Electrons and Matter Waves — Question 6

Question 6

The group velocity of a nonrelativistic matter wave follows the particle velocity

A narrow packet has dispersion $\omega(k)=\hbar k^2/(2m)$ with $k>0$. Define phase velocity $v_p=\omega/k$ and group velocity $v_g=d\omega/dk$.

1. Derive both velocities and relate them to momentum.

2. If a constant potential $V_c$ is added, find the new velocities from $\omega=(\hbar^2k^2/2m+V_c)/\hbar$.

Question 6 - Solution

The group velocity of a nonrelativistic matter wave follows the particle velocity

(a) Method and calculation

$$\boxed{v_p=\hbar k/(2m)=p/(2m)},\qquad\boxed{v_g=\hbar k/m=p/m}.$$$$\boxed{v_g=2v_p}\quad\text{for the stated zero of potential energy}.$$

(b) Interpretation and further calculation

$$\boxed{v_p^{\,(c)}=\frac{\hbar k}{2m}+\frac{V_c}{\hbar k}}.$$$$\boxed{v_g^{\,(c)}=\frac{d\omega}{dk}=\frac{\hbar k}{m}}.$$

Physical check

A constant energy offset changes the carrier phase rate but does not change packet transport or exert a force.