In: Physics
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.
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