91影视

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.

(a)

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

$E=\mathrm{鈩�}蝇$

Here, $蝇$is the angular frequency of matter wave

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

$p=\mathrm{鈩�}k$

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.

$$\begin{gathered} {\left(\mathrm{鈩�}蝇\right)}^2={\left(\mathrm{鈩�}k\right)}^2{c}^2+m^2{c}^4 \\ {\mathrm{鈩�}}^2{蝇}^2={\mathrm{鈩�}}^2{k}^2{c}^2+m^2{c}^4 \\ {蝇}^2=k^2{c}^2+\frac{m^2{c}^4}{{\mathrm{鈩�}}^2} \\ 蝇=\sqrt{k^2{c}^2+\frac{m^2{c}^4}{{\mathrm{鈩�}}^2}} \end{gathered}$$

Therefore, the correct matter wave dispersion relation is proved.

(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.