Fourier Transforms: Proving Proportionality

[smallgirl]

1. We consider the on shell wave packet:
$$\varphi(t,x)=\int\frac{dk}{2\pi}exp(-\frac{(k-k_{0})^{2}}{\Delta k^{2}}+ik(t-x))dk$$

I need to show it is proportional to:
$$exp(ik_{0}(t-x)-\frac{\triangle k^{2}}{4}(t-x)^{2})dk$$
through a Fourier transform of the gaussian


3. I used a Fourier transform of the form e^(ikx) but this doesn't seem to give me the right answer as I end up with something proportional to $$exp(-\frac{(k-k_{0})^{2}}{\triangle k^{2}}+ikt)dk$$ before integrating

 

[vela]

Show us what you think the integral for $\varphi(t,k)$ is.

 

[smallgirl]

Solved it! :-)...

However I now need to solve this:

$$\int\frac{dk}{2\pi}exp(-\frac{(k-k_{o})^{2}}{\triangle k^{2}}+ik(pt-x)$$


where $$p=1-\frac{h_{00}}{2}-h_{01}-\frac{h_{11}}{2}$$

by using Fourier transforms

 

[smallgirl]

Solved this one too now :-)

Not sure how to graph it though...

 

[paranormal]

I can provide a response to this content by explaining the concept of Fourier transforms and how they can be used to prove proportionality between two functions.

Fourier transforms are mathematical tools used to analyze functions in the frequency domain. They allow us to decompose a function into its individual frequency components, providing a better understanding of the function's behavior. In this case, we are interested in proving proportionality between two functions, which means that one function is a constant multiple of the other.

To prove proportionality between the two functions given, we can use the property of Fourier transforms that states that the transform of a product of two functions is equal to the convolution of their individual transforms. In other words, if we take the Fourier transform of the product of the two functions, we should get a result that is proportional to the transform of each individual function.

In this case, we can take the Fourier transform of the on shell wave packet function, which is given by:

\hat{\varphi}(\omega, k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \varphi(t,x) e^{-i\omega t} e^{-ikx} dt dx

Substituting the given function for \varphi(t,x) and simplifying, we get:

\hat{\varphi}(\omega, k) = \frac{\Delta k}{2\pi} e^{-\frac{\Delta k^2}{4}(\omega - k_0)^2}

On the other hand, the Fourier transform of exp(ik_0(t-x)-\frac{\triangle k^{2}}{4}(t-x)^{2})dk is given by:

\hat{\psi}(\omega, k) = \sqrt{2\pi} \delta(k-k_0) e^{-\frac{\Delta k^2}{4}(\omega - k_0)^2}

where \delta(k-k_0) is the Dirac delta function.

Comparing the two transforms, we can see that they are proportional to each other, with a constant factor of \sqrt{2\pi}\Delta k. This proves that the two functions are indeed proportional to each other.

In conclusion, by using the property of Fourier transforms, we can prove proportionality between two functions in the frequency domain. This