Answer to Question #135133 in Quantum Mechanics for shashikant meena

Question

What is Matter wave? Explain duality of particle-wave of matter. Prove that De-
Broglie matter wave is equivalent to a wave packet.

Answer 1

According to de Broglie's hypothesis, a moving particle associated with a wave which is known as de Broglie wave or matter wave.

Radiations are some times behave as a wave and at some other times as a particle , that is it have a wave particle dualism. here it should be remembered that radiation cannot exhibits its particle and wave properties simultaneously.

According to de Broglie theory, the wavelength of Broglie wavelength is

$\lambda =\frac{h}{mv}$

The propagation constant of wave is

$k= \frac{2 \pi}{ \lambda}=\frac{2 \pi mv}{h}$

The frequency of associated wave is

$\nu= \frac{E}{h}$

The de Broglie wave velocity is

$v_p=\frac{\omega}{k}$

Substitute $\omega=2 \pi \nu$ and $k= \frac{2 \pi}{ \lambda}=\frac{2 \pi mv}{h}$ in the above equation,

$v_p=\frac{c^2}{v}$

As v is less than c and hence the de Broglie wave velocity must be greater than c. So the de Broglie wave train associated with the particle would travel much faster than the particle itself. Hence the de Broglie wavelength is equivalent wave packet.