I find that students ability to work with wave

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I find that students’ ability to work with wave function improves after a brief discussion of the classical probability density for finding a particle at position x . The essential idea is that a classical particle is more likely to be found where it is moving more slowly; thus, we expect the wave function of a quantum particle to have larger amplitude where the kinetic energy is smaller, and vice versa. If we also think of a bound-state wave function as a de Broglie standing wave, then we
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Chapter 40: One-Dimensional Quantum Mechanics 40-5 expect the wavelength (i.e., the distance between the nodes) to be larger where the kinetic energy is smaller. You can make these points by asking the students to do a few graphical exercises. For example, ask them to draw a rough sketch of the n = 5 wave function for the potential energy well shown on the left below, assuming that E 5 is above the “barrier.” To do so, students need to recognize that: The wave function must be zero at the edges. The wave function will have four zero crossings and five antinodes. The wavelength will be longer in the middle, where K is smaller, and shorter at the edges. The amplitude will be larger in the middle, where K is smaller and the classical probability density is larger. x ψ ( x ) x E 5 0 L /2 L U ( x ) The inverse problem is to draw a wave-function graph such as the one shown above on the right. First ask students what the quantum number is. Then ask them to sketch a plausible potential energy function for which this is an allowed wave function. If you are covering this chapter in three days, you need to at least start the quantum harmonic oscillator on day 2. There’s not time to dwell on the quantum harmonic oscillator, but you want to note that it also has wave functions penetrating into a classically forbidden region. DAY 3: The quantum harmonic oscillator has the advantage that the first few wave functions are relatively simple and can be written explicitly. Some homework problems require calculations with the harmonic oscillator wave functions, and you might want to give hints about these. Although the calculations are straightforward, many students will have difficulty knowing just what to do unless they are pointed in the right direction. To be consistent with all other wave functions we’ve studied, this textbook defines the ground state of the harmonic oscillator to be n = 1 rather than the customary n = 0. This requires the energy levels to be written E n = ( n 1 2 ) ω rather than ( n + 1 2 ) ω . You can make a short digression to talk about molecular bonds and vibrational spectra, if time permits, since this supports the idea that quantum mechanics gives us useful information about atomic and molecular systems.
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