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Section 8.4: Exploring the Construction of a Packet
Please wait for the animation to completely load.
A localized wave packet can be constructed out of an infinite number of plane wave solutions (which are also energy eigenfunctions). In this Exploration we investigate the process of adding many time-independent plane waves together to resemble a wave packet. We will add these plane waves based on their momentum. We use a Gaussian-shaped weighting function for each plane wave based on its momentum,
Φ(p) = (α1/2/π1/4)exp(−α2(p − p0)2/2). (8.10)
You may choose how many steps, N, are taken in forming the final wave. The larger the N, the larger the number of steps and therefore smaller step widths. Pressing the "Add up the Gaussian" button will automatically do this for you. Checking the box will change the momentum distribution. In the animation, ħ = 2m = 1. Restart.
With the box unchecked and then checked, answer the following questions:
- If there are N steps. how many components are there in the sum?
- As N increases, what happens to the Φ(p) function?
- As N increases, what happens to the position-space wave function? Hint: be systematic. Try N = 2, 4, 6, 8, etc. As you do so also use the "set min/max" button to widen your field of view.
- If N → ∞, what would the Φ(p) function and position-space wave function look like?
- What are α and p0 for the two Φ(p) functions?