In Fig. So, the Bohr model works well for one-electron atoms or ions and the quantum defect-modified Bohr equation describes reasonably well some states of alkali atoms and of Rydberg molecules. The primary reason for the breakdown of the Bohr formula is the neglect of electron-electron Coulomb repulsions in its derivation, which are qualitatively corrected for by using the quantum defect parameter for Rydberg atoms and molecules.

Nevertheless, the success of the Bohr model made it clear that discrete emission spectra could only be explained by introducing the concept that not all orbits were allowed. This idea that not all energies were allowed, but only certain quantized energies could occur was essential to achieving even a qualitative sense of agreement with the experimental fact that emission spectra were discrete.

In summary, two experimental observations on the behavior of electrons that were crucial to the abandonment of Newtonian dynamics were the observations of electron diffraction and of discrete emission spectra. Both of these findings seem to suggest that electrons have some wave characteristics and that these waves have only certain allowed i.

We see that extra conditions e. However, we still are left wondering what equations can be applied to properly describe such motions and why the extra conditions are needed. It turns out that a new kind of equation based on combining wave and particle properties needed to be developed to address such issues. That is, scientists did not derive this equation; they postulated it.

## Advances in the Theory of Quantum Systems in Chemistry and Physics | Philip E. Hoggan | Springer

It should be noted that the people who worked on these problems knew a great deal about waves e. Moreover, they knew that waves could sometimes display the characteristic of quantized wavelengths or frequencies e. They knew, for example, that waves in one dimension that are constrained at two points e.

Two examples of such waves in one dimension are shown in Fig. The equation that such waves obey, called the wave equation, reads:. This speed depends on the composition of the material from which the violin string is made; stiff string material produces waves with higher speeds than for softer material. Instead, it is the condition that the wave vanish at the boundaries that generates the quantization. You will see this trend time and again throughout this text; when a wave function is subject to specific constraints at its inner or outer boundary or both , quantization will result; if these boundary conditions are not present, quantization will not occur.

Let us now return to the issue of waves that describe electrons moving. The pioneers of quantum mechanics examined functional forms similar to those shown above. Noticing that. Again the scientists who invented quantum mechanics did not derive its working equations. Instead, the equations and rules of quantum mechanics have been postulated and designed to be consistent with laboratory observations.

My students often find this to be disconcerting because they are hoping and searching for an underlying fundamental basis from which the basic laws of quantum mechanics follow logically. I try to remind them that this is not how theory works.

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Unlike relativity, however, the origins of QM cannot be attributed to any one scientist. Rather, multiple scientists contributed to a foundation of three revolutionary principles that gradually gained acceptance and experimental verification between and They are:. Quantized properties : Certain properties, such as position, speed and color, can sometimes only occur in specific, set amounts, much like a dial that "clicks" from number to number.

This challenged a fundamental assumption of classical mechanics, which said that such properties should exist on a smooth, continuous spectrum.

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To describe the idea that some properties "clicked" like a dial with specific settings, scientists coined the word "quantized. Particles of light : Light can sometimes behave as a particle. This was initially met with harsh criticism, as it ran contrary to years of experiments showing that light behaved as a wave; much like ripples on the surface of a calm lake. Light behaves similarly in that it bounces off walls and bends around corners, and that the crests and troughs of the wave can add up or cancel out. Added wave crests result in brighter light, while waves that cancel out produce darkness.

The color emitted corresponds to the distance between the crests, which is determined by the speed of the ball's rhythm. Waves of matter : Matter can also behave as a wave. This ran counter to the roughly 30 years of experiments showing that matter such as electrons exists as particles.

In , German physicist Max Planck sought to explain the distribution of colors emitted over the spectrum in the glow of red-hot and white-hot objects, such as light-bulb filaments. Somehow, colors were quantized! This was unexpected because light was understood to act as a wave, meaning that values of color should be a continuous spectrum. This seemed so strange that Planck regarded quantization as nothing more than a mathematical trick. According to Helge Kragh in his article in Physics World magazine, " Max Planck, the Reluctant Revolutionary ," "If a revolution occurred in physics in December , nobody seemed to notice it.

Planck was no exception …".

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Planck's equation also contained a number that would later become very important to future development of QM; today, it's known as "Planck's Constant. Quantization helped to explain other mysteries of physics. Alexander V.

## Advances in the Theory of Quantum Systems in Chemistry and Physics

Glushkov, Olga Yu. Khetselius, Elena P. Gurnitskaya, Andrey V. Florko, Denis E. Sukharev et al. Svetlana V.

## Frontiers in Quantum Systems in Chemistry and Physics

Malinovskaya, Alexander V. Khetselius, Andrey V. Loboda, Andrey A. Back Matter Pages