How to show speed is equal to group velocity?

In summary, in order to show that speed is equal to group velocity, you can use the equation v_g = dw/dk and solve for the momentum and velocity. Alternatively, you can also use the relativistic factor ##\gamma = 1/\sqrt {1 - (v^2/c^2)}## and the equation E = \gamma mc^2 to solve for the velocity.
  • #1
Yoki
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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.
 
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  • #2
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.
 
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Related to How to show speed is equal to group velocity?

1. What is the difference between speed and group velocity?

The speed of a wave is the rate at which the wave propagates through a medium. Group velocity, on the other hand, is the speed at which the center of mass of a wave packet moves. In other words, it is the speed at which the energy or information of the wave is being transmitted.

2. How is group velocity calculated?

Group velocity can be calculated by taking the derivative of the wave's dispersion relation with respect to its wave vector. In simpler terms, it is the slope of the curve on a graph of frequency vs. wave vector.

3. Can group velocity be greater than the speed of light?

No, according to Einstein's theory of relativity, no information or energy can travel faster than the speed of light. Therefore, group velocity cannot exceed the speed of light.

4. What factors affect the group velocity of a wave?

The group velocity of a wave can be affected by a variety of factors, including the medium through which the wave is traveling, the amplitude and frequency of the wave, and any external forces acting on the wave.

5. Why is it important to understand the concept of group velocity?

Group velocity is an important concept in physics and engineering as it helps us understand how waves behave and how information is transmitted through a medium. It is also essential in the design and development of various technologies, such as telecommunications and signal processing.

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