Fundamentals of Quantum Mechanics

Classical mechanics, introduced in the last chapter, is inadequate for describing systems composed of small particles such as electrons, atoms, and molecules. What is missing from classical mechanics is the description of wavelike properties of matter that predominates with small particles. Quantum mechanics takes into account the wavelike properties of matter when solving mechanical problems. The mathematics and laws of quantum mechanics that must be used to explain wavelike properties cause a dramatic change in the way mechanical problems must be solved. In classical mechanics, the mathematics can be directly correlated to physically measurable properties such as force, momentum, and position. In quantum mechanics, the mathematics that yields physically measurable properties is obtained from mathematical operations with an indirect physical correlation. 

At the beginning of the 20 century, experimentation revealed that electromagnetic radiation has particle-like properties (as an example, photons were shown to be deflected by gravitational fields), and as a result, it was theorized that all particles must also have wavelike properties. The idea that particles have waveUke properties resulted from the observation that a monoenergetic beam of electrons could be diffracted in the same way a monochromatic beam of light can be diffracted. The diffraction of light is a result of its wave character hence, there must be an abstract type of wave 

The constant of proportionality, h, is Planck s constant. The de Broglie relation fuses the ideas of particle-like properties (i.e. momentum) with wave-like properties (i.e. wavelength). This duality of particle and wave properties will be the theme throughout the rest of the text. 

The de Broglie relationship not only provides for a mathematical relationship for the duality of particles and waves, but it also begins to hint at the idea of quantization in mechanics. If a particle is in an orbit, the only allowed radii and momenta are those where the waves associated with the particle will interfere non-destructively as they wrap around each orbit Momenta and radii where the waves destructively interfere with one another are not allowed, as this would suggest an annihilation of the particle as it orbits through successive revolutions. 

As mentioned in the introduction to Chapter 1, for any theory to be valid it must predict classical mechanics at the limit of macroscopic particles (called the Correspondence Principle). In the de Broglie relationship, the wavelength is an indication of the degree of wave-like properties. Consider an automobile that has a mass of 1000. kg travelling at a speed of 50.0 km hr . The momentum of the automobile is 

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J. E. House, Fundamentals of Quantum Mechanic.s Academic Press, San Diego (1998). P. W. Atkins, R. S. Friedman, Molecular Quantum Mechanic.s Oxford, Oxford (1997). 

Quantum mechanically, however, the diatomic molecule and the separated atoms at infinite distance are two distinct quantum systems having their own quantum states. The physical dissociation cannot be seen as a continuous process of extending a classical spring as nearly all textbooks in chemistry, physical chemistry and quantum chemistry suggest. This is quite contrary to the fundamentals of quantum mechanics itself. Before... 

J. E. House (1998). Fundamentals of Quantum Mechanics. San Diego Academic Press. 

In the following sections it is assumed that the reader has a knowledge of the fundamentals of quantum mechanics of the type covered in any graduate course on quantum mechanics or quantum chemistry. Several aspects of quantum mechanics that are vital to the following discussions are reviewed in the appendix. 

J. E. House, Fundamental of Quantum Mechanics, Academic Press, New York, 1998. Nicholass Green, Quantum Mechanics 2 The Toolkit, Oxford University Press,... 

Now that we have reviewed some fundamentals of quantum mechanics, and laid a foundation for why bonds form, we want to turn our attention to calculating the electronic structures of atoms and molecules that are more interesting than H and H2. As always, we have to use the Schrodinger equation, but now the mathematics is much more complicated. Instead of describing all the math in detail, we touch on the fundamental math required, and we describe several of the modem techniques used in such an analysis. 

JE House, Fundamentals of Quantum Mechanics, Academic Press, San Diego, 1998. Excellent elementary yet complete discussion of the principles of quantum mechanics including particle-in-a-box, rotors, vibrations, and simple atoms. 

Quantum mechanics is based on several statements called postulates. These postulates are assumed, not proven. It may seem difficult to understand why an entire model of electrons, atoms, and molecules is based on assumptions, but the reason is simply because the statements based on these assumptions lead to predictions about atoms and molecules that agree with our observations. Not just a few isolated observations Over decades, millions of measurements on atoms and molecules have yielded data that agree with the conclusions based on the few postulates of quantum mechanics. With agreement between theory and experiment so abundant, the unproven postulates are accepted and no longer questioned. In the following discussion of the fundamentals of quantum mechanics, some of the statements may seem unusual or even contrary. However questionable they may seem at first, realize that statements and equations based on these postulates agree with experiment and so constitute an appropriate model for the description of subatomic matter, especially electrons. 

This chapter focuses on applying the fundamentals of quantum mechanics developed in the previous chapters to interpreting the vibrational and rotational transitions that occur within diatomic molecules in infrared spectroscopy. Analysis of an infrared spectrum of a diatomic molecule results in structural information about the molecule and the energy differences between the molecule s vibrational and rotational eigenstates. 

Early in the text, the fundamentals of quantum mechanics are established. This is done in a way so that students see the relevance of quantum mechanics to chemistry throughout the development of quantum theory through special boxes entitled Chemical Connection. The questions in these boxes provide an excellent basis for discussion in or out of the classroom while providing the student with insight as to how these concepts will be used later in the text when chemical models are actually developed. 

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