The authors present their studies in composing elementary wavefunctions of a hydrogen-like atom and identify several relationships between physical phenomena and musical composition that helped guide the process. The hydrogen-like atom accurately captures some of the fundamental quantum mechanical phenomena of nature and supplies the composer with a set of well-defined mathematical constraints that can create a wide variety of complex spatiotemporal patterns. The authors explore the visual appearance of time-dependent combinations of two and three eigenfunctions of an electron with spin in a hydrogen-like atom, highlighting the resulting symmetries and symmetry changes.
The authors present their studies in composing elementary wavefunctions of a hydrogen-like atom and identify several relationships between physical phenomena and musical composition that helped guide the process. The hydrogen-like atom accurately captures some of the fundamental quantum mechanical phenomena of nature and supplies the composer with a set of well-defined mathematical constraints that can create a wide variety of complex spatiotemporal patterns. The authors explore the visual appearance of time-dependent combinations of two and three eigenfunctions of an electron with spin in a hydrogen-like atom, highlighting the resulting symmetries and symmetry changes.
Many students find quantum mechanics conceptually difficult when they first encounter the subject. In this book, the postulates and key applications of quantum mechanics are well illustrated by means of a carefully chosen set of problems, complete with detailed, step-by-step solutions. Beginning with a chapter on orders of magnitude, a variety of topics are then covered, including the mathematical foundations of quantum mechanics, Schrödinger's equation, angular momentum, the hydrogen atom, the harmonic oscillator, spin, time-independent and time-dependent perturbation theory, the variational method, multielectron atoms, transitions and scattering. Throughout, the physical interpretation or application of certain results is highlighted, thereby providing useful insights into a wide range of systems and phenomena. This approach will make the book invaluable to anyone taking an undergraduate course in quantum mechanics.
The mere existence of stable atoms, and their arrangement into many-atom nanoscale structures, can only be explained by the laws of quantum mechanics. Indeed, the accumulated evidence regarding the internal structure of atoms motivated the formulation of quantum mechanics at the early twentieth century. Here we focus on single-electron (hydrogen-like) atoms. The dominance of the Coulomb central potential between the electron and nucleus, translates into a simple Schrödinger equation for their relative position distribution. Solving rigorously this equation reveals quantized energy levels, which explains the emission spectra of hydrogen-like atoms (Rydberg formula). The corresponding stationary probability density distributions for the electron around the nucleus are reminiscent of the Bohr model, where classical trajectories (orbits) are replaced by single-electron wave functions (orbitals). The degeneracies of different orbitals, associated with three quantum numbers, will later turn out to be critical for understanding many-electron atoms and the periodic table of the elements.
Introduction Helium is the natural connection between the hydrogen-like atoms (one electron outside a spherical core, such as the alkali-metal atoms of Chap. 10) and all the others in the periodic table. For these one-electron atoms the field-free description of the electron motion in terms of three quantum numbers n , ℓ , and m suffices. The other electrons are considered to form the “core” and its effect on the energy levels is reduced to the quantum defects δ ℓ , as discussed in Chap. 10. The case of helium is very different since the two ground-state electrons are in the same electronic state and differ only in the orientation of their spins. Thus spin is of paramount importance for the description of all but the hydrogenic atoms. Since helium has only two spins to deal with, their roles can be written explicitly. Because the two electrons are identical, quantum mechanics requires that the wavefunction undergoes nothing more than an unobservable phase shift when they are exchanged. This consequence is called the Pauli symmetrization principle, and is sometimes referred to as “the source of our individuality”. Moreover, it has a profound influence on the wavefunction and thus on the allowed energy states. It will be shown that this symmetry requirement dictates that two electrons cannot be in states with identical quantum numbers, and this topic will be discussed in detail in this chapter. Symmetry The profound difference between the hydrogen and helium wavefunctions arises because the two electrons are indistinguishable, and this results in a fundamental change of the energy-level structure. The concept of “spin”, introduced as a shortcut for the intrinsic magnetic moment of the electron in Chap. 7, now becomes a dominating feature of the description. Before discussing the helium energy levels, the effects of this symmetry with respect to exchange of identical particles is presented. The exchange operator In quantum mechanics, identical particles are inherently indistinguishable, so the exchange of the two electrons cannot change any observable and there should be no measurable effect on the overall wavefunction. However, there can be an overall phase change whose nature can easily be found by defining an operator P 12 that exchanges the electrons and applying it to the wavefunction. The exchange applies to all coordinates of the electron including its spin coordinate, but for simplicity only the spatial coordinate will be indicated here.
As the most basic and abundant atom in the universe, hydrogen atoms have long attracted the interest of scientists and provide the best subject for the exploration of quantum mechanics. This article provides a comprehensive study of the multifaceted characteristics and implications of the hydrogen atom. First, the research deeply studied the basic derivation of the Schrödinger equation of the hydrogen atom and provided the necessary theoretical framework. On this basis, modern computing tools are used to visualize the electron cloud of hydrogen atoms and reveal their shapes and characteristics under different orbits. In addition, the study investigated atomic transitions and associated spectra, elucidating basic quantum transition principles and serving as a benchmark for studying more complex atomic systems. Eventually, the research focus shifted to a specific direction, studying the Stark effect in hydrogen atoms, especially in the n=2 state. This research provides a novel perspective on understanding the fundamental properties of the hydrogen atom and its implications in modern physics research.
Electron spin In our development of quantum mechanics to this point, the behavior of a particle, usually an electron, is governed by a wave function that is dependent only on the cartesian coordinates x, y, z or, equivalently, on the spherical coordinates r , θ, ϕ. There are, however, experimental observations that cannot be explained by a wave function which depends on cartesian coordinates alone. In a quantum-mechanical treatment of an alkali metal atom, the lone valence electron may be considered as moving in the combined field of the nucleus and the core electrons. In contrast to the hydrogen-like atom, the energy levels of this valence electron are found to depend on both the principal and the azimuthal quantum numbers. The experimental spectral line pattern corresponding to transitions between these energy levels, although more complex than the pattern for the hydrogen-like atom, is readily explained. However, in a highly resolved spectrum, an additional complexity is observed; most of the spectral lines are actually composed of two lines with nearly identical wave numbers. In an alkaline-earth metal atom, which has two valence electrons, many of the lines in a highly resolved spectrum are split into three closely spaced lines. The spectral lines for the hydrogen atom, as discussed in Section 6.5, are again observed to be composed of several very closely spaced lines, with equation (6.83) giving the average wave number of each grouping.