that nanosize physical systems are quantum-mechanical systems, inasmuch as their sizes are comparable to typical de Broglie wavelengths of the particles composing these systems. In dealing with quantum-mechanical systems, one aims at determining the wavefunction of a single particle or of the whole system. As we will demonstrate in the subsequent discussion, knowledge of the wavefunction in quantum mechanics is sufficient to describe completely a particle or even a system of particles.
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.
We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the ε j 's and g j 's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics. We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.
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.
3.4. SummaryIn this Chapter, we have shown the limitations of classical mechanics and the success of quantum mechanics. The basic concepts and formalism of quantum mechanics have been exposed, including the quantized nature of the electromagnetic field, the wave-particle duality, the probability of presence of a particle, the wavefunction, and the Schrödinger equation. Simple quantum mechanical systems have been analyzed to understand these novel concepts, including an infinite and a finite potential well. Through these, some aspects of quantum mechanics have been discussed, including the quantization of energy levels and momenta, and tunneling effects.
This chapter starts with the historical development of quantum mechanics and explanations of key concepts behind fundamental discoveries such as Planck's hypothesis, the photoelectric effect, Bohr's atomic model, Compton scattering, the de Broglie hypothesis, the Pauli exclusion principle, the Schrödinger equation, Heisenberg quantum mechanics and the uncertainty principle, the Born interpretation of wave function, the Davisson and Germer experiment, and Dirac relativistic quantum mechanics, which made important contributions to the foundation of modern quantum mechanics. Concepts like the particle properties of radiation, the wave properties of matter, quantized energy and momentum, and the idea that one can no longer know exactly where a single particle like an electron is at any instant of time are brought out clearly through this chapter. A list of important quantum mechanical parameters along with solved examples and unsolved exercises on the key concepts are included towards the end.