Particle through a slit Uncertainty of position

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SUMMARY

The discussion revolves around calculating the spread of a beam of 1.0-g particles shot at 100 m/s through a 2.0 mm diameter hole, using the uncertainty principle defined by the equation ΔxΔp ≥ ħ/2. The user attempts to derive the uncertainty in position (Δx) and arrives at a value of 5.275 x 10-34 m, indicating confusion regarding the application of the hole diameter and the relevance of concepts like Fraunhofer Diffraction and de Broglie waves. The discussion highlights the need for clarity on how to incorporate these factors into the uncertainty calculation.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically the uncertainty principle.
  • Familiarity with the concept of momentum (p = mv) in physics.
  • Knowledge of diffraction patterns, particularly Fraunhofer Diffraction.
  • Basic comprehension of wave-particle duality and de Broglie waves.
NEXT STEPS
  • Study the implications of the Heisenberg Uncertainty Principle in practical scenarios.
  • Research Fraunhofer Diffraction and its mathematical formulation.
  • Explore de Broglie wavelength calculations for particles of varying masses.
  • Investigate the relationship between particle size, velocity, and beam spread in quantum mechanics.
USEFUL FOR

Students and educators in physics, particularly those focusing on quantum mechanics and wave-particle duality, as well as anyone interested in the practical applications of the uncertainty principle in particle physics.

darkfall13
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Homework Statement



An air rifle is used to shoot 1.0-g particles at 100 m/s through a hole of diameter 2.0 mm. How far from the rifle must an observer be to see the beam spread by 1.0 cm because of the uncertainty principle? Compare this answer with the diameter of the Universe (2x10^26 m)

Homework Equations



[tex]\Delta{x}\Delta{p} \geq \frac{\bar{h}}{2}[/tex]

The Attempt at a Solution



If we solve for [tex]\Delta{x}[/tex] and expand [tex]p[/tex] to [tex]mv[/tex] and plug in we receive [tex]\Delta{x} = 5.275\cdot{10}^{-34}[/tex]. So I believe I'm on the wrong track, how do we use the hole diameter (if even needed) and what else should I look at to begin this problem? Fraunhofer Diffraction? de Broglie waves?

Thank you!
 
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