Angular Frequency and Group Velocity of a Relativistic Particle

Introduction:
In relativistic physics, the energy (E) of a particle is related to its momentum (p) by the famous equation E^2 = (mc^2)^2 + (pc)^2, where m is the rest mass of the particle, c is the speed of light, and p is the momentum. This equation is derived from the relativistic energy-momentum relation.

Derivation:
Let's consider a relativistic particle with mass m and momentum p. The de Broglie wavelength (λ) of this particle is given by the equation λ = h/p, where h is the Planck's constant. The phase velocity (v_phase) of the particle is given by v_phase = λf, where f is the frequency of the particle.

Angular Frequency:
The angular frequency (ω) is defined as the rate at which the phase of a periodic waveform advances in radians per second. It is related to the frequency (f) by the equation ω = 2πf. Let's calculate the angular frequency of the relativistic particle.

Substituting λ = h/p into v_phase = λf, we get v_phase = h/p * f. Using the relation E = hf, where E is the energy of the particle, we can write f = E/h. Substituting this into the previous equation, we get v_phase = h/p * E/h. Canceling out the h terms, we have v_phase = E/p.

Phase Velocity and Group Velocity:
The phase velocity (v_phase) represents the speed at which the phase of the particle's waveform propagates. On the other hand, the group velocity (v_group) represents the speed at which the particle itself propagates.

In the given problem, the phase velocity is given as 2c, where c is the speed of light. We need to find the group velocity.

Calculation:
Using the relation v_phase = E/p and the relativistic energy-momentum relation E^2 = (mc^2)^2 + (pc)^2, we can write v_phase = sqrt((mc^2)^2 + (pc)^2)/p.

Given that v_phase = 2c, we have 2c = sqrt((mc^2)^2 + (pc)^2)/p. Squaring both sides and rearranging the terms, we get 4c^2 = ((mc^2)^2 + (pc)^2)/p^2.

Multiplying both sides by p^2 and rearranging the terms, we have 4c^2p^2 = (mc^2)^2 + (pc)^2. Substituting (pc)^2 = (mc^2)^2 - (mc^2)^2, we get 4c^2p^2 = (mc^2)^2 - (mc^2)^2 + (pc)^2.

Simplifying the equation, we have 4c^2p^2 = (mc^2)^2 - (mc^2)^2 + p^2c^2. Canceling out the common terms, we get 4c^2p^2 = p^2c^2.

Dividing both sides by p^2c^2, we have 4