Electromagnetic radiation and high-energy electrons exhibit behaviors that require an understanding of the elementary quantum concept of the wave-particle duality. Other selected, relevant topics in quantum mechanics such as the tunnel effect, the uncertainty principle, the exclusion principle, atomic structure, and blackbody radiation are also briefly reviewed here. Students who are already familiar with quantum mechanics may elect to bypass this section. On the other hand, those who are not yet acquainted with these concepts may benefit from this review, perhaps even before delving into chapters 5 and 6.
Electromagnetic radiation and high-energy electrons exhibit behaviors that require an understanding of the elementary quantum concept of the wave-particle duality. Other selected, relevant topics in quantum mechanics such as the tunnel effect, the uncertainty principle, the exclusion principle, atomic structure, and blackbody radiation are also briefly reviewed here. Students who are already familiar with quantum mechanics may elect to bypass this section. On the other hand, those who are not yet acquainted with these concepts may benefit from this review, perhaps even before delving into chapters 5 and 6.
This chapter should be treated as an appendix. It is included for completeness and covers certain fundamentals of quantum mechanics that will aid the understanding of topics discussed in this textbook. The reader is urged to refer to many of the good introductory textbooks on the subject to learn or revisit the fundamental concepts [1, 2, 3, 4]. First, we briefly discuss three key ingredients of quantum mechanics, i.e., the blackbody radiation spectrum, the concept of the photon and de Broglie wavelength, and the related wave-particle duality concept. Next, we briefly restate the four major postulates of quantum mechanics used in the so-called orthodox or Copenhagen interpretation of quantum mechanics that is in vogue. We continue on to describe a generalized form of the Heisenberg Uncertainty Principle which is illustrated with a few familiar bound state problems (particle-in-a-box and one-dimensional harmonic oscillator) and a few spin related problems.
Assuming a background in basic classical physics, multivariable calculus, and differential equations, A Concise Introduction to Quantum Mechanics provides a self-contained presentation of the mathematics and physics of quantum mechanics. The relevant aspects of classical mechanics and electrodynamics are reviewed, and the basic concepts of wave-particle duality are developed as a logical outgrowth of experiments involving blackbody radiation, the photoelectric effect, and electron diffraction. The Copenhagen interpretation of the wave function and its relation to the particle probability density is presented in conjunction with Fourier analysis and its generalization to function spaces. These concepts are combined to analyze the system consisting of a particle confined to a box, developing the probabilistic interpretation of observations and their associated expectation values. The Schrödinger equation is then derived by using these results and demanding both Galilean invariance of the probability density and Newtonian energy-momentum relations. The general properties of the Schrödinger equation and its solutions are analyzed, and the theory of observables is developed along with the associated Heisenberg uncertainty principle. Basic applications of wave mechanics are made to free wave packet spreading, barrier penetration, the simple harmonic oscillator, the Hydrogen atom, and an electric charge in a uniform magnetic field. In addition, Dirac notation, elements of Hilbert space theory, operator techniques, and matrix algebra are presented and used to analyze coherent states, the linear potential, two state oscillations, and electron diffraction. Applications are made to photon and electron spin and the addition of angular momentum, and direct product multiparticle states are used to formulate both the Pauli exclusion principle and quantum decoherence. The book concludes with an introduction to the rotation group and the general properties of angular momentum.
Because atoms and molecules are small, their behavior can only be understood using quantum mechanics, rather than classical mechanics. Quantum mechanics amounts to studying the behavior of electrons and nuclei and their interactions. We will limit ourselves to slowly moving electrons and nuclei, and thus will not need to discuss relativistic quantum mechanics. Large particles also obey quantum mechanics, but the relevant wavelengths are so small that we don't notice them. Small particles absolutely require the use of quantum mechanics for any meaningful understanding. This treatment is not merely an inconvenience: it leads to useful and interesting phenomena such as the Pauli exclusion principle, the Aufbau principle, Hund's rules, the Heisenberg uncertainty principle, wave-particle duality, the photoelectric effect, Ehrenfest's theorem, and the correspondence principle.
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.
Classical mechanics, which gives a fairly accurate description of large systems (e.g., solar system) as also of mechanical systems in our every day life, however, breaks down when applied to small (microscopic) systems such as molecules, atoms and nuclei. For example, (1) classical mechanics cannot even explain why the atoms are stable at all. A classical atom with electrons moving in circular or elliptic orbits around the nucleus would continuously radiate energy in the form of electromagnetic radiation because an accelerated charge does radiate energy. As a result the radius of the orbit would become smaller and smaller, resulting in instability of the atom. On the other hand, the atoms are found to be remarkably stable in practice. (2) Another fact of observation that classical mechanics fails to explain is wave particle duality in radiation as well as in material particles. It is well known that light exhibits the phenomena of interference, diffraction and polarization which can be easily understood on the basis of wave aspect of radiation. But light also exhibits the phenomena of photo-electric effect, Compton effect and Raman effect which can only be understood in terms of corpuscular or quantum aspect of radiation. The dual behavior of light, or radiation cannot be consistently understood on the basis of classical concepts alone or explained away by saying that light behaves as wave or particle depending on the kind of experiment we do with it (complementarity). Moreover, a beam of material particles, like electrons and neutrons, demonstrates wave-like properties (e.g., diffraction). A brief outline of phenomena that require quantum mechanics for their understanding follows.
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.