Quantum Theory of the Hydrogen Atom

For a particle in a one-dimensional box, in which state (value of n) is the greatest probability of finding the particle at one-quarter the length of the box from either end  

Before we begin, let s summarize a few key ideas from Section 8-5. We learned that if a particle is confined to a one-dimensional box, the energy of the particle is quantized. That is, the particle can possess only certain quantities of energy. In addition, we learned that the state of the particle, or the matter wave associated with it, can be characterized by a quantum number, n, and described by a wave function, i/r , that can be analyzed to reveal certain general features. For the particle in a box, not only do we find that if/ has n — 1 nodes, but also we discover an interesting correlation between the energy of each state and the number of nodes in the associated wave function The energy of the particle increases with the number of nodes. 

How does the system of a particle in a box help us understand the hydrogen atom The electron in a hydrogen atom is also confined, not literally by impenetrable walls but in principle because of its attraction to the nucleus. If we accept the basic idea that the electron in a hydrogen atom is confined by its attraction to the nucleus, then it should come as no surprise that the energy of the hydrogen atom is also quantized. The allowed energies will not be the same as for the particle in a box, but the energies will be restricted to certain values nonetheless. We should also expect that the state of the electron will be characterized by quantum numbers and described by a wave function that can be analyzed to reveal certain important features. By the end of the next section, we will see that all these assertions are true. 

Solving the Schrodinger equation for the hydrogen atom gives the same expression for the energy levels, equation (8.5), that we encountered previously  

The Schrodinger equation is accepted as a basic postulate of quantum mechanics. It cannot be derived from other equations. However, the form of the Schrodinger equation can be justified as follows. We start with the equation for a standing wave in one dimension  

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By the end of the 1920s, Bohr s institute (the official name was the University of Copenhagen Institute of Theoretical Physics, but everyone called it Bohr s institute instead) had become the world s most famous center for research in physics. It was visited at one time or another by many of the most notable physicists of the first half of the twentieth century, and many noted physicists of the second half of the century did postdoctoral work there. Physicists often visited the institute to have discussions with Bohr, who was recognized as a leading physicist within a few years of the publication of his quantum theory of the hydrogen atom. And in 1922 he was awarded the Nobel Prize. Bohr and Einstein received the prize the same year. But it wasn t a shared prize Einstein s was for 1921. 

Danish physicist Niels Bohr (1885-1962) proposed a quantum theory of the hydrogen atom by suggesting that the electron moves about its nucleus in discrete quanta (the energies of electrons are restricted to having only certain values, quanta, much as stairs do as opposed to a ramp), establishing a balance between the electron s centrifugal force and its attraction for the nucleus. It was not until 1927 that covalent bonding was properly... 

Wave Mechanics Quantum Theory of the Hydrogen Atom Periodic Table... 

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. 

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. 

The paper that reported these results ended with the recognition that there was a problem Whether the failure of theory and experiment to agree is because of some unknown factor in the theory of the hydrogen atom or simply an error in the estimate of one of the natural constants, such as [the fine structure constant], only further experiment can decide. This was the result that Rabi conveyed to the physicists at Shelter Island. Rabi s reputation as an experimentalist brought credibility to the measured results and issued a challenge to the theorists. As with the Lamb shift, it was quantum electrodynamics that was brought to bear on... 

This is an instance of a fundamental result in quantum mechanics, that any measured component of orbital angular momentum is restricted to integral multiples of h. The Bohr theory of the hydrogen atom, to be discussed in the next chapter, can be derived from this asssumption alone. 

The year 1913 marks a major climax in the history of science. The application of Planck s quantum hypothesis to blackbody radiation, and later by Einstein to the photoelectric effect, had met with disbelief and in some quarters even with scorn. Bohr s application to the theory of the hydrogen atom compelled belief and worked a revolution in thought. In the following ten years this new knowledge was quickly assimilated and applied with spectacular success to the interpretation of spectra and chemical periodicity. 

Rydberg constant /rid-berg/ A constant that occurs in formula for the frequencies of spectral lines in atomic spectra. For the hydrogen atom it has the value 1.0968 x 10 m"i. The value of the Rydberg constant can be calculated from the BOHR THEORY of the hydrogen atom and from quantum mechanics. These calculations showed that ... 

Quantum mechanics is basically statistical in nature. Knowing the state, we cannot predict the result of a position measurement with certainty we can only predict the probabilities of various possible results. The Bohr theory of the hydrogen atom specified the precise path of the electron and is therefore not a correct quantum-mechanical picture. 

This is the Rydberg constant, R, from the hydrogen atom spectrum. Quantum mechanics therefore predicts the experimentally determined hydrogen atom spectrum. At this point, quantum mechanics predicts everything that Bohr s theory did and more, and so supersedes the Bohr theory of the hydrogen atom. 

The old quantum theory includes Planck s black-body radiation theory, Einstein s theory of the photoelectric effect, and Bohr s theory of the hydrogen atom. 

In 1913 Bohr published a theory of the hydrogen atom, based on unproved assumptions. We regard his theory as the third part of the old quantum theory. A simplified version of Bohr s assumptions is ... 

The quantum mechanical theory of the hydrogen atom is given below (see Chapter 7.5) The Bohr model of the hydrogen atom is a transition from purely classical presentations to quantum mechanical ones the motion of electrons along the orbits is accepted however not all orbits are permitted, the angular momentum is accepted, though its values and orientations are subject to strict limitation. One can consider the Bohr model as the transition from classical mechanics to quantum mechanics with the preservation of many its attributes. As a result, many of the ideas of the Bohr model will often be met in order to simplify the students understanding. 

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