derive de -broglie relation?
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Answer:
De Broglie proposed that as light exhibits both wave-like and particle-like properties, matter to exhibit wave-like and particle-like properties. ... On the basis of his observations, de Broglie derived a relationship between wavelength and momentum of matter. This relationship is known as the de Broglie relationship.
De Broglie first used Einstein's famous equation relating matter and energy:
E=mc2(1)
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=hν(2)
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:
mc2=hν(3)
Because real particles do not travel at the speed of light, De Broglie submitted velocity ( v ) for the speed of light ( c ).
mv2=hν(4)
Through the equation λ , de Broglie substituted v/λ for ν and arrived at the final expression that relates wavelength and particle with speed.
mv2=hvλ(5)
Hence
λ=hvmv2=hmv(6)
A majority of Wave-Particle Duality problems are simple plug and chug via Equation 6 with some variation of canceling out units
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⭐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:
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