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Use uncertainty principle to obtain the result of Bohr's Model

  1. Sep 29, 2013 #1
    Problem
    Find the minimum energy of the hydrogen atom by using uncertainty principle

    a. Take the uncertainty of the position Δr of the electron to be approximately equal to r
    b. Approximate the momentum p of the electron as Δp
    c. Treat the atom as a 1-D system


    My step

    1. Δr Δp ≥ h/4(pi)
    Δp ≥ h/4(pi)r

    2. Total energy E = (p^2 /2m) - ke^2 /r

    ≥ (h^2/8(pi)^2 m r^2) - (ke^2)/r

    3. rearranging the term

    (Emin)r^2 + (ke^2)r - (h^2 /8(pi)^2 m ) = 0

    Require Δ = 0 for the quadratic equation

    I obtain E = -54.7 eV ≠ -13.6 eV


    If I replace Δr by (1/2)Δr, I can obtain the correct result. But I don't know why.
     
  2. jcsd
  3. Oct 1, 2013 #2

    BruceW

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    Homework Helper

    you have p=h/4(pi)r, but then you seem to say p^2 /2m = h^2/8(pi)^2 m r^2 there is a mistake in this step
     
  4. Jan 23, 2015 #3
    woah there! you can't shift Δr to the RHS in case there is a > sign,right? if Δr.Δp=h/4π, ⇒Δp=h/4πr
     
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