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# Section 9.5: Exploring the Addition of Two Plane Waves

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During the quantum-mechanical scattering of plane waves, we must consider the superposition of the right-moving incident wave and the left-moving reflected wave results in Region I. In this Exploration we investigate how two time-dependent plane waves can be added together to resemble a scattering situation. **In the animation, ħ = 2m = 1.** Restart.

- With the default settings, explain why the arguments of the cosines and sines are of the form (5*x-25*t) and (-5*x-25*t). In other words, what does the ±5 signify and what does the 25 signify? Remember that
*ħ*= 2m = 1 in this animation. - With the default settings, describe the sum of the two plane waves. Look at the real and imaginary parts of the wave functions to verify your conjecture.
- With the default settings, if this animation were showing the wave function in Region I of a scattering problem, what would
*T*and*R*be? - Now set re2 = -0.9*cos(-5*x-25*t) and im2 = -0.9*sin(-5*x-25*t) for Wave Function 2. What results? Now change the number multiplying plane wave 2 to -0.8, -0.7, -0.6, -0.5, -0.4, -0.3, -0.2, -0.1, and finally 0. What happens to the resulting wave as you change this amplitude? If Wave Function 1 represented Ψ
_{inc}and Wave Function 2 represented Ψ_{refl}, what would happen to the transmission coefficient as you change the amplitude of Wave Function 2?

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