Question

The matter wave dispersion relation given in equation (6-23) is correct only at low speed and when mass/internal energy is ignored.

(a) Using the relativistically correct relationship among energy, momentum and mass, show that the correct dispersion relation is

$\mathbf{\omega }\mathbf{=}\sqrt{{\mathbf{k}}^{\mathbf{2}}{\mathbf{c}}^{\mathbf{2}}\mathbf{+}\frac{{\mathbf{m}}^{\mathbf{2}}{\mathbf{c}}^{\mathbf{4}}}{{\hslash }^{\mathbf{2}}}}$

(b) Show that in the limit of low speed (small p and k) and ignoring mass/internal energy, this expression aggress with that of equation (6-23).

Short answer

(a) The correct matter wave dispersion relation $\omega =\sqrt{k^2{c}^2+\frac{m^2{c}^4}{{\hslash }^2}}$ is proved.

(b) The equation (6-23) agrees with correct dispersion relationship at low speed.

Step-by-step solution

Matter Waves

The matter waves are used to relationship between momentum and wavelength of particle. If the wavelength of the particle is high then momentum of particle is low.

Proof for correct dispersion relation

(a)

The energy of the particle of matter wave is given as:

$E=\hslash \omega$

Here, $\omega$is the angular frequency of matter wave

The momentum of the particle of matter wave is given as:

$p=\hslash k$

Here, k is the wave number for matter wave

The relativistically correct relation among energy, momentum and mass is given as:

$E^2=p^2{c}^2+m^2{c}^4$

Substitute all the values in the above equation.

$$\begin{gathered} {\left(\hslash \omega \right)}^2={\left(\hslash k\right)}^2{c}^2+m^2{c}^4 \\ {\hslash }^2{\omega }^2={\hslash }^2{k}^2{c}^2+m^2{c}^4 \\ {\omega }^2=k^2{c}^2+\frac{m^2{c}^4}{{\hslash }^2} \\ \omega =\sqrt{k^2{c}^2+\frac{m^2{c}^4}{{\hslash }^2}} \end{gathered}$$

Therefore, the correct matter wave dispersion relation is proved.

Proof for matter waves at low speed

(b)

The wavelength of the matter wave becomes higher if momentum is low at low speed. The internal energy of the wave is also ignored so the correct dispersion relationship holds good and equation (6-23) agrees with correct dispersion relationship at low speed.

Therefore, the equation (6-23) agrees with correct dispersion relationship at low speed.