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Unformatted text preview: 03-07Jan2011 Chemistry 21b Molecular Structure & Spectroscopy Week # 1 The Electronic Structure of Atoms In this quarter of Ch21, we will be primarily concerned with spectroscopy the interaction of light with matter and what it can tell us about molecules and materials. As we will see in future lectures, perturbation theory is the simplest route to thinking about this fundamental interaction. In setting up many of the situations we will encounter, the so called matrix form of quantum mechanics will be helpful, and so before turning to the nature of the electronic structure of atoms these notes will present a brief overview of the quantum mechanical underpinnings of molecular systems that are hopefully familiar from Ch21a along with a brief review of the notation that will be followed in these notes. The Basic Postulates of Quantum Mechanics As with all scientific theories, the test of quantum mechanics lies in the rigorous comparison of its predictions with observations. Building up these predictions rests on five basic postulates, that we briefly review here (in addition to Chapter 1 of the class text (Atkins & Friedman), the text by Engel has a more extensive discussion of these postulates, see Quantum Chemistry & Spectroscopy , Ch. 3): Postulate 1 There exists a wave function ( r , t ) that completely specifies the state of a quantum mechanical system. It depends on both the spatial coordinates of the particle(s) and time, and has the property that ( r , t )( r , t ) dx dy dz denotes the probability that there is a particle located at r in a volume dx dy dz and at time t , where ( r , t ) is the complex conjugate of ( r , t ). Postulate 2 The observables in classical mechanics (position, linear and angular momentum, energy, etc.) are represented quantum mechanically by Hermitian operators. More on this next. Postulate 3 For each and every Hermitian operator A associated with an observable, only certain values of this observable are possible. In particular, they must satisfy the eigenvalue equation A = a , where is the wave function of the system. Because the operators are Hermitian, the eigenvalues are real and for non-degenerate systems the eigenfunctions (or state functions) are orthogonal. Because the wavefunctions are also probability distributions, they are best constructed to be orthonormal as well (or, integraltext ( r , t )( r , t ) dx dy dz = 1). Finally, because the operators are Hermitian, it can be proven that the eigenfunctions are complete meaning that any state function of the system can be decomposed into an appropriate sum of the eigenfunctions. Postulate 4 The expectation, or average, value of an observable at time t that is represented 1 by an operator A is given by the integral < a > = integraldisplay ( r , t ) A ( r , t ) d r where ( r , t ) is the state function of the system....
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