String Waves and Matter Waves — Question 7

Question 7

Two measured wave numbers reveal whether the evolution is dispersive

Two narrow matter-wave components have wave numbers k and $2k$, both positive, with $\omega=\hbar k^2/(2m)$. For comparison a string has $\omega=vk$. Define a two-component envelope speed as $\Delta\omega/\Delta k$.

1. Find that envelope speed for the matter wave.

2. Compare it with the group velocity at the midpoint wave number and with the string envelope speed.

Question 7 - Solution

Two measured wave numbers reveal whether the evolution is dispersive

(a) Method and calculation

$$\Delta\omega=\frac{\hbar}{2m}[(2k)^2-k^2]=\frac{3\hbar k^2}{2m},\quad\Delta k=k.$$$$\boxed{v_{\mathrm{env}}=3\hbar k/(2m)}.$$

(b) Interpretation and further calculation

$$k_c=3k/2:\quad\boxed{v_g(k_c)=\left.\frac{d\omega}{dk}\right|_{k_c}=3\hbar k/(2m)}.$$$$\omega_s=vk\Rightarrow\boxed{\Delta\omega_s/\Delta k=v}.$$

Physical check

For a quadratic dispersion the secant slope equals the tangent slope at the midpoint. This equality would not hold exactly for an arbitrary dispersion law.