Introduction to Symmetry in Quantum Mechanics

Symmetry is one of the most powerful tools that can be applied to quantum mechanics and wavefunctions. Most people are generally aware of the concept of symmetry An object is round or square, or the left side is the same as the right side, or maybe they are mirror images. All of these statements imply a recognition of symmetry, a spatial similarity due to the shape of an object. But more technically, symmetry is a powerful mathematical tool that can potentially simplify our study of quantum mechanics. 

Such comparisons apply in quantum mechanics, too. Recognizing the symmetry of an atomic or molecular system allows one to simplify the quantum mechanics, sometimes dramatically. We have already seen some aspects of symmetry odd and even functions, the spherical nature of the hydrogen atom s Is orbital, the cylindrical shape of H2 and H2. All these are applications of symmetry. In this chapter, we will develop a general understanding of symmetry using a mathematical tool called group theory. Then, we can see how symmetry applies to some aspects of quantum mechanics. 

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

FIGURE 13.2 A rectangle has reflection planes of symmetry, labeled a. Upon reflection of all points of the rectangle through the plane of symmetry, the original object is reproduced. The reflection of only one point on the rectangle is shown. Can you find two other reflection planes of symmetry for the rectangle  

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This section gives an introduction to the effects of symmetry and quantum mechanical tunneling on chemical reactions in general and hydrogen transfer in particular. 

Frequently the work involved conjugated molecules to which Electronic population analysis was usually added to the energy calculations and a theoretical dipole moment was obtained that could be compared with the experimental data. With the advent of NMR. and ESR. spectroscopy other observables became available, and theory was successfully applied to the interpretation of these spectra. However, very little was done in the field of real chemistry, that is, in the study of reaction mechanisms and reaction rates. Over the last decade the availability of large electronic computers, the introduction of approximate but reliable quantum mechanical methods which include all the electrons, or at least all valence electrons in a molecular system and the discovery of the rules of orbital symmetry have led to a significant change of the situation. 

There is no evidence that any classical attribute of a molecule has quantum-mechanical meaning. The quantum molecule is a partially holistic unit, fully characterized by means of a molecular wave function, that allows a projection of derived properties such as electron density, quanmm potential and quantum torque. There is no operator to define those properties that feature in molecular mechanics. Manual introduction of these classical variables into a quantum system is an unwarranted abstraction that distorts the non-classical picture irretrievably. Operations such as orbital hybridization, LCAO and Bom-Oppenheimer separation of electrons and nuclei break the quantum symmetry to yield a purely classical picture. No amount of computation can repair the damage. 

However, classifying the wave function of polymerization by the spatial symmetry features and taking into account the above specificity of addition polymerization, it is advisable for simplicity to introduce as a supplement to LCAO the approximation of the polymerization wave function in the form of a linear combination of molecular orbitals of fragments (LCMOF). The validity of introduction of this approximation is based on the general quantum-mechanical principle of superposition. 

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