The Old Quantum Theory

Gas molecules can alter their states of vibration and rotation by exchanging energy with the radiation field. This exchange occurs in discrete quantities, resulting in modifications to the field at specific frequencies associated with resonances in the molecular structure. As a consequence, molecules absorb and emit radiation in a complex pattern of discrete lines that deviate significantly from a blackbody spectrum. Because each type of molecule has a unique structure and, therefore, unique energies of motion, the pattern of observed lines is characteristic of the matter present. 

The Bohr rule that the angular momentum must be an integral multiple of h 2n can, therefore, be interpreted as a requirement that the electron orbital circumference must equal an integral number of wavelengths. In other words, after each orbital revolution the wave that describes the electron motion must constmctively interfere with itself. This is a strong indication that the motion of material particles has wavelike properties. 

The Bohr theory, and its interpretation in terms of waves by de Broglie, was successful in accounting for the spectra of atoms with only one electron, and it worked approximately for alkali elements (Li, Na, Rb, Cs). However, it could not be applied to complex systems, such as molecules, and it did not constitute a dynamical theory of particle motion, nor did it provide an understanding of the interaction of radiation with matter. It did, however, set the stage for the development of modem quantum theory. 

What would you have to do to produce the correct pitch at a different temperature  

12 The speed of sound in sea water at 25°C is 1531 ms T If a SONAR system on a submarine uses a frequency of 500 s find the wavelength of the sound waves. 

13 Assume that the fundamental and the first overtone are simultaneously excited in a fiexible string such that 

Construct a graph representing the shape of the string at time t = 0, at time t = LI 4c), and at time t = 3L/(4c). Comment on the differences in the shapes. 

14 Show that the relationship v = c/k for a traveling wave also holds for the standing wave in Eq. (14.3-39). 

It is commonly accepted that the old quantum theory era spans from the birth of Planck s quantum hypothesis to the formulation of Schrodinger s equation. This section describes the old quantum theory in three parts the failure of classical mechanics, the birth of the quantum theory, and the completion of wave mechanics.5 8) This century obviously began with the birth of quantum theory. Many researchers appeared on the scene of quantum theory at the time, but we remember mostly the contributions of four researchers Max Planck (1901), Albert Einstein (1905), Niels Bohr (1913), and de Broglie (1923). Then Schrodinger proposed the new wave equation to conclude the age of the old quantum theory. Heisenberg established matrix mechanics and formulated the uncertainty principle. 

R ( r ) Laguene s polynomial, 0 ) Legendre s polynomial, P ( 4 ) Z component of angular momentum Equality of particles 

SCF method (repeat the calculation with a trial function until getting self-consistent solution) -77.879eV 

The wave theory of light reached its climax with Hertz s contributions around 1888. Physics was now in a state of turmoil, tom apart by the wave-particle duality caused by the quantum theory. As a result, research in photo- and opto-related areas was divided into two streams quantum electrodynamics and molecular spectroscopy. 

There is no question that the three reports, the photoelectric effect, the special relativity and the Brownian motion published by Einstein in 1905 changed the traditional consciousness of researchers in the 20th century. The report on the photoelectric effect clarified an old problem in the wave theory of light by linking discussions on the nature of light and the quantum hypothesis originating in Planck s study of specific heat. The report on special relativity refuted the ether theory and created an entirely new field. The report on Brownian motion, aimed at the very existence of atoms and molecules, created the theory 

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Einstein A 1917 On the quantum theory of radiation Phys. Z. 18 121 Reprinted ter Haar D 1967 The Old Quantum Theory (New York Pergamon)... 

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]). 

This is a crudal and frequently overlooked point about electronic configurations. They are far from being based in quantum mechanics it is precisely this theory that shows them to be an inadequate concept The notion that electron orbits and configurations really exist or "refer" is a relic of the old quantum theory and of Pauli s introduction of the exclusion prind-ple in its original and now strictly incorrect... 

Only with Bohr s 1913-1923 introduction of the "old quantum theory" (itself strongly inspired by chemical periodicity patterns vide infra) and the final discovery of Schrodinger s wave mechanics in 1925 would the periodic table be supplanted as the deepest expression of current chemical understanding ([21], p 2). 

The energy values correponding to the various stationary states are found from the wave equation to be those deduced originally by Bohr with the old quantum theory namely,... 

These results were true for classical mechanics and the old quantum theory, and had been assumed without proof by many people before the work of Born and Oppenheimer was published. 

In recent years the old quantum theory, associated principally with the names of Bohr and Sommerfeld, encountered a large number of difficulties, all of which vanished before the new quantum mechanics of Heisenberg. Because of its abstruse and difficultly interpretable mathematical foundation, Heisenberg s quantum mechanics cannot be easily applied to the relatively complicated problems of the structures and properties of many-electron atoms and of molecules in particular is this true for chemical problems, which usually do not permit simple dynamical formulation in terms of nuclei and electrons, but instead require to be treated with the aid of atomic and molecular models. Accordingly, it is especially gratifying that Schrodinger s interpretation of his wave mechanics3 provides a simple and satisfactory atomic model, more closely related to the chemist s atom than to that of the old quantum theory. 

