Instructors_Guide_Ch40 - 40 One-Dimensional Quantum...

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40-1 40 One-Dimensional Quantum Mechanics Recommended class days: 3 Background Information At this point, students have seen that: • Atomic particles exhibit wave-like behavior. • Standing de Broglie waves lead to quantization of energy. • The Bohr model suggests that real atoms have quantized energy levels. •Wave-particle duality can be understood with a probabilistic wave function. Now we need to assemble these bits and pieces into an actual theory of matter on the atomic scale. The goals of this chapter are quite limited. Physics majors and those engineers who might use quantum mechanics extensively will go on to take a modern physics course and perhaps a full course in quantum mechanics. For them, this chapter is an introduction. A larger group of students needs to understand the significance of energy levels and wave functions, particularly as they apply to issues such as molecular bonds or tunneling, but they’ll never have to solve the Schrödinger equation for themselves. Hence the focus of this chapter is on: •Modeling quantum systems. • Interpreting and using solutions of the Schrödinger equation. • Understanding quantum phenomena such as tunneling. These goals can be met with one-dimensional quantum mechanics as applied to time- independent, bound-state problems with real-valued solutions. Students are not expected or asked to solve differential equations, but they should learn to recognize and interpret the major features of a solution of the Schrödinger equation. Classical mechanics analyzes phenomena by developing a model of the forces acting on a particle. Some forces may be accurately known, others are reasonable approximations. Once the forces of the model are identified, the particle’s trajectory can be found by solving Newton’s second law. We also use models in quantum mechanics, but now the models are expressed in terms of a potential energy function. This is much harder for students to deal with. Although potential energy diagrams were emphasized in Chapter 10, most students will need to review how potential energy diagrams are interpreted classically to yield turning points, kinetic energies, forces, and so on. You will also need to be very explicit with your explanations of how a particle confined to a certain region can be modeled as being in a potential energy well.
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40-2 Instructor’s Guide Unpublished research at Kansas State University has found that students often interpret graphs and diagrams very differently than we intended. For example, we like to show wave functions “in” the potential well, each oscillating about its energy level. Students find these graphs to be confusing because we’re graphing both energy and wave functions on the vertical axis. Conse- quently, some students interpret this graph as saying that ψ 3 is always larger in value than ψ 1 because it is “above,” and hence more positive, than ψ 1 . They don’t recognize, and textbooks rarely state, that each wave function is drawn such that the energy-level line about which it oscillates is the “zero” for that wave function.
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This note was uploaded on 01/14/2011 for the course CD 254 taught by Professor Kant during the Spring '10 term at Central Oregon Community College.

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Instructors_Guide_Ch40 - 40 One-Dimensional Quantum...

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