Chapter 1 #43 [ENDORSED]

[MichelleTran3I]

What is the minimum uncertainty in the speed of an electron con ned to within a lead atom of diameter 350. pm? Model the atom as a one-dimensional box with a length equal to the diameter of the actual atom.

What does this question ask for specifically when it says to model the atom as a one-dimensional box?

[Leon Popa]

Hi Michelle,

I did some research into the one-dimensional box concept - here's what I understood.

Apparently, the problem is asking us to the imagine the atom itself as a box or container in which a single particle is confined. In this case that particle is the electron. Ultimately we are finding the translational motion of the electron in this 1-D box from which it cannot escape. The box is like an infinitely deep well, and the electron can never get out. This seems to be analogy for the electron's motion within an atom. Because the atom has a diameter of 350. pm, we should imagine that the box has a diameter of 350. pm.

In the end I don't think the box analogy has much to do with finding the solution. If anything though, hope this description was clarifying.

- Leon.

[Remi Lathrop 1G]

As a follow-up question about this question, in the solution manual they use an h value of (1.054 x 10^-34 Js) which is not the usual value we have been using in class? Can someone explain this difference?

[Diego Zavala 2I]

The value for h is still
Js, but in Heisenberg's Indeterminacy Equation which is equal to
1.054x

[Diego Zavala 2I]

* 6.626x10^-34 Js

[Humza_Khan_2J]

The h is different in the Uncertainty Equation, it's called "h bar." "H bar" is h/2pi. This is why it's "h bar"/2 instead of h/4pi.

[Alissa Stanley 3G]

Can someone explain why you do h/2pi? Are there any other situations that we need to be aware of where h is divided by 2pi?

[Diego Zavala 2I]

Alissa, I believe the 2pi comes from derivation of the formula that is outside the scope of the course. However, you may prefer to read "Uncertainty Principle Equation" in wikipedia for more information about how the equation was derived.