Video Transcript
If a muon and a proton have the same de Broglie wavelength, which particle has the greater speed?
In order to answer this question, we will need to know how to express the de Broglie wavelength of a particle in terms of its speed and also particular properties of that particle like, say, its mass. In general, the de Broglie wavelength of a particle is equal to the Planck constant divided by the particleโs momentum. We know that for particles with mass, like muons and protons, the momentum is related to their speed. So this relationship is very close to giving us what we need. In particular, using what we know about the de Broglie wavelength of our muon and our proton, we can use this formula to relate their momenta and then relate their speeds.
We can write the wavelength of the muon as ๐ minus equals โ divided by ๐ minus and the wavelength of the proton as ๐ plus equals โ divided by ๐ plus. Here, weโve used the subscripts minus and plus to represent the muon and the proton because the charge of the muon is negative and the charge of the proton is positive. We havenโt used letters to avoid confusion with the letters that are already part of our formula. We are told in the question that the two particles have the same de Broglie wavelength. That is, ๐ plus equals ๐ minus. Expressing this equality in terms of the momenta using the de Broglie relation, we have that โ divided by ๐ minus is equal to โ divided by ๐ plus.
Since both sides of this equality have the same numerator, the denominators must also be the same. That is, the muon and the proton must have exactly the same momentum. Now for particles with mass, there are two formulas we have for calculating momenta. For nonrelativistic particles, the momentum is the rest mass times the speed. And for relativistic particles, the momentum is the relativistic gamma factor times the rest mass times the speed. Regardless of which formula applies, we see that particles with larger rest masses have larger momenta and with greater speeds have larger momenta.
Now we recall that the rest mass of a proton is approximately 10 times larger than the rest mass of a muon. Since the proton is more massive than the muon, to have the momentum of the muon be equal to the momentum of the proton, the speed of the proton must be smaller than the speed of the muon. That way, the lesser speed of the proton will offset its greater mass when calculating the momentum. And the converse is also true. The greater speed of the muon will offset its lesser mass. The end result of these two offsetting effects is that we can have two momenta that are the same. We, therefore, conclude that it is the muon that has the greater speed.
