Determining Uncertainty in Position using Heinsburg's Uncertainty Principle

In summary, the problem involves a 52.9 g ball moving at 12.2 m/s with an accuracy of 0.04%. Using the uncertainty of the speed and the equation Δp*Δx = ℏ/2, we can determine the minimum uncertainty in the ball's position to be a very small value due to the small value of Planck's constant.
  • #1
aquella7
1
0

Homework Statement



A 52.9 g ball moves at 12.2 m/s.
If its speed is measured to an accuracy of
0.04%, what is the minimum uncertainty in
its position? Answer in units of m.



Homework Equations



delta x times delta p = 1/2 (h/2pi)


The Attempt at a Solution



I used the uncertainty of the speed to determine the minimum and maximum momentums for the ball. I used these to determine what delta p is equal to (0.6195648) and then substituted that number into the equation to solve for delta x. I used kilograms for the mass, which I believe are the correct units as opposed to just grams. If you could help me determine what I am doing wrong, that would be great! Thanks!

Homework Statement

 
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  • #2
aquella7 said:

Homework Statement



A 52.9 g ball moves at 12.2 m/s.
If its speed is measured to an accuracy of
0.04%, what is the minimum uncertainty in
its position? Answer in units of m.

Homework Equations



delta x times delta p = 1/2 (h/2pi)

The Attempt at a Solution



I used the uncertainty of the speed to determine the minimum and maximum momentums for the ball. I used these to determine what delta p is equal to (0.6195648) and then substituted that number into the equation to solve for delta x. I used kilograms for the mass, which I believe are the correct units as opposed to just grams. If you could help me determine what I am doing wrong, that would be great! Thanks!

Homework Statement


Welcome to PF.

Δp*Δx ≥ ℏ/2

Planck's constant is pretty small, so I'd think you should get a very small number here.

h = 6.6*10−34

And at that you are using ℏ which is h/2π.

Your Δp = .0004*.6344 already, so Δx must be quite small as far as the minimum allowed uncertainty in Δx
 
  • #3


The Heinsburg's Uncertainty Principle states that the product of the uncertainty in position (delta x) and the uncertainty in momentum (delta p) is equal to or greater than 1/2(h/2pi), where h is Planck's constant.

Using this principle, we can determine the minimum uncertainty in position for the given scenario.

First, we need to convert the mass of the ball from grams to kilograms. 52.9 g is equal to 0.0529 kg.

Next, we can calculate the minimum and maximum momentums for the ball using the given speed and uncertainty.

Minimum momentum (p_min) = m*v - (0.04% of m*v)

= (0.0529 kg)*(12.2 m/s) - (0.04%)*(0.0529 kg)*(12.2 m/s)

= 0.6195 kg*m/s

Maximum momentum (p_max) = m*v + (0.04% of m*v)

= (0.0529 kg)*(12.2 m/s) + (0.04%)*(0.0529 kg)*(12.2 m/s)

= 0.6196 kg*m/s

Now, we can use these values to calculate delta p.

delta p = p_max - p_min

= 0.6196 kg*m/s - 0.6195 kg*m/s

= 0.0001 kg*m/s

Finally, we can use the Heinsburg's Uncertainty Principle to calculate the minimum uncertainty in position.

delta x = (1/2(h/2pi)) / delta p

= (1/2(6.626 x 10^-34 J*s / 2*pi)) / (0.0001 kg*m/s)

= 3.98 x 10^-34 m

Therefore, the minimum uncertainty in position is 3.98 x 10^-34 m.
 

1. What is Heisenberg's uncertainty principle?

Heisenberg's uncertainty principle is a fundamental principle in quantum mechanics that states it is impossible to know both the exact position and momentum of a particle at the same time. This principle is based on the idea that the act of measuring one property of a particle will inherently disturb the other property, making it impossible to have precise knowledge of both at the same time.

2. How is uncertainty in position determined using Heisenberg's uncertainty principle?

In order to determine uncertainty in position using Heisenberg's uncertainty principle, the momentum of the particle must be measured. The more precise the measurement of momentum, the larger the uncertainty in position will be. This is because the act of measuring momentum will inevitably disturb the particle's position, making it impossible to know both the exact position and momentum simultaneously.

3. Is Heisenberg's uncertainty principle a limitation of our technology or a fundamental principle of nature?

Heisenberg's uncertainty principle is a fundamental principle of nature, not a limitation of our technology. It is a direct consequence of the quantum nature of particles and the inherent uncertainty in their properties. No matter how advanced our technology becomes, it will always be impossible to know both the exact position and momentum of a particle at the same time.

4. Can Heisenberg's uncertainty principle be applied to macroscopic objects?

No, Heisenberg's uncertainty principle only applies to particles at the quantum level. Macroscopic objects, such as everyday objects, do not exhibit quantum behavior and are not subject to the uncertainty principle. This principle is only relevant in the microscopic world of atoms and subatomic particles.

5. How does Heisenberg's uncertainty principle impact our understanding of the physical world?

Heisenberg's uncertainty principle has had a profound impact on our understanding of the physical world. It challenges our classical intuition and forces us to re-evaluate our understanding of particles and their properties. It also has implications for the fundamental concepts of causality and determinism, as the uncertainty principle suggests that certain properties of particles cannot be determined with absolute certainty.

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