Abstract Basic experimental evidence is sketched: the black body radiation spectrum, the photoeffect, Compton scattering and electron diffraction; the Bohr model of the atom. Quantum mechanics is reviewed using the Copenhagen interpretation: eigenstates, observables, hermitian operators and expectation values are explained. Wave-particle duality, Schrödinger’s equation, and expressions for particle density and current are described. The uncertainty principle, the collapse of the wavefunction, Schrödinger’s cat and the no-cloning theorem are discussed. Dirac delta functions and the usage of wavepackets are explained. An introduction to state vectors in Hilbert space and the bra-ket notation is given. Abstracts of special relativity and Lorentz invariants follow. Minimal electromagnetic coupling and the gauge transformations are explained.
Abstract Basic experimental evidence is sketched: the black body radiation spectrum, the photoeffect, Compton scattering and electron diffraction; the Bohr model of the atom. Quantum mechanics is reviewed using the Copenhagen interpretation: eigenstates, observables, hermitian operators and expectation values are explained. Wave-particle duality, Schrödinger’s equation, and expressions for particle density and current are described. The uncertainty principle, the collapse of the wavefunction, Schrödinger’s cat and the no-cloning theorem are discussed. Dirac delta functions and the usage of wavepackets are explained. An introduction to state vectors in Hilbert space and the bra-ket notation is given. Abstracts of special relativity and Lorentz invariants follow. Minimal electromagnetic coupling and the gauge transformations are explained.
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.
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.
Quantum physics is the most successful description of the physical world we live in. Quantum theoretical predictions have been experimentally tested and confirmed up to the astonishing 14 digits of precision in terms of the anomalous magnetic moment of the electron. The bra-ket notation, composed of angle brackets and vertical bars, enhances the clarity of mathematical operations and improves the comprehension of quantum mechanical expressions. Quantum wave functions participate in interference phenomenona in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. When a quantum system is measured, it undergoes indeterministic transition that actualizes only one of the states from the set of all physically possible future states of the system. The Hilbert vector space structure of the space of quantum states follows from the linearity of the Schrodinger equation. The quantum state vector and the quantum wave function in a Hilbert space are abstract mathematical concepts that are basis independent.
This extended and updated second edition course text presents a logical and concise introduction to the basic concepts, applications, and physical meaning of quantum mechanics. An initial review of classical mechanics and electromagnetism provides the reader with the context for quantum mechanics. Starting with atomic level experimental results, the probabilistic nature of quantum mechanics is derived, with the wave function related to the statistical theory of random variables. Applying the requirement of Galilean invariance yields the Schrödinger equation, and the Copenhagen interpretation of the wave function is discussed. After numerous basic applications of wave mechanics, including the hydrogen atom and the harmonic oscillator, the text then presents Dirac notation and Hilbert space theory. New chapters discuss perturbation theory, path integrals, scattering theory, and quantum entanglement. Key features • Extended and updated edition of a well-used textbook. • Offers a thorough connection to classical mechanics and electromagnetism. • Provides a logical development of basic quantum mechanical concepts and a connection to formal quantum mechanics. • Acts as a complete introduction to the most important quantum mechanical results in basic systems. • An introductory text that provides connections to the current research in quantum computing and measurement theory.
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.
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.