Graphs showing ψ x and ψ x 2 on the same set of

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line about which it oscillates is the “zero” for that wave function. Graphs showing ψ ( x ) and | ψ ( x )| 2 on the same set of axes can be equally confusing to students. Instructors need to be very explicit about what they’re graphing and need to ask students to interpret what they’re seeing. Student Learning Objectives To introduce the Schrödinger equation as the “law” of quantum mechanics. To recognize that solutions of the Schrödinger equation give the allowed energies and wave functions for a physical situation that is modeled by the potential energy function U ( x ). To interpret wave functions and energy levels. To understand quantum phenomena such as bonding and tunneling. Pedagogical Approach Quantum-mechanical modeling requires finding an appropriate potential energy function rather than, as in classical mechanics, an appropriate set of forces. Physicists may consider it plausible, at least as a first approximation, to model a neutron in a nucleus or an electron in a layered device as a particle in a finite, square-well potential, but this is totally incomprehensible to most beginning students. For quantum mechanics to be meaningful to students, you need to focus lots of attention on interpreting the potential energy diagram. It’s often useful to start with a classical model of forces on a particle, then recast this in terms of potential energy. For example, an electron in a quantum-well device can move freely in the GaAs layer where it’s part of the sea of electrons. At the edge of the layer, the electron experiences strong forces that pull it back in. In energy terms, this suggests a flat-bottomed potential with steep walls. We model this as a finite square-well potential. Students can understand this if you lead them through the reasoning, but few will realize on their own why the square-well potential is chosen. The Schrödinger equation was solved numerically for most of the potential wells of this chapter. Students see only the energies and wave functions that come out of the numerical solution. As noted above, only a small fraction of the students will ever have any need to solve the Schrödinger equation themselves, but they may need to interpret solutions. Thus the main emphasize of this chapter should be on what it means to have solved the Schrödinger equation. n = 1 E 1 n = 2 E 2 = 4 E 1 E 3 = 9 E 1 n = 3 0 L x U ( x )
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Chapter 40: One-Dimensional Quantum Mechanics 40-3 Students should learn that: Wave function solutions exist only for certain “allowed” energies. These are the stationary states that Bohr postulated but was unable to find. The wave function is continuous and oscillates inside the potential well, with n antinodes and n 1 zero crossings for a wave function with quantum number n . The wave function can be thought of as a de Broglie standing wave of variable wavelength. The wavelength increases as K = E U decreases.
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