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How to show speed is equal to group velocity?

  • #1
4
1

Homework Statement


My question is, how do I show that speed is equal to group velocity? More information at https://imgur.com/a/m6FwNaG

Homework Equations


v_g = dw/dk

The Attempt at a Solution


Part a is substitution, part b uses v_g = dw/dk, part c is multiplication by h-bar, but I am stuck at part d. We are dealing with a relativistic energy equation, but the question says "the speed v" as if that has some meaning. What am I missing? I have a feeling I am overlooking something incredibly simple.
 

Answers and Replies

  • #2
RPinPA
Science Advisor
Homework Helper
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The speed v is the speed that occurs in the relativistic factor ##\gamma = 1/\sqrt {1 - (v^2/c^2)}##. And momentum depends on velocity, ##p = \gamma mv##. So it looks like you need to solve for the momentum, and from that solve for the velocity.

Edit: I realized on thinking about this that it would be even easier to simply use the fact that ##E = \gamma mc^2## and solve for ##\gamma## and v that way.
 
Last edited:

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