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

  1. 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. jcsd
  3. vela

    vela 12,971
    Staff Emeritus
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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].
  4. 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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