Relativistic Group Velocity Calculation

[neelakash]

Homework Statement

we are given that an electron and a proton have the same KE.We are to compare their phase and group velocity...

Homework Equations

The Attempt at a Solution

K=(γ₁-1)m₁c²=(γ₂-1)m₂c²
Now, I found it very problematic to extract the ratio of v₁/v₂ in terms of m₁ and m₂
So,I expanded the γ s binomially where the major contribution comes from the first few terms...It follows that group velocity if proton is much less than that of the electron...

Please tell me if I am correct and sggest any other possible ways...

Regards,
neelakash

 

[Meir Achuz]

Don't use v.
Use E=T+M and p=\sqrt{T^2+2M}.

 

[neelakash]

but how would you compae between the group velocities?

Don't use v.
Use E=T+M and p=\sqrt{T^2+2M}.

I hope your formula is pc=√[K(K+2mc²)] where K is the KE

But, p=γmv...so that you are to know γ if you want to know v
γ s are different for e and p...

 

[Meir Achuz]

but how would you compare between the group velocities?



I hope your formula is pc=√[K(K+2mc²)] where K is the KE

But, p=γmv...so that you are to know γ if you want to know v
γ s are different for e and p...

Sorry, I should have had T=\sqrt{T^2+2MT}.
I use T for KE, which is more common, and relativistic units with c absent.
v_P=E/p, and v_g=dE/dp=p/E.
You don't need gamma or v, but they are gamma=E/M and v=p/E.

 

[neelakash]

Buddy,what you are using seems not quite effective here...Remember we are to compare between group and phase velocities of an e and a p whose KE are the same...And you have not used the fact that their KE are the same...

I am referring to another method...It is no approximation..stands on sheer logic...

1 stands for e and 2 stands for p

K=(γ₁-1)m₁c²=(γ₂-1)m₂c²
Now,(γ₂-1)/(γ₁-1)=m₁/m₂

=>(γ₂-γ₁)/(γ₁+γ₂-2)=(m₁-m₂)/(m₁+m₂)<0
Also,(γ₁+γ₂)>2

=> γ₂<γ₁
From which you can deduce the relation between group and phase velocity...