Solving Wave Packet Spreading Homework

[Math Jeans]

Homework Statement

So I already finished most of this problem, but I'm having trouble figuring out the very last part second part.

The last part of the problem is:
"Finally, take one additional term in the Taylor series expression of $$\omega(k)$$ and show that $$\sigma$$ is now replaced by a complex quantity. Find the expression of the 1/e width of the packet as a function of time for this case and show that the packet moves with the same group velocity as before but spreads in width as it moves. Illustrate this result with a sketch."

I found the complex quantity, and it is the second part I'm having issues with.

Homework Equations

The 1/e width is such that at $$k = k_0 \pm \frac{1}{\sqrt{\sigma}}$$, the amplitude distribution is 1/e of its maximum value $$A(k_0)$$.
The 1/e width is defined as $$\frac{2}{\sqrt{\sigma}}$$.

The complex expression for $$\sigma$$ is $$\sigma - \frac{1}{2}i\omega''_0 t$$

The Attempt at a Solution

Well, the implication of this is that:
$$\frac{2}{\sqrt{\sigma - \frac{1}{2}i\omega''_0 t}}$$

Since this is the 1/e width, I had thought that it should be increasing in order to imply spreading, however, when I graph the real component of this equation with respect to time, I always get a decreasing trajectory for t>0. Would this not imply that it is contracting?

Well, I then went ahead and graphed my wave equation, and I did get some spreading (in that the oscillations remained visible for a larger width, however, the width of each curve was the same, but this is fine due to non-variable frequency).

How do I get my expression for $$\sigma$$ to correctly demonstrate the spreading effect?

thanks,
Jeans

 

[Gokul43201]

Homework Statement

So I already finished most of this problem, but I'm having trouble figuring out the very last part second part.

The last part of the problem is:

Please also type out the entire question, so the reader has the correct context.

 

[Math Jeans]

Please also type out the entire question, so the reader has the correct context.

Actually, you're timing is impeccable because I just figured it out.

The 1/e width refers to width in terms of wave number, so if spreading is in terms of the x-coordinate, then it will become larger as opposed to smaller.