If an electron and a muon have the same speed, which particle has the greater de Broglie wavelength?
To answer this question, we’ll need to know how the de Broglie wavelength of a particle is related to its speed. We will also need to know how the wavelength depends on properties of the particular particle in order to compare electrons and muons. 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. The Planck constant is, of course, the same for all particles. But the momentum depends on both the speed of the particle and its properties, specifically mass. 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.
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. The important observation is that 𝛾 only depends on the speed of the particle, not on any of its other properties. So any two particles with the same speed have the same relativistic 𝛾 factor. Now, remember, our electron and muon have the same speed. This means that the particle with the greater momentum will be the one with the greater mass. 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.
Turning back to the de Broglie relation, wavelength is inversely proportional to momentum. And this means that the particle with the smaller momentum will be the particle with the greater de Broglie wavelength. 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. At this point, we recall that muons are significantly more massive than electrons by a factor of about 200.
So of our two particles, it is the electron that has the greater de Broglie wavelength.