## Section 9.5: Exploring the Addition of Two Plane Waves

re1 | im1

re2 | im2

click to see the real and imaginary parts of the wave functions.

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.

1. 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.
2. 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.
3. 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?
4. 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 Ψinc and Wave Function 2 represented Ψrefl, what would happen to the transmission coefficient as you change the amplitude of Wave Function 2?

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