Wave packet width given a wave function

In summary, the conversation discusses finding the wave packet Ψ(x, t) with given conditions and a system's dispersion relation. The solution for Ψ(x, t) is presented using an integral and it is not clear how to find the wave packet's width. There is also a question about deriving the wave equation from the dispersion equation.
  • #1
Safder Aree
42
1

Homework Statement


Find the wave packet Ψ(x, t) if φ(k) = A for k0 − ∆k ≤ k ≤ k0 + ∆k and φ(k) = 0 for all other k. The system’s dispersion relation is ω = vk, where v is a constant. What is the wave packet’s width?

Homework Equations


[/B]
I solved for Ψ(x, t):

$$\Psi(x,t) = \frac{1}{\pi\sqrt{K_0+\Delta K}} \int_{-\infty}^{\infty} \frac{sin(k(k_0 + \Delta K))}{k} e^{i(kx-hk^2/2m)t}dk$$

The Attempt at a Solution



How would I go about finding the wave packet's width. I'm not even sure what this means. Thank you for any guidance.
 
Last edited:
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  • #2
I don't understand why you have an integral there. You should be able to find an analytical form for Ψ.
 
  • #3
DrClaude said:
I don't understand why you have an integral there. You should be able to find an analytical form for Ψ.

Am I supposed to be deriving the the wave equation from the dispersion equation?
 

Related to Wave packet width given a wave function

1. What is a wave packet width?

A wave packet width refers to the measure of the spatial extent of a wave function. It is a measure of how spread out the wave is in space.

2. How is the wave packet width calculated?

The wave packet width can be calculated by taking the square root of the variance of the wave function. This can be done using mathematical techniques such as integration or Fourier analysis.

3. What factors affect the wave packet width?

The wave packet width can be affected by various factors, including the shape and amplitude of the wave function, as well as the energy and momentum of the particles associated with the wave.

4. Why is the wave packet width important in quantum mechanics?

The wave packet width is important in quantum mechanics as it gives insight into the localization and momentum distribution of particles. It is also used in calculations to determine the probability of finding a particle at a certain position.

5. How does the wave packet width change over time?

The wave packet width can change over time due to the wave function evolving according to the Schrödinger equation. As the wave function spreads out over time, the wave packet width will increase.

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