Uncertainty Principle and Fourier Transforms

[Cawb07]

Hey I'm hoping for some help in understanding the relationship between $\delta x \delta p \geq h$ and $\delta k \delta x \approx 1$.

I know the second describes the relationship between span of a wave packet and the frequencies used in a series representation of a Gaussian function.

I understand the uncertainty principle to mean that if $\delta x$ is infinite, then $\delta p$ takes on one value and is represented by a sine or cosine - so the transform looks like a dirac function at $\pm p_0$ for some $\sin p_0x$ or $\cos p_0x$? Now in the other direction, if we know the position of the particle, the momentum has no particular values that it's likely to take on, so does the transform look like $a_n(p) = 0$ or $a_n(p) =$ constant? Or is it more likely to take on smaller values of p...?

Does the uncertainty principle have a similar Gaussian relationship?

 

[DrClaude]

I understand the uncertainty principle to mean that if $\delta x$ is infinite, then $\delta p$ takes on one value and is represented by a sine or cosine - so the transform looks like a dirac function at $\pm p_0$ for some $\sin p_0x$ or $\cos p_0x$? Now in the other direction, if we know the position of the particle, the momentum has no particular values that it's likely to take on, so does the transform look like $a_n(p) = 0$ or $a_n(p) =$ constant? Or is it more likely to take on smaller values of p...?

Be careful. The Fourier transform is complex, so the FT of $\sin p_0x$ or $\cos p_0x$ gives two Dirac deltas, one at $p_0$ and the other at $-p_0$. The FT of a single Dirac delta gives a constant function

Does the uncertainty principle have a similar Gaussian relationship?

A Gaussian wave packet is a minimum uncertainty wave function (it minimises the uncertainty relation).