How Does Quantum Uncertainty Affect Dust Particle Trajectories?

[James98765]

Homework Statement

1.0 micrometer diameter dust particles (m=1.0 * 10^-15 kg) are in a vacuum chamber. The dust particles are released from rest above a 1.0 micrometer diameter hole, fall through the hole, and land on a detector at distance d below.

Part A:
If the particles were purely classical, they would all land in the same 1.0 micrometer diameter circle. But quantum effects don't allow this. If d = 1.0 m, by how much does the diameter of the circle in which most dust particles land exceed 1.0 micrometers? Is this increase in diameter likely to be detectable?

Part B:
Quantum effects would be noticeable if the detection circle diameter increased by 10% to 1.1 micrometers. At what distance d would the detector need to be placed to observe this increase in the diameter?

Homework Equations

Heisenberg Uncertainty:
dX*dV<=h/2

The Attempt at a Solution

In all honesty, I haven't been able to get started on this problem. I'm not exactly sure how the diameter of the detection circle relates to the vertical distance d.

Should I be using a single slit diffraction analysis to explain the situation?
I considered it but determined that this wasn't the correct model. If single slit diffraction was occurring, wouldn't the dust particles be detected anywhere on the detection screen, not just within a certain diameter.

Does the Heisenberg uncertainty principle explain the larger detection circle diameter?
I believe that dX (uncertainty of position) could be equal to the detection circle diameter. For me to determine dX, I must know dV (uncertainty of velocity). But since I don't know dV, how can I find dX?

Those are basically my two attempts to this problem and both have been unsuccessfull. Am I way off, and if so, can I get a suggesstion? I'm not looking for an entire solution to the problem. I just need a starting point.

Thanks a lot for your help.

 

[phinds]

My fairly uninformed belief is that while the probability of a particle landing at the radius of a HUGE circle is small, it is not zero (and it doesn't matter how big you define HUGE to be, it's just that the bigger it is, the smaller the probability).

 

[Redbelly98]

Homework Equations

Heisenberg Uncertainty:
dX*dV<=h/2

Actually, the Heisenberg uncertainty relates position and momentum, not velocity. So the equation is slightly different than what you wrote.

The Attempt at a Solution

In all honesty, I haven't been able to get started on this problem. I'm not exactly sure how the diameter of the detection circle relates to the vertical distance d.

Should I be using a single slit diffraction analysis to explain the situation?
I considered it but determined that this wasn't the correct model. If single slit diffraction was occurring, wouldn't the dust particles be detected anywhere on the detection screen, not just within a certain diameter.

Does the Heisenberg uncertainty principle explain the larger detection circle diameter?
I believe that dX (uncertainty of position) could be equal to the detection circle diameter. For me to determine dX, I must know dV (uncertainty of velocity). But since I don't know dV, how can I find dX?

dx is the uncertainty in position as the dust particle falls through the 1.0 micrometer hole. Technically, there is zero uncertainty if a sphere falls through a hole of the exact same size, but that would lead to infinite uncertainty in the momentum and this problem would make no sense. So, it seems they want you to use 1.0 micrometers as the uncertainty dx. From that you can use the Heisenberg relation to find the uncertainty in momentum.

 

[James98765]

Thanks for your help. After reading your suggestion I'm relatively certain you're correct.