- #1

HAMJOOP

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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.