Use Uncertainty Principle to estimate the size of the H atom

[samsun2024]

Homework Statement

In Bohr model of the hydrogen atom, the max. kinetic energy that an electron can have is around 13.6 eV, as it would otherwise not be confined to the atom.

Uncertainty Principle: ΔxΔy ≥ h/2π

a) Find the max. momentum of such confined electron.
b) Use Uncertainty Principle to estimate the size of the H atom.

Homework Equations

The Attempt at a Solution

for part a)
13.6 (eV) ≈ 2.18E-18 (J)

use E = 0.5 m v^2

i found v ≈ 2.2E6 (m/s)

use p = mv

i found the max. momentum of the e- is 2E-24 (Ns)

for part b)

i knew the max. energy the the e- can have is 13.6 eV,
therefore it can have energy from 0 - 13.6 eV,
therefore Δp = 2E-24
use the Uncertainty Principle,

Δx = \frac{h}{2πΔp}

and i put all the number in, and found
Δx = 5.28E-11

however the correct answer is 2.6E-11

i don't know why i need to divide the Δx by half to get the answer.

 

[ehild]

Uncertainty Principle: ΔxΔy ≥ h/2π

An 1/2 is missing: ΔxΔy ≥ \hbar/2

See:
http://en.wikipedia.org/wiki/Uncertainty_principle

ehild

 

[samsun2024]

But my past exam paper said the equation is ΔxΔy ≥ h/2pi ...

 

[ehild]

But my past exam paper said the equation is ΔxΔy ≥ h/2pi ...

Well, there is some confusion in using the uncertainty principle.

ehild

 

[samsun2024]

so, is the way that i am doing correct?
cos i have done few past exam papers and all of them have one question about uncertainty principle...

 

[ehild]

I think your derivation is correct.

ehild

 

[samsun2024]

my worry are gone now, thanks a lot =]