Uncertainty Principle and dart of mass

• teme92
In summary, the conversation discusses the uncertainty principle and its application to a dart being dropped from a height. The formula for the time it takes to drop is given as well as the formula for the distance at which the dart hits the ground with uncertainty. The conversation then references a formula for the uncertainty in the lateral position, ##\Delta x##, and the minimum possible value for the distance, ##R##, which is derived from the uncertainty principle. The source of the formula for ##\Delta x## is not clear.
teme92

Homework Statement

A dart of mass ##m## is dropped from a height ##l##. Formulate the uncertainty principle and estimate the minimum limitations, set by the uncertainty principle, of the accuracy that can be achieved in the lateral ##x## position after falling, given an original uncertainty ##\Delta x##, and no original uncertainty in y.

The Attempt at a Solution

The time to drop is ##\sqrt{2l/g}##. With uncertainty the dart hits the ground at a distance ##R## where:

##R=\Delta x+\sqrt{2l/g}\space \Delta p/m \geq\Delta x +\sqrt{2l/g}\space \hbar/2m\Delta x##

This next part I found from a book but I don't understand how its got from above:

##\Delta x=(l\hbar^2/2m^2g)^{\frac{1}{4}}##

with ##R\geq(8l\hbar^2/m^2g)^{\frac{1}{4}}##

Could anyone clarify how ##\Delta x## and ##R## were found?

Last edited:
Please review the statement of the problem, it seems that the last sentence is not finished and maybe we miss important data.

Hey soarce I edited the omission.

Your derivation seems correct. Can you give us the reference from where you took the formula for ##\Delta x##?

1. What is the Uncertainty Principle?

The Uncertainty Principle, also known as Heisenberg's Uncertainty Principle, is a fundamental principle in quantum mechanics that states that it is impossible to know the exact position and momentum of a particle at the same time. This is due to the inherent probabilistic nature of particles at the quantum level.

2. How does the Uncertainty Principle relate to the dart of mass?

The Uncertainty Principle applies to all particles, including the dart of mass. This means that it is impossible to know the exact position and momentum of the dart of mass at the same time. This is due to the fact that the dart of mass is made up of particles at the quantum level.

3. Can the Uncertainty Principle be violated?

No, the Uncertainty Principle is a fundamental principle in quantum mechanics and has been extensively tested and proven to be true. It is a key aspect of our understanding of the behavior of particles at the quantum level.

4. How does the Uncertainty Principle impact our everyday lives?

The Uncertainty Principle may not have a direct impact on our everyday lives, as it is most noticeable at the quantum level. However, it does play a crucial role in the development of technologies such as transistors and lasers, which have revolutionized modern technology.

5. What are the practical applications of the Uncertainty Principle?

The Uncertainty Principle has practical applications in fields such as quantum computing, cryptography, and medical imaging. It also helps us understand the behavior of subatomic particles and the fundamental laws of nature at the quantum level.

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