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

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.

## Answers and Replies

BruceW
Homework Helper
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

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