Heisenberg Indeterminacy Principle Assessment #18

[Stella Nguyen 1J]

Hi everyone!

I'm a little confused on where to even start this problem. Could you guys help me with this?

Problem 18: The hydrogen atom has a radius of approximately 0.05 nm. Assume that we know the position of an electron to an accuracy of 1 % of the hydrogen radius, calculate the uncertainty in the speed of the electron using the Heisenberg uncertainty principle.
Comment on your value obtained.

Answer Options:
A. Delta v <= 4 m/s, Delta v is low
B. Delta v >= 10^4 m/s, Delta v is high
C. Delta v <= 8 m/s, Delta v is low
D. Delta v >= 10^8 m/s, Delta v is very high. In knowing with certainty the position of the electron to within 1 % of the hydrogen atom radius the resulting uncertainty in the electron's speed is so high that we essentially do not know its speed at the same time that we know its position.
E. None of the above

Thank you!

[Sydney Lam_2I]

Hi!

So first things first we want to look at the uncertain equation to figure out what we need and what we're looking for. When you look at the equation you realize that you'll need to get know mass and position in order to find the velocity (indeterminacy in momentum*indeterminacy in position)= h/4pi (velocity is found in momentum in which is mass*velocity). They already stated that there is an indeterminacy in the position (numerically 0.01*radius of the hydrogen atom) and remember to multiply that by 2 since it is plus or minus that amount. From there you look up the mass of the hydrogen atom and solve for the velocity.

Hope this helps!

[Asia Yamada 2B]

First, you would convert the radius into meters which would be 5x10^-11m. You have to multiply the radius by 0.01 because the radius is known to an accuracy of 1%. You should get 5x10^-13m. This is the uncertainty in position and from there you can just plug the uncertainty in position into Heisenberg’s Indeterminacy Equation (Δx • Δp ≥ h/4pi). This will give you the uncertainty in momentum. Since p=mv, and you know the mass of an electron (9.109x10^-31kg), you can solve for uncertainty in velocity. I believe the correct answer is D, Δv ≥ 10^8 m/s.