Question

The velocity of matter waves is _____________.

Answer 1

1. According to the De Broglie Hypothesis, there is a wave associated with moving particles called matter waves or De Broglie waves.
2. The wavelength of matter-wave is given by$\lambda =\frac{h}{mv}$ ……equation(1) where h is Planck's constant,m is mass and v is the velocity of the particle.
3. Energy is given by $E=h\nu \Rightarrow \nu =\frac{E}{h}$…….equation(2) where $\nu$ is the frequency of the wave.
4. According to Mass-Energy relations,$E=M{c}^2$……equation(3) where M is the mass and c is the speed of light.
5. Equating equation(3) and (2), we get : $\nu =\frac{E}{h}=\frac{M{c}^2}{h}$……equation (4)
6. The speed of waves is given by $c=\nu \lambda$,……..equation (5) ,$\nu and\lambda$ are frequency and wavelength respectively.
7. Expression for matter's wave's velocity ($v_p$) =$\nu \lambda$…….equation (6)
8. Using equations (1) and (2) in (6), We get :
9. $$\begin{gathered} v_{p=}\nu \lambda \Rightarrow v_p=\frac{M{c}^2}{h}\times \frac{h}{mv}=\frac{c^2}{v} \\ {v}_p=\frac{c^2}{v} \end{gathered}$$
10. This clearly implies that the velocity of matter waves is greater than speed of light.