Do Muons and Electrons with the Same Kinetic Energy Share Identical Wavelengths?

AI Thread Summary
Muons and electrons with the same kinetic energy do not share identical wavelengths due to their differing masses. The relationship between energy and wavelength, E=hc/λ, applies primarily to massless particles, while the kinetic energy for massive particles is given by p²/(2m). The discussion emphasizes that the total relativistic energy, E, must be considered for high-speed particles, which complicates direct comparisons. Additionally, the de Broglie wavelength formula indicates that λ varies with mass, reinforcing that muons and electrons cannot have the same wavelength even if their kinetic energies are equal. Understanding these principles is crucial for accurately addressing the problem.
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Homework Statement



Consider the statement below made by a student: "Muons have a higher mass than electrons, but because the energy, E, is related to the wavelength by E=hc/\lambda, muons that have the same kinetic energy as electrons will also have the same wavelength."

Do you agree or disagree with this statement? Explain your reasoning.

Homework Equations



\lambda=h/p (de Broglie wavelength)

The Attempt at a Solution



The statement seems wrong to me. If you substitute in for \lambda in the first equation, you get cp, but kinetic energy is p^2/(2m) and those two can't be the same (solving for c gives c=v/2).

I'd never seen the first equation before, but looking in my textbook it looks like E is the change in energy of an atom when a photon is absorbed or emitted and I don't know how you could apply it to an electron/muon (can you?)
 
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\lambda is different for different objects, so why should E be the same?

E is the total relativistic energy, not p^2/2m. Also, E = hf, where f is the frequency associated with the de Broglie wave.

Just do the simple math. Also, reading up on de Broglie wavelength would be a good idea.

EDIT:
-----

Are you talking about the muons and the electrons having the same energy, by any chance? This is not mentioned in the problem, so I assumed not.

Also, E=hc/\lambda is valid only for massless particles which travel at speed c.
 
Last edited:
>\lambda is different for different objects, so why should E be the same?
The question is assuming you have two particles with the same kinetic energy.

>For high speed particles, E is the total relativistic energy, not p^2/2m.
Ok, but I still doubt that cp=Ek for an electron. Is that wrong?

>Also, E = hf, where f is the frequency associated with the de Broglie wave.
And if f=v/\lambda, then E = vp. But v can't be c for an electron, so the equation can't work here (?)
 
Sorry, you posted while I was replying. Yes, that's the problem:
"muons that have the same kinetic energy as electrons..."

>Also, is valid only for massless particles which travel at speed c.

I'm guessing this is essentially the answer to the question
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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