Basic Heisenberg uncertainty principle stuff

In summary: I'm just starting to learn LaTeX so I'm not too familiar with all the commands yet. In summary, the uncertainty in the momentum of the bullet perpendicular to its motion would be determined by using the equation ##\Delta y \Delta p_{y}\approx\frac{\hbar}{2}##. For question 2, the maximum deviation from a pinpoint target 300 m away would be determined by using the uncertainty in the y component velocity found using ##p=mv## and the distance traveled in time ##t=\frac{d}{v}##.
  • #1
rtsswmdktbmhw
38
2

Homework Statement


A bullet is shot from a rifle.

1) if the position of the centre of mass of the bullet perpendicular to its motion is known to have an accuracy of 0.01 cm, what is the corresponding uncertainty in its momentum?

2) If the accuracy of the bullet were determined only by the uncertainty principle, by how much might the centre of the bullet miss a pinpoint target 300 m away?

Homework Equations


##\Delta x \Delta p_{x}\geq\frac{\bar h}{2}## (or ##\Delta x \Delta p_{x}\approx\frac{\bar h}{2}##?)

The Attempt at a Solution


1) If the uncertainty is perpendicular to the motion, then can't we have 0 uncertainty? But the question asks for 'corresponding uncertainty' so it sounds like I'm supposed to get some answer. I'm not sure whether to plug into the equation and get a number. But that doesn't sound right because then I would be getting ##\Delta p_{y}## if using 0.01 cm, right?

2) Since the position of the bullet has an uncertainty of 0.01 cm, would this simply be how much the centre of the bullet might miss the pinpoint target?

Sorry if there is anything obvious or simple. I have just done what seems like a crash course on introductory QM in the past week or two. Things might not have clicked properly yet...
 
Physics news on Phys.org
  • #2
##\Delta x \Delta p_{x}\geq\frac{\hbar}{2}## (or ##\Delta x \Delta p_{x}\approx\frac{\hbar}{2}##?)

(tip: in LaTeX, use "\hbar" for ##\hbar##.)

That's just along the x direction. There are exactly similar equations for the y and z directions: ##\Delta y \Delta p_y \geq\frac{\hbar}{2}## and ##\Delta z \Delta p_z \geq\frac{\hbar}{2}## Remember momentum is a vector quantity, with x, y and z components.

If you assume the gun is aimed along the x direction, then you can use either y or z for the perpendicular direction.

1) if the position of the centre of mass of the bullet perpendicular to its motion is known to have an accuracy of 0.01 cm, what is the corresponding uncertainty in its momentum?

Here the "corresponding uncertainty in its momentum" refers to the component pependicular to the direction that the gun is aimed.
 
  • #3
Oh, okay, so I should be finding the y component uncertainty. So it's simple plug and chug with the equation? Is it correct to replace the inequality with the approximately equal sign, or should I be leaving it as an inequality?

For question 2, then, is it a matter of finding y component velocity using p=mv, then simply finding the maximum inaccuracy (deviation from pinpoint target) at 300 m using dist=vel*time?
 
  • #4
Right on both counts. Strictly speaking it should be kept as an uncertainty, although most of the time we're just interested in the order of magnitude (power of ten), so lots of times people use the "approximately equals." The actual uncertainty is likely to be somewhat bigger than what ##\hbar/2## gives, but not as much as ten times bigger.
 
  • #5
Okay, thanks for your explanations! And also for the tip for ##\hbar##.
 

1. What is the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle is a fundamental principle in quantum mechanics that states that the more precisely the position of a particle is known, the less precisely its momentum can be known, and vice versa. In other words, it is impossible to know both the position and momentum of a particle with absolute certainty.

2. How does the Heisenberg uncertainty principle relate to the concept of measurement?

The Heisenberg uncertainty principle applies to the act of measurement itself. When a quantum system is measured, the act of measurement disturbs the system, making it impossible to know its exact properties. This is due to the fact that the act of measurement requires an exchange of energy, which can affect the position and momentum of the particle being measured.

3. Does the Heisenberg uncertainty principle apply to all particles?

Yes, the Heisenberg uncertainty principle applies to all particles, including subatomic particles such as electrons, protons, and neutrons. It also applies to larger particles, but the effects are so small that they are not noticeable in everyday life.

4. How is the Heisenberg uncertainty principle related to the wave-particle duality of particles?

The Heisenberg uncertainty principle is closely linked to the wave-particle duality of particles. It states that particles can exhibit both wave-like and particle-like behavior, and the uncertainty principle is a result of this duality. The more accurately the position of a particle is known, the more localized it becomes, resembling a particle. Conversely, the less accurately its position is known, the more spread out it becomes, resembling a wave.

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

The Heisenberg uncertainty principle has had a significant impact on our understanding of the physical world, particularly in the field of quantum mechanics. It has challenged traditional notions of causality and determinism, showing that the behavior of particles at the subatomic level is inherently unpredictable. It also plays a crucial role in the development of technologies such as microscopes and MRI machines, which rely on the principles of quantum mechanics to function.

Similar threads

  • Introductory Physics Homework Help
Replies
7
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
664
  • Introductory Physics Homework Help
Replies
21
Views
1K
  • Quantum Interpretations and Foundations
Replies
23
Views
4K
  • Introductory Physics Homework Help
Replies
2
Views
1K
Replies
2
Views
209
Replies
10
Views
1K
  • Quantum Physics
Replies
16
Views
863
Replies
3
Views
282
  • Introductory Physics Homework Help
Replies
9
Views
5K
Back
Top