The object of this paper is to give an account of the fundamental principles of wave mechanics in a manner which shall make clear the physical significance of all the quantities and processes involved. The principles are illustrated by discussions of the propagation of free electric waves in uniform electromagnetic fields and of bound electric waves in the hydrogen atom. The paper concludes with relativistic wave mechanics (prior to the work of Dirac and Darwin) and a short account of the Compton effect.
The object of this paper is to give an account of the fundamental principles of wave mechanics in a manner which shall make clear the physical significance of all the quantities and processes involved. The principles are illustrated by discussions of the propagation of free electric waves in uniform electromagnetic fields and of bound electric waves in the hydrogen atom. The paper concludes with relativistic wave mechanics (prior to the work of Dirac and Darwin) and a short account of the Compton effect.
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.
This text presents a rigorous mathematical account of the principles of quantum mechanics, in particular as applied to chemistry and chemical physics. Applications are used as illustrations of the basic theory. The first two chapters serve as an introduction to quantum theory, although it is assumed that the reader has been exposed to elementary quantum mechanics as part of an undergraduate physical chemistry or atomic physics course. Following a discussion of wave motion leading to Schrödinger's wave mechanics, the postulates of quantum mechanics are presented along with essential mathematical concepts and techniques. The postulates are rigorously applied to the harmonic oscillator, angular momentum, the hydrogen atom, the variation method, perturbation theory, and nuclear motion. Modern theoretical concepts such as hermitian operators, Hilbert space, Dirac notation, and ladder operators are introduced and used throughout. This text is appropriate for beginning graduate students in chemistry, chemical physics, molecular physics and materials science.
The quantum theory of radiation, formulated in wave-mechanical language but in terms of running waves, is shown to lead under suitable conditions to the classical theory of emission by a particle as a first approximation. Non-relativistic mechanics is used, and the limitations of the method are brought out in terms of classical theory. The characteristic quantum phenomenon of the Compton effect requires that the Compton recoil shall exceed the spread in momentum of the particle, and so that the wave-packet shall greatly exceed in diameter the wave-length of the incident radiation; this condition is realizable on any scale of magnitude. For quasi-classical motion just the reverse is required; furthermore, if the motion is to last for a number of waveperiods, the wave-length must greatly exceed mch (m=massoftheparticle). The classical packet-motion arises from combinations between quantum-states of the particle-field system; this fact suggests that the photon is not a fundamental constituent of the field but only one possible aspect of its action. The infinite "zero-energy" associated apparently even with the lowest energy-state of the field implies no electromagnetic field intensities at any point. Whereas for problems on energy or momentum it is customary to represent an incident wave-train by a single typical quantum state, to obtain motional phenomena of classical type a more complete expression such as that given in the paper must be used. The physically empty field likewise corresponds to a mixture of quantum states but without correlation among their amplitudes.
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.
Quantum mechanics has evolved from a subject of study in pure physics to one with a wide range of applications in many diverse fields. The basic concepts of quantum mechanics are explained in this book in a concise and easy-to-read manner, leading toward applications in solid-state electronics and optics. Following a logical sequence, the book focuses on key ideas and is conceptually and mathematically self-contained. The fundamental principles of quantum mechanics are illustrated by showing their application to systems such as the hydrogen atom, multi-electron ions and atoms, the formation of simple organic molecules and crystalline solids of practical importance. It leads on from these basic concepts to discuss some of the most significant applications in semiconductor electronics and optics. Containing many homework problems, the book is suitable for senior-level undergraduate and graduate-level students in electrical engineering, material sciences, applied physics and chemistry.
The paper gives an account of the author's work on a new form of quantum theory. \textsection{}1. The Hamiltonian analogy between mechanics and optics. \textsection{}2. The analogy is to be extended to include real "physical" or "undulatory" mechanics instead of mere geometrical mechanics. \textsection{}3. The significance of wave-length; macro-mechanical and micro-mechanical problems. \textsection{}4. The wave-equation and its application to the hydrogen atom. \textsection{}5. The intrinsic reason for the appearance of discrete characteristic frequencies. \textsection{}6. Other problems; intensity of emitted light. \textsection{}7. The wave-equation derived from a Hamiltonian variation-principle; generalization to an arbitrary conservative system. \textsection{}8. The wave-function physically means and determines a continuous distribution of electricity in space, the fluctuations of which determine the radiation by the laws of ordinary electrodynamics. \textsection{}9. Non-conservative systems. Theory of dispersion and scattering and of the "transitions" between the "stationary states." \textsection{}10. The question of relativity and the action of a magnetic field. Incompleteness of that part of the theory.