Comparison of width of a wavefunction in real space and momentum space

[BasharTeg]

Hello, I have a slight problem with Quantumtheory here.

Homework Statement

I have solved the schrödinger equation in the momentum space for a delta potential and also transferred it into real space. So now I have to find the correlation between the width of the wavefunction in both spaces (and then motivate it physically) and I am stuck here because I don't even know where to start.

Homework Equations

$\Psi (x) = \sqrt{\kappa}e^{- \kappa |x|}$

$\Psi (p) = \frac{\sqrt{2 ( \hbar \kappa)^3}}{\sqrt{\pi}(p^2 + (\hbar \kappa)^2)}$

The Attempt at a Solution

I was thinking about maybe the uncertainty relation of momentum and space would help here, but I am stuck where to start.


Hope someone can help or give a hint.

 

[vela]

Just looking at the functions, you can approximate the characteristic width of the wave functions in position space by using $\kappa x \approx 1$ and in momentum space by using $p / \hbar \kappa \approx 1$.

If you want to be more precise, calculate $\Delta x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}$ and $\Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2}$.

 

[BasharTeg]

Thanks I will look into it. I guess I have to calculate Δx and Δp since I need a correlation how the width in momentum space affects the width in real space and vice versa.