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Unformatted text preview: CHEM 356, Lecture 7, Fall 2009 1 II. The Postulates of Quantum Mechanics. The de Broglie pilot wave concept leads logically to the as- sociation of a particle with a function, traditionally denoted by ( r , ? ), that is analogous to the classical displacement function ( r , ? ). For a bound particle, which is equivalent to classical standing waves, we might reasonably expect to be able to separate the r and ? dependence of ( r , ? ). These expectations are gathered together into Postulate 1 . The state of a single particle in a particular system is de- scribed as fully as possible by an appropriate state function or wavefunction (r , ? ), which may be expressed in certain cases as the product ( r ) ( ? ). Both and must be single valued, continuous, and finite for all values of their coor- dinates, and they must be smoothlyvarying within their boundaries, at which they vanish. CHEM 356, Lecture 7, Fall 2009 2 The TimeIndependent Schrodinger Equation. Let us consider the case that can be expressed as the prod- uct function ( r , ? ) = ( r ) ( ? ), and let us assume for the moment that obeys the timeindependent classical wave equation. We would have 2 = 4 2 2 = 4 2 2 ? 2 2 , where in the second step we have employed the de Broglie relation = /? . Let us now rearrange the right-hand side a little, to obtain 2 = 8 2 2 ( ? 2 2 ) = 8 2 2 , with being the KE of the particle. If we rearrange this expression once more to give ( 2 8 2 2 ) = , we may identify the kinetic energy operator as 2 2 2 . CHEM 356, Lecture 7, Fall 2009...
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- Fall '09