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Comparison of width of a wavefunction in real space and momentum space 
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#1
Oct1311, 01:57 PM

P: 5

Hello, I have a slight problem with Quantumtheory here.
1. The problem statement, all variables and given/known data I have solved the schrödinger equation in the momentum space for a delta potential and also transfered 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. 2. Relevant equations [itex]\Psi (x) = \sqrt{\kappa}e^{ \kappa x}[/itex] [itex]\Psi (p) = \frac{\sqrt{2 ( \hbar \kappa)^3}}{\sqrt{\pi}(p^2 + (\hbar \kappa)^2)}[/itex] 3. 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. 


#2
Oct1311, 04:02 PM

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Just looking at the functions, you can approximate the characteristic width of the wave functions in position space by using [itex]\kappa x \approx 1[/itex] and in momentum space by using [itex]p / \hbar \kappa \approx 1[/itex].
If you want to be more precise, calculate [itex]\Delta x = \sqrt{\langle x^2 \rangle  \langle x \rangle^2}[/itex] and [itex]\Delta p = \sqrt{\langle p^2 \rangle  \langle p \rangle^2}[/itex]. 


#3
Oct1311, 04:47 PM

P: 5

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.



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