Heisenberg Uncertainty Principle homework Question

[kongieieie]

Sorry about not using symbols but I haven't learned how to do that yet.

1. Homework Statement

A woman is on a ladder of height H. She drops small rocks of mass m toward a point target on the floor.

Show that according to the Heisenberg Uncertainty Principle, the average miss distance must be at least

delta(x final) = sqrt [h/pi*m] * sqrt sqrt [2H/g]

where h is the Planck's constant,
pi is 3.14
m is the mass of the rock
H is the height from which the rock is dropped
g is the acceleration due to gravity.

Assume that delta(xfinal ) = delta(x initial) + (delta(v))*t
Also justify the assumption.

2. Homework Equations

delta(x) * delta(p) < or = [h/4*pi]

delta(p) = m * delta(v)

v= u +at
s= 0.5(u + v)t
s= ut +0.5at^2
v^2= u^2 +2as

3. The Attempt at a Solution

I did a couple of substitutions and got something like t=sqrt[2H/g] and delta(v)=sqrt[2gH] but I can't seem to get the equation needed. Tried for 2 hours and can't seem to understand it all. Please help? Or at least give some hints. Also, I don't know how to justify the assumption above.

 

[kongieieie]

Nvm. I found a similar answer in another thread.

 

[tomcue]

As a scientist, it is important to understand the principles and equations behind a problem before attempting to solve it. In this case, the Heisenberg Uncertainty Principle states that the product of the uncertainties in position and momentum of a particle must be greater than or equal to Planck's constant divided by 4 times pi. This principle applies to all particles, including the small rocks being dropped by the woman on the ladder.

To solve this problem, we can start by considering the initial position and momentum of the rock. The initial position is at the top of the ladder, at a height H, and the initial momentum is zero as the rock is not moving. As the rock falls, its position and momentum will change. However, we cannot know both the exact position and momentum of the rock at any given time due to the Heisenberg Uncertainty Principle.

Based on the given equations, we can calculate the uncertainty in position (delta(x)) and momentum (delta(p)) as follows:

delta(x) = delta(v) * t = sqrt[2gH] * t
delta(p) = m * delta(v) = m * sqrt[2gH]

Substituting these values into the Heisenberg Uncertainty Principle equation, we get:

delta(x) * delta(p) = (sqrt[2gH] * t) * (m * sqrt[2gH]) = 2gHmt

Since this must be greater than or equal to h/4*pi, we can set up the following inequality:

2gHmt >= h/4*pi

Solving for t, we get:

t >= h/(8*g*H*m*pi)

This is the minimum amount of time required for the rock to reach the target on the floor. Now, we can use the equations of motion to calculate the average miss distance (delta(xfinal)).

We know that the average miss distance is equal to the initial position (H) plus the product of the average velocity (delta(v)) and the minimum time (t) required for the rock to reach the target.

Therefore, delta(xfinal) = H + (delta(v)) * t

Substituting our previously calculated values for t and delta(v), we get:

delta(xfinal) = H + (sqrt[2gH]) * (h/(8*g*H*m*pi)) = H + (h/(8*m*pi))

Simplifying further, we get:

delta