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In: Physics

Exercise 2: Using the Relativistic Energy Momentum relationship ?2 = ?2?2 + ?2?4 and the de-Broglie...

Exercise 2: Using the Relativistic Energy Momentum relationship ?2 = ?2?2 + ?2?4 and the de-Broglie relationships ? = ℏ? and ? = ℏ? with ? the angular frequency and ? the wave number, find the group ?? and the phase velocity ??ℎ of the de-Broglie waves and show that their product is equal to ?2.

Solutions

Expert Solution

Given the energy momentum relationship
   
And the de Brogile relationships
   
we have
  

  
  The group velocity is defined as
  
  

   

And the ohase velocity is defined as
   
And so,
we have the product of the phase velocity and the group velocity as
  
  
  
  


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