Because of considerable mathematical difficulties the Schrödinger equation can be solved exactly for only a few simple quantum systems. In particular for atoms with at least two electrons no such exact solutions exist. The most important system where an exact solution is indeed possible consists of a single electron with charge q 1 = - e in motion under the influence of a Coulomb potential about a nucleus with a positive charge q 2 = Ζe , for Ζ ≥ 1 an integer. Such a system is called a hydrogenic or hydrogen-like atom , with atomic number Z . One may view the nucleus as consisting of Ζ protons. For Ζ = 1 we obtain the hydrogen atom. For Ζ = 2, for example, the system is singly ionized helium He + ; this is helium with its two protons as usual, but stripped of one of its two electrons. Since it has only one electron but two protons it is positively charged, as indicated by the plus sign +. Similarly the lithium atom Li consists of three electrons and three protons (and four neutrons). If we strip off two electrons we obtain the ion Li ++ —doubly ionized lithium, which is the case Ζ = 3. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Because of considerable mathematical difficulties the Schrödinger equation can be solved exactly for only a few simple quantum systems. In particular for atoms with at least two electrons no such exact solutions exist. The most important system where an exact solution is indeed possible consists of a single electron with charge q 1 = - e in motion under the influence of a Coulomb potential about a nucleus with a positive charge q 2 = Ζe , for Ζ ≥ 1 an integer. Such a system is called a hydrogenic or hydrogen-like atom , with atomic number Z . One may view the nucleus as consisting of Ζ protons. For Ζ = 1 we obtain the hydrogen atom. For Ζ = 2, for example, the system is singly ionized helium He + ; this is helium with its two protons as usual, but stripped of one of its two electrons. Since it has only one electron but two protons it is positively charged, as indicated by the plus sign +. Similarly the lithium atom Li consists of three electrons and three protons (and four neutrons). If we strip off two electrons we obtain the ion Li ++ —doubly ionized lithium, which is the case Ζ = 3. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge -e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the centre of mass coordinate and the relative coordinates. The existence of degenerate energy eigenvalues implies that linear combination of the corresponding eigenfunctions are solutions of the wave equation with the same energy. The isotopes of atomic hydrogen, deuterium and tritium also provide examples of hydrogenic systems. Since the mass of deuterium is md˜2mp and that of tritium is mt˜3mp, the reduced masses of them will be slightly different from that of hydrogen. In addition to deuterium and tritium, there exists other hydrogenic systems in which the role of proton is played by another positive particle.
Abstract This chapter reiterates the quantum numbers for atomic orbitals, known from general chemistry, and places them into the context developed so far. It is sketched how the Schrodinger equation (SE) for the hydrogen atom hydrogen-like systems (one electron plus a nucleus of charge Z) is set up. When the nucleus is treated as a fixed point charge, the SE is only for the electron. The solutions of the SE can be obtained by switching to spherical polar coordinates, such that the variables are separable in terms of the electron distance from the nucleus, r, and two angles. The kinetic energy of the electron then has a radial component, and an angular component. The latter is associated with the angular momentum quantum number, which is codified by the letters s, p, d, f, and so forth. A step by step solution of the SE is provided later, in chapter 19.
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.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.
This chapter presents a detailed analysis of the hydrogen-like atom within the framework of quantum mechanics. The radial Schrödinger equation is solved for the Coulomb potential, yielding the quantized energy spectrum, energy eigenstates, and the associated degeneracy of levels. The expectation value of the electron's radius is calculated, and a comparison is drawn between quantum predictions and classical circular orbits. The quantum two-body problem is introduced to account for the motion of the nucleus, revealing its influence on the energy levels of the electron. This correction underpins the operation of the mass spectrometer and supports the spectroscopic identification of hydrogen isotopes such as deuterium and tritium. The analysis also explains how similar effects confirm the presence of helium in the solar spectrum through astrophysical observations. Metadata/Keywords: hydrogen-like atom, Coulomb potential, energy spectrum, quantum orbits, nucleus motion, two-body problem, mass spectrometer, hydrogen isotopes, helium, solar spectrum
An atom consists of a nucleus, containing protons and neutrons, with various electrons swarming around the outskirts. The electrons are held in the vicinity of the nucleus as they are negatively charged and hence attracted by the positively charged protons. Although atoms are complicated objects, there are certain situations where they can be treated as simple lumps of matter. Their internal structure has no relevance to the details of what is happening. This sort of situation is exemplified by experiments in which atoms are deliberately smashed into one another at comparatively low energy and as such provide an ideal start in a study of the quantum physics of identical particles. When we are dealing with atoms, there is something more to being ‘identical’ than we can appreciate with larger-scale objects. Atoms of the same type are really identical, measurably identical if you like, which can make a considerable difference in some physically important situations.