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

[HAMJOOP]

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.

 

[BruceW]

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

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

 

[Paradox101]

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