Equations, mathematical Bohr model

Each solution to the equation (that is, each energy state of the atom) is associated with a given wave function, also called an atomic orbital. It s important to keep in mind that an orbital in the quantum-mechanical model bears no resemblance to an orbit in the Bohr model an orbit was, supposedly, an electron s path around the nucleus, whereas an orbital is a mathematical function with no direct physical meaning. 

When the Schrodinger equation is solved, it yields many solutions— many possible wave functions. The wave functions themselves are fairly comphcated mathematical functions, and we do not examine them in detail in this book. Instead, we will introduce graphical representations (or plots) of the orbitals that correspond to the wave functions. Each orbital is specified by three interrelated quantum numbers n, the principal quantum number I, the angular momentum quantum number (sometimes called the azimuthal quantum number) and mi, the magnetic quantum number. These quantum numbers all have integer values, as had been hinted at by both the Rydberg equation and Bohr s model. A fourth quantum number, nis, the spin quantum number, specifies the orientation of the spin of the electron. We examine each of these quantum numbers individually. 

In order to understand these observations it is necessary to resort to quantum mechanics, based on Planck s postulate that energy is quantized in units of E = hv and the Bohr frequency condition that requires an exact match between level spacings and the frequency of emitted radiation, hv = Eupper — Ei0wer. The mathematical models are comparatively simple and in all cases appropriate energy levels can be obtained from one-dimensional wave equations. 

There are some scientists and philosophers who still claim that a model by definition "furnishes a concrete image" and "does not constitute a theory." 10 But if the model is the mathematical description, then the question of whether the model is the theory appears to become moot, since most people accept the view that rigorous mathematical deduction constitutes theory. For others, like Hesse and Kuhn, even if the model is a concrete image leading to the mathematical description, it still has explanatory or theoretical meaning, for, as Kuhn put it, "it is to Bohr s model, not to nature, that the various terms of the Schrodinger equation refer." 11 Indeed, as is especially clear from a consideration of mathematical models in social science, where social forces are modeled by functional relations or sets of mathematical entities, the mathematical model turns out to be so much simpler than the original that one immediately sees the gap between a "best theory" and the "real world." 12... 

It is worth noting at this point that the various scientific theories that quantitatively and mathematically formulate natural phenomena are in fact mathematical models of nature. Such, for example, are the kinetic theory of gases and rubber elasticity, Bohr s atomic model, molecular theories of polymer solutions, and even the equations of transport phenomena cited earlier in this chapter. Not unlike the engineering mathematical models, they contain simplifying assumptions. For example, the transport equations involve the assumption that matter can be viewed as a continuum and that even in fast, irreversible processes, local equilibrium can be achieved. The paramount difference between a mathematical model of a natural process and that of an engineering system is the required level of accuracy and, of course, the generality of the phenomena involved. 

Although the four quantum numbers n, 1, m, and s, the Pauli Exclusion Principle, and Hund s rules were developed in the context of the Bohr-Sommerfeld model, they all found immediate application to Schrodinger s new quantum mechanics. The first three numbers specified atomic orbitals (replacing Bohr s orbits). Physicist Max Bom (1882-1970) equated the square of the wave functions, to regions of probability for finding electrons in each orbital. Werner Heisenberg (1901-76), whose mathematics provide the foundation of quantum mechanics, developed the uncertainty principle the product of the uncertainty in position (Ax) of a tiny particle such as an atom (or an electron) and the uncertainty in its momentum (Ap) is larger than the quantum (h/47t) ... 

In Bohr s model of the hydrogen atom, only one number, n, was necessary to describe the location of the electron. In quantum mechanics, three quantum numbers are required to describe the distribution of electron density- in an atom. These numbers are derived from the mathematical solution of Schrbdinger s equation for the hydrogen atom. They are called the principal quantum number, the angular momentum quantum number, and the magnetic quantum number. Each atomic orbital in an atom is characterized by a unique set of these three quantum numbers. 

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