Classical mechanics failure

The failure of classical mechanics in the analysis of physical phenomena, such as black-body radiation, is routinely discussed in elementary texts to emphasize the need of a quantum theory. The failure of classical mechanics to deal correctly with simple chemical systems, although rarely stated, is equally dramatic. 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

Although the Bohr theory satisfactorily explained the spectra of hydrogen and of other species containing one electron (He, Li +, etc.) the wavelengths in the observed spectra of more complex species could not be calculated. Bohr s assumption of circular orbits was modified in 1916 by Arnold Sommerfeld (1868-1951), who assumed elliptical orbits. Even so, the Bohr approach was doomed to failure, because it modified classical mechanics to solve a problem that could not be solved by classical mechanics. It was a contrived solution. This failure of classical mechanics set the stage for the development of a new physics, quantum mechanics, to deal with small particles. The Bohr theory, however, did introduce the ideas that only certain energy levels are possible, that these energy levels are... 

It has been found, however, that the numbers obtained in this way do not agree with experimental determinations of the ionisation and excitation potentials. On this account wc shall refrain from discussing more fully this model for Ha+. At present the reason for the failure of the theory is by no means clear. We shall see later that the treatment of atomic problems with the help of classical mechanics... 

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. 

We have compared the birth of quantum theory to a drama written and acted by Planck, Einstein, Bohr, and de Broglie. Indeed they inherited a critical mind with a deep insight into the quantum theory and they pursued continuously the development of this concept. From their research in that fertile period of time, we can study the inner workings of their considerations as human beings. Taking advantage of the failure of classical mechanics, they created a new concept. In the formation process of the quantum theory, they never denied the system of classical physics. The new physical system which they constructed included classical physics and grew to such a scale that it exerted a tremendous influence upon all science fields for nearly a century. 

This has led many researchers to posmlate a simple shear stress criterion for failure. This idea persists to this day despite some early analysis, such as that of Goland and Reissner who used classical mechanics of material concepts to demonstrate that the joint includes shear, bending, and cleavage stresses [5], This is more recently substantiated by several numerical analyses, some of which are cited in [1]. 

In classical terms, the mechanical properties of elastic solids can be described by Hooke s law, which states that an applied stress is proportional to the resultant strain but is independent of the rate of strain. For liquids, the corresponding statement is known as Newton s law, with the stress now independent of the strain but proportional to the rate of strain. Both are hmiting laws, valid only for small strains or rates of strain, and although it is essential that conditions involving large stresses, leading to eventual mechanical failure, be smdied, it is also important to examine the response to small mechanical stresses. Both laws can prove useful under these circumstances. 

The theoretical basis for classical mechanics, in the form we know it today, was laid by Newton in the seventeenth century. The theory had an unprecedented success in explaining virtually all observed phenomena pertaining to macroscopic systems. The failure of Newtonian mechanics to describe systems on an atomic and molecular scale was not realized to its full extent until toward the end of the nineteenth century, when the atomic structure of... 

Part 1 of this textbook presents the study of the macroscopic properties of material systems of many molecules, based on thermodynamics. This part of the textbook presents the study of individual atoms and molecules. This study is based on quantum mechanics. Classical mechanics pre-dated quantum mechanics, and this chapter presents both classical mechanics and the so-called Old Quantum Theory, which consists of several theories contrived to explain the failure in the late 1800s of classical mechanics to describe or explain certain molecular phenomena. 

Classical mechanics was invented by Sir Isaac Newton to describe and predict the motions of objects such as the planets as they move about the sun. Although classical mechanics was a great success when applied to objects much larger than atoms, it was a complete failure when applied to atoms and molecules. It was superseded by quantum mechanics, which has enjoyed great success in explaining and predicting atomic and molecular properties. However, quantum mechanics was built upon classical mechanics, and someone has said that if classical mechanics had not been discovered prior to quantum mechanics, it would have had to be invented in order to construct quantum mechanics. 

One way to understand this qualitative failure of classical mechanics is to note that the origin of the problem is our attempt to observe very small deflections as a function of b. As we saw, a deflection arises from momentum transfer in the direction of b. So what we are seeking to do is to measure simultaneously momentum and position in the same direction. The Heisenberg uncertainty principle puts a limit on our inherent ability to do so. When can we no longer... 

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