This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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 right-hand 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...
View Full Document
- Fall '09
- Ψ, time–dependent Schr¨dinger equation