Abstract De Broglie’s wavelength for a particle is used to explain the Heisenberg Uncertainty principle. Determinism is replaced by wave mechanics based on the probability of decay through exit channels. The concepts of spin and parity are explained with contributions from Pauli, Ralph de Laer Kronig, Samual Goudsmit, and George Uhlenbeck. Sommerfeld uses quantum numbers to explain atomic spectra. Pauli uses observations of the anomalous Zeeman effect to suggest new quantum numbers based on the electron having an additional two-value property, spin. Lee and Yang show parity is not always conserved. The theory of alpha decay using George Gamow’s tunnelling explanation and Fermi’s explanation of beta decay following Pauli’s suggestion of spin and the prediction of the neutrino complete the explanation of radioactivity. Dirac predicts antimatter and positron decay is observed. James Chadwick discovers the neutron.
Abstract De Broglie’s wavelength for a particle is used to explain the Heisenberg Uncertainty principle. Determinism is replaced by wave mechanics based on the probability of decay through exit channels. The concepts of spin and parity are explained with contributions from Pauli, Ralph de Laer Kronig, Samual Goudsmit, and George Uhlenbeck. Sommerfeld uses quantum numbers to explain atomic spectra. Pauli uses observations of the anomalous Zeeman effect to suggest new quantum numbers based on the electron having an additional two-value property, spin. Lee and Yang show parity is not always conserved. The theory of alpha decay using George Gamow’s tunnelling explanation and Fermi’s explanation of beta decay following Pauli’s suggestion of spin and the prediction of the neutrino complete the explanation of radioactivity. Dirac predicts antimatter and positron decay is observed. James Chadwick discovers the neutron.
The new quantum theory was a mathematical description of the way in which an electron behaves when bounded by some kind of potential – such as being in an electric field, acting as a harmonic oscillator or simply being confined to a certain area of space. This mathematical description was then interpreted to be a description of the electron as being probabilistic. It was Schrödinger’s wave equation, together with de Broglie’s concept of a matter wave, that led to various important concepts like the Heisenberg uncertainty principle, an explanation of the emission and absorption of photons in atoms, quantum tunnelling and so on. This chapter explains what Schrödinger did – the marriage of waves and particles.
The new quantum theory was a mathematical description of the way in which an electron behaves when bounded by some kind of potential – such as being in an electric field, acting as a harmonic oscillator or simply being confined to a certain area of space. This mathematical description was then interpreted to be a description of the electron as being probabilistic. It was Schrödinger’s wave equation, together with de Broglie’s concept of a matter wave, that led to various important concepts like the Heisenberg uncertainty principle, an explanation of the emission and absorption of photons in atoms, quantum tunnelling and so on. This chapter explains what Schrödinger did – the marriage of waves and particles.
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.
In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom, which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized,” meaning that only certain sizes, shapes, and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2)~. The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra. The new quantum theory appeared to be more revolutionary and more powerful. It was triggered by Heisenberg’s discovery of matrix mechanics and Schro¨dinger’s discovery of wave mechanics. These two formalisms would not only predict the quantization of energy and provide a prescription to determine the energy difference between the levels (and thus explain the multiplicity of atomic spectra), but also allow one to calculate easily probabilities of transitions between different quantized energy states. At first, matrix mechanics and wave mechanics looked entirely different in their mathematical appearance and physical meaning. However, Schro¨dinger and Eckart [1] independently showed that the two theories are mathematically equivalent. Toward the end of 1926, Dirac unified the two theories using the concept of state vector and thus established the transformation theory of quantum mechanics. This ultimately had a profound implication for the quantum mechanical (mathematical) recipe to treat spin, as we will show in this chapter. The transformation theory is the mathematical recipe to handle modern quantum mechanics. In Heisenberg’s matrix mechanics, a physical quantity is expressed by a matrix, whereas in Schro¨dinger’s wave mechanics, a physical quantity is expressed by a linear operator. In the unified transformation theory, physical quantities are represented by abstract linear operators called Dirac’s q-numbers, which are linear operators in an infinite-dimensional linear space. Depending upon which types of orthogonal coordinate systems are used in this linear space, either matrix mechanics or wave mechanics emerges.
Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.
Abstract In addition to recounting some contemporary scientific history, Chapter 6 describes the hypothesis that matter, like light, can display wavelike properties, and the creation of the various formulations of quantum mechanics. That matter could have a wavelength was proposed in 1924 by Louis de Broglie, who presented a specific formula for calculating it, one that was verified experimentally in 1927. However, de Broglie’s hypothesis was overshadowed by the creation of three versions of quantum mechanics in 1925/26. The first, denoted matrix mechanics, was proposed by Werner Heisenberg. It was quickly and successfully applied by Wolfgang Pauli to the hydrogen atom. Paul Dirac introduced the next version, which was followed by that of Erwin Schrödinger via a wave equation whose solutions, denoted wave functions, were soon interpreted byMax Born to be related to the probability that certain outcomes or events will occur: classical-physics determinism was thereby removed from quantum mechanics.