Use Uncertainty Principle to estimate the size of the H atom

In summary, the conversation discusses the Bohr model of the hydrogen atom and the Uncertainty Principle. The maximum kinetic energy of an electron in this model is around 13.6 eV, and the maximum momentum is found to be 2E-24 (Ns). The Uncertainty Principle is then used to estimate the size of the hydrogen atom, with a resulting value of Δx = 5.28E-11. However, the correct answer is 2.6E-11, as there is a missing 1/2 in the equation. The student's derivation is deemed correct, despite some confusion in using the Uncertainty Principle.
  • #1
samsun2024
5
0

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 = [itex]\frac{h}{2πΔp}[/itex]

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.
 
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  • #3
But my past exam paper said the equation is ΔxΔy ≥ h/2pi ...
 
  • #4
samsun2024 said:
But my past exam paper said the equation is ΔxΔy ≥ h/2pi ...

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

ehild
 
  • #5
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...
 
  • #6
I think your derivation is correct.

ehild
 
  • #7
my worry are gone now, thanks a lot =]
 

1. How does the Uncertainty Principle relate to the size of the H atom?

The Uncertainty Principle states that it is impossible to know both the exact position and momentum of a particle at the same time. This applies to the electrons in the H atom, which constantly move and therefore have uncertain positions. This uncertainty in position makes it difficult to accurately measure the size of the H atom.

2. Can the Uncertainty Principle be used to directly measure the size of the H atom?

No, the Uncertainty Principle cannot be used to directly measure the size of the H atom. As mentioned before, it only states that there is a limit to how accurately we can know the position and momentum of a particle, not the exact size of the atom.

3. How can the Uncertainty Principle be used to estimate the size of the H atom?

Since we cannot directly measure the size of the H atom using the Uncertainty Principle, we can use it to estimate the size by considering the range of possible positions for the electron in the atom. The larger the range of possible positions, the larger the atom is likely to be.

4. What other factors can affect the accuracy of using the Uncertainty Principle to estimate the size of the H atom?

There are other factors that can affect the accuracy of this estimation, such as the energy level of the electron and the shape of the electron's orbital. These factors can also affect the uncertainty in the electron's position and therefore impact the estimation of the atom's size.

5. Can the Uncertainty Principle be used to estimate the size of other atoms or particles?

Yes, the Uncertainty Principle can be applied to any particle, including other atoms, to estimate their size. However, the accuracy of the estimation may vary depending on the specific properties of the particle.

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