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.
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.
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