Question

derive the de brogile wave equation what is meant by dual nature of matter​

Answer 1

Explanation:

De Broglie first used Einstein's famous equation relating matter and energy:

$e = m {c}^{2}$

with

E = energy,

m = mass,

c = speed of light

Using Planck's theory which states every quantum of a wave has a discrete amount of energy given by Planck's equation:

$e = h \nu$

with

E = energy,

h = Plank's constant (6.62607 x 10-34 J s),

ν= frequency

Since de Broglie believed particles and wave have the same traits, he hypothesized that the two energies would be equal:

$m {c}^{2} = h \nu$

Because real particles do not travel at the speed of light, De Broglie submitted velocity (v) for the speed of light (c).

$m {v}^{2} = h \nu$

Through the equation λ, de Broglie substituted v/λ for ν and arrived at the final expression that relates wavelength and particle with speed.

$m {v}^{2} = \frac{hv}{ \lambda}$

Hence

$\lambda = \frac{hv}{m {v}^{2} } = \frac{h}{mv}$

This is known as de Broglie wave equation and λ is known as de Broglie wavelength.

Answer 2

★☆〖Qบęຮτ ı¨ ø nˇ〗☆★

⭐The Dual Nature of Matter⭐

=> de-Broglie's Principle states that "All material particles in motion possess wave characteristics..."

=> de-Broglie's Relationship can be derived by combining the mass and energy relationships proposed by Max Planck, and Albert Einstein...

E = ∫c²dm = Σc²Δm = mc²

E = hν

=> The combination of these two yielded the desired result:

λ = h/mc

=> The above equation is valid for a Photon(γ⁰)

=> The same relation can be extended to every particle of this universe, if the speed of light in vacua(c) is replaced by the ordinary velocity of the particle:

$\: \: \: \: \: \: \: \: \: λ = \frac{h}{ \: mv⃗}$

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