Question

what do you mean by dual nature of matter? show that de Broglie's wave length is inversely proportional to the velocity of the particle.​

Answer 1

Answer:

Dual nature of matter implies on matter showing two behaviors:

1. As a particle
2. As wave

Matter shows both of these behaviors under certain conditions.  As a particle is generally on Newtonian scale and as aa wave is on quantum scale .

Broglie's principle says that wavelength of a mass is inversely proportional to the momentum of the body. It's given by,

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

where h=  plank's constant.

Therefore clearly,

$\lambda$ ∝$\dfrac{1}{v}$

Answer 2

Question :

⠀▪︎What do you mean by dual nature of matter ? Show that de Broglie's wave length is inversely proportional to the velocity of the particle .

⠀⠀⠀⠀Ans : According to the concept of De Broglie's concept of matter shows their two properties according to the conditions :

1. As Wave ( in state of moving ).
2. As Praticle ( in state of rest ) .

⠀⠀⠀⠀⠀━ So , that's why Matter has dual nature .

⠀⠀⠀⠀▪︎ According to the De Broglie's equation suggests that the wave length of any object is inversely proportional to its momentum ( or product of mass & Velocity )

Therefore,

• Representation in Mathematical Eq. :

$\qquad \dag\:\:\bigg\lgroup \sf{ \lambda \:= \dfrac{h}{mv}}\bigg\rgroup$

⠀⠀⠀⠀Here , h is the Plank's Quantity , mv is the momentum ( product of mass(m) and Velocity(v) is known as momentum ) & $\bf \lambda$ is the wavelength.

Now , We can clearly see that ,

⠀⠀⠀⠀▪︎ Wave length is inversely proportional to the velocity of the particle.

$\qquad \dag\:\:\bigg\lgroup \sf{ \lambda \propto \dfrac{1}{v}}\bigg\rgroup$

⠀⠀⠀⠀Here , $\bf \lambda$ is the wavelength and v is the velocity.

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⠀⠀⠀⠀⠀More To Know about Dual nature of Matter :

• It was proposed by De Broglie .
• It was proposed in 1923
• It was verified by Davisson & Gerber by the Diffraction.

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