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If an electron and a muon have the same speed, which particle has the greater de Broglie wavelength?
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To answer this question, we’ll need to know how the de Broglie wavelength of a particle is related to its speed.
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We will also need to know how the wavelength depends on properties of the particular particle in order to compare electrons and muons.
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The de Broglie wavelength formula has that 𝜆, the de Broglie wavelength of the particle, is equal to ℎ, the Planck constant, divided by 𝑝, the momentum of the particle.
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The Planck constant is, of course, the same for all particles.
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But the momentum depends on both the speed of the particle and its properties, specifically mass.
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In particular, for speeds much smaller than the speed of light, the momentum is mass times velocity, while for speeds approaching that of the speed of light, the momentum is 𝛾 times the mass times the velocity, where 𝛾 is the relativistic factor that depends on velocity.
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𝛾 is defined as one divided by the square root of one minus 𝑣 squared over 𝑐 squared, where 𝑣 is the speed of the particle, and 𝑐 is the speed of light.
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The important observation is that 𝛾 only depends on the speed of the particle, not on any of its other properties.
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So any two particles with the same speed have the same relativistic 𝛾 factor.
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Now, remember, our electron and muon have the same speed.
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This means that the particle with the greater momentum will be the one with the greater mass.
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Because the other factors used to calculate momentum, whether speed for nonrelativistic speeds or speed times 𝛾 for relativistic speeds, will be the same for both particles since they have the same speed.
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Turning back to the de Broglie relation, wavelength is inversely proportional to momentum.
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And this means that the particle with the smaller momentum will be the particle with the greater de Broglie wavelength.
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So we are looking for the particle with the smallest momentum, which, because our two particles are moving at the same speed, will be the particle with the smaller rest mass.
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At this point, we recall that muons are significantly more massive than electrons by a factor of about 200.
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So of our two particles, it is the electron that has the greater de Broglie wavelength.