Math Problem Statement
A particle of mass 0.5 MeV/c 2 has kinetic energy 100 eV. Find its de -Broglie wavelength and group velocity, where c is the velocity of light.
Solution
To find the de Broglie wavelength and group velocity of a particle given its mass and kinetic energy, we'll follow these steps:
Given Data:
- Mass () = 0.5 MeV/c²
- Kinetic Energy () = 100 eV
- Speed of light () = m/s
1. Convert mass from MeV/c² to kg:
The conversion factor is .
2. Convert kinetic energy from eV to joules:
The conversion factor is .
3. Find the particle’s momentum :
The non-relativistic kinetic energy formula can be rearranged to find :
4. Calculate the de Broglie wavelength :
The de Broglie wavelength is given by: where is Planck’s constant ().
5. Calculate the group velocity :
The group velocity for a non-relativistic particle can be approximated by the classical velocity :
Summary:
- de Broglie wavelength:
- Group velocity:
Do you want details or have any questions?
Here are some related questions:
- How is the de Broglie wavelength affected by the particle's mass?
- What would happen to the de Broglie wavelength if the kinetic energy were doubled?
- How does the de Broglie wavelength relate to the particle's wave-particle duality?
- What is the significance of the group velocity in wave mechanics?
- How would the group velocity change if the mass of the particle were halved?
- What is the difference between phase velocity and group velocity?
- How is the concept of de Broglie wavelength used in electron microscopy?
- Can you explain the concept of wave-particle duality with another example?
Tip: When dealing with particle physics, always ensure the units are correctly converted to maintain consistency in calculations.
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Math Problem Analysis
Mathematical Concepts
Quantum Mechanics
De Broglie Wavelength
Group Velocity
Formulas
de Broglie wavelength formula: λ = h / p
Non-relativistic kinetic energy: KE = p^2 / (2m)
Group velocity approximation: vg ≈ √(2KE/m)
Theorems
de Broglie hypothesis
Suitable Grade Level
Advanced Physics
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