de broglie wave equation - derivation by sk

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WAVE-PARTICLE EQUATIONSK Deriving de Broglie’s


de Broglie Wave Equation

Let us have a fresh look at

de Broglie’s wavelength equation which is:

λ = ℎ/𝑝𝑝

In the earlier days of de Broglie’s theory, the electron (𝑒𝑒−) is the main particle in concern. So we start our discussion with the electron.

The Eskade Postulate 01 stated that all particle motions are instigated by phonons or photons, be it electrons or any other matter.

λ = ℎ/𝑝𝑝

λ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = ℎ/𝑝𝑝𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

λ𝑒𝑒 = ℎ/𝑝𝑝𝑒𝑒


Eskade Carrier Postulate

By the carrier postulate (Eskade postulate 01), and by the principle of the conservation of energy, the energy of the moving electron comes entirely from the incorporated photon. So the energy of the composition resides with photons:

Electron kinetic energy = Photon kinetic energy

12𝑚𝑚𝑒𝑒𝑣𝑣𝑒𝑒2 =

12𝑚𝑚𝛾𝛾𝑐𝑐𝛾𝛾2

𝑚𝑚𝑒𝑒𝑣𝑣𝑒𝑒2 = 𝑚𝑚𝛾𝛾𝑐𝑐𝛾𝛾2

12𝑚𝑚𝑒𝑒𝑣𝑣𝑒𝑒2 =

12𝑚𝑚𝛾𝛾𝑐𝑐𝛾𝛾2

Kinetic energy of particle

Kinetic energy of photon


Eliminating ½ from both sides


Momentum of Wave

Now 𝑚𝑚𝑒𝑒𝑣𝑣𝑒𝑒 = 𝑝𝑝𝑒𝑒 is the momentum of the electron; and 𝑚𝑚𝛾𝛾𝑐𝑐𝛾𝛾2 = 𝑓𝑓ℎ is the energy of the photon. So:


Becomes:𝑝𝑝𝑒𝑒𝑣𝑣𝑒𝑒 = 𝑓𝑓𝛾𝛾ℎ

Swapping the relevant items, we have:

𝑣𝑣𝑒𝑒/𝑓𝑓𝛾𝛾 = ℎ/𝑝𝑝𝑒𝑒

This is in accordance with the principle of energy conservation. The photon is still vibrating at the same frequency.


2 x Kinetic energy of photon

Planck energy of photon


𝑝𝑝𝑒𝑒𝑣𝑣𝑒𝑒 = 𝑓𝑓𝛾𝛾ℎ𝑚𝑚𝑒𝑒𝑣𝑣𝑒𝑒 = 𝑝𝑝𝑒𝑒 is the momentum of the electron


The de Broglie Wave Equation

The equation 𝑣𝑣𝑒𝑒/𝑓𝑓𝛾𝛾 gives us the wavelength 𝜆𝜆𝑒𝑒:

𝑣𝑣𝑒𝑒𝑓𝑓𝛾𝛾

= 𝜆𝜆𝑒𝑒

Thus the de Broglie wavelength equation is:

𝜆𝜆𝑒𝑒 =ℎ𝑝𝑝𝑒𝑒

Which was proposed by de Broglie in 1924.


𝜆𝜆𝑒𝑒 = ℎ/𝑝𝑝𝑒𝑒


Summary of de Broglie particle-waves.

1) An electron can move because of carrier particles such as photons or phonons.

2) A moving electron is wave-like because of the oscillating carrier.

3) The momentum of the electron is the momentum of the carrier.

4) The kinetic energy of the electron is the energy of the carrier.

5) The wavelength of a moving electron is the shortened wavelength of the carrier because of the heavier electron.

Louis de Broglie (1892-1987)


Summary of Eskade Postulates

1) Particles are moved by carrier particles mainly photons or phonons.

2) The vibrating nature of matter are due to the photons or phonons as in de Broglie waves.

3) Photons retain their vibration in free or bound state, leading to the principle of energy and momentum conservations in low energy cases.


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de broglie wave equation - derivation by sk

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