This chapter discusses the physical properties of atoms and molecules, which are quantized. This means that they can only take certain discrete values rather than a full range of continuously varying values. The chapter examines the quantized energy levels of a molecule. These are called stationary states, which describe an energy that is independent of time. The chapter also outlines the procedure for constructing a quantum mechanical Hamiltonian, which includes writing down the complete and classical expression for the energy of the system. The chapter reviews the general approach to the determination of energy levels of a physical system. This involves two stages: First, the appropriate Hamiltonian is defined and, in a second step, the eigenfunctions and eigenvalues of the Hamiltonian are determined. The chapter also discusses the Schrödinger equation, which can only be solved for one-body systems.
This chapter discusses the physical properties of atoms and molecules, which are quantized. This means that they can only take certain discrete values rather than a full range of continuously varying values. The chapter examines the quantized energy levels of a molecule. These are called stationary states, which describe an energy that is independent of time. The chapter also outlines the procedure for constructing a quantum mechanical Hamiltonian, which includes writing down the complete and classical expression for the energy of the system. The chapter reviews the general approach to the determination of energy levels of a physical system. This involves two stages: First, the appropriate Hamiltonian is defined and, in a second step, the eigenfunctions and eigenvalues of the Hamiltonian are determined. The chapter also discusses the Schrödinger equation, which can only be solved for one-body systems.
This chapter addresses quantum mechanics and spectroscopy. Two key ideas emerge from the theory of quantum mechanics: the first is that the electron is described by a wavefunction, and the second is that the energy of the electron is quantized. The wavefunction is important since its square gives the probability density of the electron. Quantization means that the energy can only have certain discrete values, rather than being allowed to vary continuously. The chapter looks at how these wavefunctions and energy levels arise in quantum mechanics, and how they can actually be computed in some simple cases. This will involve setting up and then solving the famous Schrödinger equation. Quantum mechanics is not restricted to the description of electrons in atoms and molecules, but can also be used to describe the motion of molecules in space (translation), as well as the rotation and vibration of molecules.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract For macroscopic systems, we take the continuity of many variables for granted, including the continuity in energy. For example, the heat flux along a rod through conduction, according to the Fourier law, can be continuously varied to any desired value by controlling the temperature difference and the material properties. The microscopic picture of energy, however, is entirely different. According to quantum mechanical principles, the permissible energy levels of matter (electrons, crystals, molecules, and so on) are often discontinuous. Differences in allowable energy levels among materials are major factors that distinguish them from each other. For example, why is glass transparent in the visible light range but not silicon, and why are some materials electrical insulators but others are conductors? In this chapter, we introduce the basic quantum mechanical concepts necessary to appreciate various energy states found in different materials. It should be remembered that these energy states represent the range of possibilities for the matter but do not tell which state the matter will be in. The latter depends on the temperature, a topic we will discuss in chapter 4. Important concepts that should be mastered through this chapter include the wave-particle duality, the Schrodinger equation and the meaning of the wave function, the Pauli exclusion principle, quantum states, and degeneracy. Solutions of the Schr6dinger equation for various simple yet very common potentials will be given. Key concepts and results of this chapter are summarized in the last section of the chapter.
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 postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
The "quantum equation" should "carry the quantum conditions in itself" (Second Commun., p. 511); 2. There should be a special mathematical method for solving the stationary Schrödinger equation, which accounts for the non-classical character of the quantization problem, ie, which is different from the classical eigenvalue methods for calculating the (eigen)modes of strings, resonators and so on (ibid., p. 513);3. The derivation should uniquely decide, that the energy rather than the frequency values are discretized (ibid., pp.511, 519), since only the former means true quantization, while the latter corresponds to the classical discretization mentioned in requirement 2;4. The use of the classical expressions for the kinetic and potential energies should be justified (Fourth Commun., p. 113).Schrödinger's requirements 1 and 2 mean, that -contrary to the very title of the papersquantization is actually not an eigenvalue problem.For in the latter the discretization is imposed not by the differential equation itself, but by the boundary conditions, and this is the classical discretization for standing waves in organ pipes etc.I will fulfill all four requirements by treating quantization as a selection problem.The number of stationary states of a quantum system is smaller than that of a classical system [6].I will describe, (i), the selection of quantum systems out of the set of all mechanical systems and, (ii), the selection of their stationary states out of the continuum of classical stationary states.Earlier arguing [19][10] is improved and extended in several essential points. Elements of an Eulerian representation of classical mechanics Euler's axiomsLeonhard Euler [20][21][22][23][24][25] was the first to apply the calculus to all areas of mathematics and mechanics of his time, and he developed novel methods, such as the calculus of variations and topology.Moreover, he worked out an axiomatic of mechanics, where only Newton's Law (axiom) 1 concerning the conservation of stationary states is retained as an axiom, while Newton's Laws (axioms) 2 and 3 concerning the change of stationary states are treated as problems to be solved (for a detailed account, see [19][10][45]).This allows for introducing alternative equations of motion without loosing the contact to CM.The existence of stationary states is postulated in the following axioms (as in Newton's axioms, rotatory motion is not considered). Axiom E0: Every body is either resting or moving.This means, that the subsequent axioms E1 and E2 are not independent of each other; they exclude each other and, at once, they are in harmony with each other [22].