Hydrogen atom quantum theory

Keywords Atomic spectroscopy Balmer formula - Hydrogen spectrum Quantum theory - Rydberg constant... 

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

Among the few systems that can be solved exactly are the particle in a onedimensional box, the hydrogen atom, and the hydrogen molecule ion Hj. Although of limited interest chemically, these systems are part of the foundation of the quantum mechanics we wish to apply to atomic and molecular theory. They also serve as benchmarks for the approximate methods we will use to treat larger systems. 

The concept of chemical periodicity is central to the study of inorganic chemistry. No other generalization rivals the periodic table of the elements in its ability to systematize and rationalize known chemical facts or to predict new ones and suggest fruitful areas for further study. Chemical periodicity and the periodic table now find their natural interpretation in the detailed electronic structure of the atom indeed, they played a major role at the turn of the century in elucidating the mysterious phenomena of radioactivity and the quantum effects which led ultimately to Bohr s theory of the hydrogen atom. Because of this central position it is perhaps not surprising that innumerable articles and books have been written on the subject since the seminal papers by Mendeleev in 1869, and some 700 forms of the periodic table (classified into 146 different types or subtypes) have been proposed. A brief historical survey of these developments is summarized in the Panel opposite. 

Bohr s treatment gave spectacularly good agreement with the observed fact that a hydrogen atom is stable, and also with the values of the spectral lines. This theory gave a single quantum number, n. Bohr s treatment failed miserably when it came to predictions of the intensities of the observed spectral lines, and more to the point, the stability (or otherwise) of a many-electron system such as He. 

The quantum number ms was introduced to make theory consistent with experiment. In that sense, it differs from the first three quantum numbers, which came from the solution to the Schrodinger wave equation for the hydrogen atom. This quantum number is not related to n, , or mi. It can have either of two possible values ... 

I restrict my attention to non-relativistic pioneer quantum mechanics of 1925-6, and even further to the time independent formulation. Numerous other developments have taken place in quantum theory, such as Dirac s relativistic treatment of the hydrogen atom (Dirac [1928]) and various modern quantum field theories have been constructed (Redhead [1986]). Also, much work has been done in the philosophy of quantum theory such as the question of E.P.R. correlations (Bell [1966]). However, it seems fair to say that no fundamental change has occurred in quantum mechanics since the pioneer version was established. The version of quantum mechanics used on a day-to-day basis by most chemists and physicists remains as the 1925-6 version (Heisenberg [1925], Schrodinger [1926]). 

This adiabatic principle was one of the corner-stones of the old quantum theory. It allowed one to find the quantum conditions when an adiabatic change was imposed on a system. It was used successfully to account for the Stark and Zeeman effects in the spectrum of atomic hydrogen, resulting from the application of an electric and magnetic field respectively (Schwartzchild [1916] Epstein [1916]). 

Figure 5. Niels Bohr came up with the idea that the energy of orbiting electrons would be in discrete amounts, or quanta. This enabled him to successfully describe the hydrogen atom, with its single electron, In developing the remainder of his first table of electron configurations, however, Bohr clearly relied on chemical properties, rather than quantum theory, to assign electrons to shells. In this segment of his configuration table, one can see that Bohr adjusted the number of electrons in nitrogen s inner shell in order to make the outer shell, or the reactive shell, reflect the element s known trivalency.

If they were to account for the spectrum of atomic hydrogen and then atoms of the other elements, scientists of the early twentieth century had to revise the nineteenth-century description of matter to take into account wave-particle duality. One of the first people to formulate a successful theory (in 1927) was the Austrian scientist Erwin Schrodinger (Fig. 1.23), who introduced a central concept of quantum theory. 

With the development of the quantum-mechanical theory of valence it was recognized5 that a hydrogen atom, with only one stable orbital, cannot form more than one pure covalent bond6 and that the attraction... 

Hitherto it has been assumed that Tg corresponds to the classical equilibrium (or quantum-mechanical average) distance between the non-bonded atoms in the absence of interaction. It is inherent in the proper application of first-order perturbation theory that the perturbation is assumed to be small. In the case of the hindered biphenyls, however, it is known from the calculations cited in the introduction that the transition state is distorted to a considerable extent. The hydrogen atom does not occupy the same position relative to the bromine atom that it... 

The study of the hydrogen atom also played an important role in the development of quantum theory. The Lyman, Balmer, and Paschen series of spectral lines observed in incandescent atomic hydrogen were found to obey the empirical equation... 

In order to apply quantum-mechanical theory to the hydrogen atom, we first need to find the appropriate Hamiltonian operator and Schrodinger equation. As preparation for establishing the Hamiltonian operator, we consider a classical system of two interacting point particles with masses mi and m2 and instantaneous positions ri and V2 as shown in Figure 6.1. In terms of their cartesian components, these position vectors are... 

The units we use in daily life, such as kilogram (or pound) and meter (or inch) are tailored to the human scale. In the world of quantum mechanics, however, these units would lead to inconvenient numbers. For example, the mass of the electron is 9.1095 X J0 31 kg and the radius of the first circular orbit of the hydrogen atom in Bohr s theory, the Bohr radius, is 5.2918 X 10 11 m. Atomic units, usually abbreviated as au, are introduced to eliminate the need to work with these awkward numbers, which result from the arbitrary units of our macroscopic world. The atomic unit of length is equal to the length of the Bohr radius, that is, 5.2918 X 10 n m, and is called the bohr. Thus 1 bohr = 5.2918 X 10"11 m. The atomic unit of mass is the rest mass of the electron, and the atomic unit of charge is the charge of an electron. Atomic units for these and some other quantities and their values in SI units are summarized in the accompanying table. 

The fact that quantum observables are represented by matrices immediately suggests problems of non-commutation. For instance, the observables can be measured at the same time only if they have a complete orthogonal set of eigenvectors in common. This happens only when they commute, i.e. XY = YX, or the commutator [X, Y] = XY — YX = 0. This is a central feature of the matrix formulation of quantum theory discovered by Heisenberg, Born and Jordan while trying to explain the observed spectral transitions of the hydrogen atom in a more fundamental way than the quantization... 

The first application of quantum theory to a problem in chemistry was to account for the emission spectrum of hydrogen and at the same time explain the stability of the nuclear atom, which seemed to require accelerated electrons in orbital motion. This planetary model is rendered unstable by continuous radiation of energy. The Bohr postulate that electronic angular momentum should be quantized in order to stabilize unique orbits solved both problems in principle. The Bohr condition requires that... 

Despite the complication due to the interdependence of orbital and spin angular momenta, the Dirac equation for a central field can be separated in spherical polar coordinates [63]. The energy eigenvalues for the hydrogen atom (V(r) = e2/r, in electrostatic units), are equivalent to the relativistic terms of the old quantum theory [64]... 

The quantum theory of the previous chapter may well appear to be of limited relevance to chemistry. As a matter of fact, nothing that pertains to either chemical reactivity or interaction has emerged. Only background material has been developed and the quantum behaviour of real chemical systems remains to be explored. If quantum theory is to elucidate chemical effects it should go beyond an analysis of atomic hydrogen. It should deal with all types of atom, molecules and ions, explain their interaction with each other and predict the course of chemical reactions as a function of environmental factors. It is not the same as providing the classical models of chemistry with a quantum-mechanical gloss a theme not without some common-sense appeal, but destined to obscure the non-classical features of molecular systems. 

The initial purpose of pioneer quantum mechanics was to provide the theoretical framework to account for the structure of hydrogen and the nuclear model of atoms in general. The final result, a quantum theory of atomic structure can be discussed in terms of the time-independent Schrodinger equation, in its most general form... 

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