Wave-Particle Duality

In summary, to find the kinetic energy at which a particle's DeBroglie wavelength equals its Compton Wavelength, you need to use the relativistic expression for kinetic energy and set the equations for DeBroglie and Compton wavelengths equal to each other. The correct expression for kinetic energy is E = sqrt{(mc^2)^2 + (mc^2)^2}.
  • #1
hxcguitar101
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Homework Statement


At what Kinetic energy will a particle's debroglie wavelength equal its Compton Wavelength


Homework Equations


DeBroglie
λ = h/mv

Compton
λ = h/mc

The Attempt at a Solution



Setting the two equations equal to each other, I got v = c, then said KE = (1/2)mc^2, but somehow that just doesn't sound right. What do you think?
 
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  • #2
Nope, that's not right, as you suspected. The DeBroglie wavelength is actually [itex]\lambda = h/p[/itex] where p is the momentum of the particle. You need to use the relativistic expression for the kinetic energy to get the correct answer.
 
  • #3
So, it's when p = mc, so would you use E = Sqrt((pc)^2+(mc^2)^2) = sqrt((mc)^2+(mc^2))?
 
  • #4
Essentially, yes, but you need to get the algebra right. You have [itex]p = mc[/itex] so [itex]pc = mc^2[/itex] and
[tex]E=\sqrt{(pc)^2+(mc^2)^2} = \sqrt{(mc^2)^2+(mc^2)^2} = \cdots[/tex]Also, remember E gives the total energy, not the kinetic energy.
 
  • #5


Your attempt at a solution is correct. When the velocity of the particle is equal to the speed of light, the kinetic energy will be equal to its rest energy (m*c^2). This is because at this velocity, the particle is behaving as a massless particle, and its DeBroglie wavelength will be equal to its Compton wavelength. This is an interesting aspect of wave-particle duality, where a particle can exhibit both particle-like and wave-like behavior depending on its properties and conditions. This duality is a fundamental concept in quantum mechanics and has been experimentally verified through various experiments.
 

1. What is wave-particle duality?

Wave-particle duality is a principle in quantum mechanics that states that particles can exhibit both wave-like and particle-like properties depending on how they are observed or measured.

2. How was wave-particle duality discovered?

The concept of wave-particle duality was first proposed by physicist Louis de Broglie in 1924, based on the work of Max Planck and Albert Einstein on the photoelectric effect and the quantization of energy.

3. What experiments demonstrate wave-particle duality?

One of the most famous experiments that demonstrated wave-particle duality was the double-slit experiment, where particles (such as electrons) were observed to behave as both waves and particles when passing through a barrier with two slits.

4. What are the implications of wave-particle duality?

The concept of wave-particle duality challenges our traditional understanding of the nature of matter and raises questions about the fundamental nature of reality. It also plays a crucial role in our understanding of quantum mechanics and the behavior of subatomic particles.

5. How is wave-particle duality relevant to everyday life?

Although the effects of wave-particle duality are typically only observable on a microscopic scale, it has important practical applications in fields such as electronics, where the behavior of electrons as both waves and particles is crucial to understanding and developing new technologies.

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