Question

Derive de broglie equation for microscopic particles

Answer 1

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

E=$mc^{2}$            (1.1)

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                 (1.2)

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                (1.3)

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

mv2=hν                 (1.4)

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                 (1.5)

Hence

λ=hvmv2=hmv           (1.6)

A majority of Wave-Particle Duality problems are simple plug and chug via Equation  1.6  with some variation of canceling out units

Answer 2

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

E=            (1.1)

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                 (1.2)

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                (1.3)

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

mv2=hν                 (1.4)

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                 (1.5)

Hence

λ=hvmv2=hmv           (1.6)

A majority of Wave-Particle Duality problems are simple plug and chug via Equation  1.6  with some variation of canceling out units

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