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Lecture12Notes - Chem 120A Spring 2006 READING Particle in...

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Chem 120A Particle in a Box, Continued 02/13/06 Spring 2006 Lecture 12 READING: Engel (QCS): Chapter 4.1, Chapter 5.1-5.3 Solutions for a particle in a box in one-dimension Recall from last time that we had a 1D system with potential energy, V ( x ) = 0 0 x a otherwise . (1) The 1D Schr¨odinger equation is given by, ˆ H Ψ = E Ψ ¯ h 2 2 m d 2 Ψ ( x ) dx 2 + V ( x ) Ψ ( x ) = E Ψ ( x ) . (2) We divided this system into regions I, II, and III, as shown in Lecture 10, and applied the condition Ψ I ( x ) = Ψ III ( x ) = 0 (3) in order to solve the ordinary differential equation in Equation2. The result is a set of solutions, also called ’eigenfunctions,’ that depend on a quantum number, n : Ψ n ( x ) = 2 a sin n π x a , (4) where we found the constant 2 / a by applying the normalization condition, a 0 Ψ ( x ) Ψ ( x ) dx = 1 . (5) Each of these solutions corresponds to a unique energy, also called an ’eigenvalue,’ which depends on the quantum number, n : E n = h 2 n 2 8 ma 2 . (6) Note that these energies are discrete and quantized by their dependence on n . Also note that the first energy level is not at E = 0. An application of particle in a box: spectra of conjugated molecules The particle in a box model is useful pedagogically because it is one of the few systems for which the Schr¨odinger equation is exactly solvable. It is also useful in scientific applications where it is used to Chem 120A, Spring 2006, Lecture 12 1
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Figure 1: Butadiene a a b c c Length of box = 2a+b+2*(0.5*c) = 2(1.54)+1.35+1.54 L = 5.78 Angstroms calculate energies that agree fairly well with experimentally observed values. Here, we will examine its application to the butadiene molecule. Butadiene is an organic conjugated molecule, as shown in Figure. Because the electrons in the π bonds of the molecule are delocalized over the backbone of the molecule, we assume that the electrons are confined to be within a box defined by the molecule. Using the bond lengths for the C-C single and double bonds and the C-H bond (see Figure , we can estimate the length, L of the box, we then use this length to calculate the energies of the levels for a box of this length. There are 4 electrons in the pi system of butadiene. We
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