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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 Ψ( r , ? ), that is analogous to the classical displacement function Φ( 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 Ψ( 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 Ψ(r , ? ), which may be expressed in certain cases as the product ( r ) ( ? ). Both Ψ and must be single– valued, continuous, and finite for all values of their coor dinates, and they must be smoothly–varying within their boundaries, at which they vanish. CHEM 356, Lecture 7, Fall 2009 2 The Time–Independent Schr¨odinger Equation. Let us consider the case that Ψ can be expressed as the prod uct function Ψ( r , ? ) = ( r ) ( ? ), and let us assume for the moment that obeys the time–independent classical wave equation. We would have ∇ 2 = − 4 2 2 = − 4 2 2 ? 2 ℎ 2 , where in the second step we have employed the de Broglie relation = ℎ/? . Let us now rearrange the righthand side a little, to obtain ∇ 2 = − 8 2 ℎ 2 ( ? 2 2 ) = − 8 2 ℎ 2 , with being the KE of the particle. If we rearrange this expression once more to give ( − ℎ 2 8 2 ∇ 2 ) = , we may identify the kinetic energy operator ˆ as ˆ ≡ − ℏ 2 2 ∇ 2 . CHEM 356, Lecture 7, Fall 2009...
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 Fall '09
 Prof.Iaskjd
 Ψ, time–dependent Schr¨dinger equation

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