This equation, including succeeding terms, was obtained originally by Sommerfeld from relativistic considerations with the old quantum theory the first term, except for the screening constant sQ> has now been derived by Heisenberg and Jordan] with the use of the quantum mechanics and the idea of the spinning electron. The value of the screening constant is known for a number of doublets, and it is found empirically not to vary with Z. 

It has been found possible to evaluate s0 theoretically by means of the following treatment (1) Each electron shell within the atom is idealised as a uniform surface charge of electricity of amount — zte on a sphere whose radius is equal to the average value of the electron-nucleus distance of the electrons in the shell. (2) The motion of the electron under consideration is then determined by the use of the old quantum theory, the azimuthal quantum number being chosen so as to produce the closest approximation to the quantum... 

It will be observed that the function differs appreciably from zero only within a radius of the order of magnitude of the major axis of the corresponding ellipses of the old quantum theory namely, r = 2a0w2/Z, or = in, as was remarked by Schrodinger (I). In fig. 2 are given values of D as a function of... 

Jones, 1 Roy. Soo. Proc., A, vol. 105, p. 650 (1924). The first attempt to calculate the mole refraction from the quadratic Stark effect formula was made by Lennard-Jones (Jones), with the old quantum theory, f Z. f. Physik, vol. 38, p. 518 (1926). 

According to the old quantum theory, the orbit of an electron moving in such a field consists of a number of elliptical segments. Each segment can be characterized by a segmentary quantum number n, in addition to the azimuthal quantum number Ic, which is the same for all segments. In all cases it is found that about half of the entire orbit lies in the outermost (j.th) region. 

The first attempts to rationalize the magnetic properties of rare earth compounds date back to Hund [10], who analysed the magnetic moment observed at room temperature in the framework of the old quantum theory, finding a remarkable agreement with predictions, except for Eu3+ and Sm3+ compounds. The inclusion by Laporte [11] of the contribution of excited multiplets for these ions did not provide the correct estimate of the magnetic properties at room temperature, and it was not until Van Vleck [12] introduced second-order effects that agreement could be obtained also for these two ions. 

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]... 

Pauling, who previously had heard Sommerfeld lecture on the old quantum theory at Caltech, now heard Sommerfeld lecture in Munich on the new wave mechanics. Pauling instantly liked the physicists new interpretation of the electron. The hydrogen atom could be thought of as... 

Discussing physics with Bohr was sometimes quite exhausting. In 1925, while working at Bohr s institute, Heisenberg discovered quantum mechanics, the theory that superseded the old quantum theory that had been developed by Bohr and his colleagues. Then, in 1926, a theory that looked much different from Heisenberg s, but which turned out to be mathematically equivalent, was propounded by the Austrian physicist Erwin Schrodinger. 

By the time Bohr turned his attention to the problem, significant advances had been made. Physicists working with the old quantum theory had developed a number of rules about the manner in which electrons interacted with one another. Bohr realized that these rules could be used to confirm Kossel s hypothesis and to make informed guesses about the atomic structure of the elements. For example, hydrogen has one electron, placed in the innermost shell. Helium, having two electrons, has this shell filled up. Thus lithium, the third element, has to have two electrons in an inner shell and one with an... 

Quantum mechanics, which replaced the old quantum theory, was not easy to interpret. It conceived of both light and particles as having a dual nature. They were sometimes observed to be waves and sometimes particles, depending on the type of experiment that one performed. For example, the electron seemed sometimes to be a particle and sometimes a packet of waves. Furthermore, quantum mechanics described the subatomic world in terms of probabilities. 

Atomic. From spectroscopic studies, it is known lliat when an electron is bound to a positively charged nucleus only certain fixed energy levels are accessible to the electron. Before 1926, the old quantum theory considered that the motion of the electrons could be described by classical Newtonian mechanics in which the electrons move in well defined circular or elliptical orbits around the nucleus. However, the theory encountered numerous difficulties and in many instances there arose serious discrepancies between its predictions and experimental fact. 

Bohr s theory was extended in various ways, especially by Somerfeld, who showed how to deal with elliptical orbits. There was a certain amount of qualitative success in applying the theory to atoms with several electrons. These developments in what is now called the old quantum theory were important as they laid much of the groundwork necessary for a correct theory. Ultimately, they were unsuccessful. Bohr s theory did not really explain what is going on why should only some orbits be allowed Where does the quantization condition (eqn 4.12) come from Following the developments of... 

The surprising conclusion to be drawn from this analysis is that a quantitative description of orbital angular momentum, as detailed as the wave-mechanical model, can be obtained within the old quantum theory. 

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