Electrons and Matter Waves — Question 7

Question 7

Relativistic carrier phase can move faster than light without superluminal transport

Use a free massive particle wave with total-energy convention $\omega=E/\hbar$, $k=p/\hbar$ and $E^2=p^2c^2+m^2c^4$. The particle speed is $v=.600c$.

1. Derive the group and phase velocities.

2. Evaluate both and verify their product.

Question 7 - Solution

Relativistic carrier phase can move faster than light without superluminal transport

(a) Method and calculation

$$v_g=\frac{dE}{dp}=\frac{pc^2}{E}=v.$$$$\boxed{v_p=\frac Ep=\frac{c^2}{v}}.$$

(b) Interpretation and further calculation

$$\boxed{v_g=.600c,\qquad v_p=\frac53c}.$$$$\boxed{v_pv_g=c^2}.$$

Physical check

The carrier phase is not an independently controllable signal. A localized packet propagates at its group velocity, which is subluminal for a massive particle.