# De Droglie particle-wave equation - Derivation by de Broglie

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ABCC Australia 2015 www.new-physics.com

WAVE EQUATIONde Broglies derivation of his:

ABCC Australia 2015 www.new-physics.com

Determining the Matter Wave Equation

To determine the wavelength of the wavy electron, de Broglie made use of the relations between the energy , the velocity of light , the momentum and the frequency of a photon or particle established by Planck and Einstein at the time.

To start with, de Broglie first employed Einsteins relativistic energy equation.

Light =

Light f =

=

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Classical Momentum

In classical mechanics, the momentum of a particle is equal to the product of its mass and velocity , or = . If the speed is so high as close to the speed of light (relativistic speed), its momentum will be governed by Einsteins relativistic equation.

Classical Newtonian Einsteinan

Your need to use my equations

Velocity of particle Velocity of light

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Einsteins Energy Equation

Einsteins equation for the energy of a particle at high speed is written as:

2 = 22 + (2)2

Taking the square roots on both sides, we have:

= 22 + (2)2

At the same time, Einstein's theory of relativity pointed out that for a particle like a photon of zero rest mass = 0.So we can neglect the (2)2 term and the relativistic energy becomes:

= 22 + (2)2

= 22 =

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Plancks Equation

On the other hand, according to Planck, the energy of a photon is related to its frequency and Plancks constant by the famous Plancks equation:

=

where is Planck's constant; is the frequency of the radiation or photon.

Photon frequency

gamma - symbol for photon h Plancks constant

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Speed & Wavelength

In radiation (light), the frequency of a photon is related to its velocity and wave length by:

=

=

So in terms of , the Plancks energy relationship can be written as:

= = /Or:

= /

= /

c

= /

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Planck + Einstein

Linking up Plancks formulae with Einsteins energy equation, de Broglie had:

= =

= or:

=

That is: Plancks frequency energy= Einsteins relativistic energy

Kinetic energy of photon

Frequency energy of photon

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Wavelength and Momentum

By manipulating the equation a little bit in moving the terms on both sides, we have a new equation which finally becomes:

= /

As seen in previous page / = .

=

/ = /

= /

Swap side

Swap side

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De Broglie Hypothesis

At this point, de Broglie made an ingenious intuitive guess that if the electron is also a wave particle, its formulae should also be like that of a photon wave. That is, the same formula works also for the electron:

=

=

Photonwave

Electronwave

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de Broglie equation

This relation between the wavelength and the momentum of the electron later became known as the famous de Broglie equation. is called the de Broglie wavelength of the electron:

=

So the particle bursts open and becomes a wave-particle. It is an assumption that if an electron is free, it would behave like a photon.

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DERIVATION BY SKTo be continued on:

ABCC

The physical origin of de Broglies particle-wave equation

Wave equationDetermining the Matter Wave EquationClassical MomentumEinsteins Energy EquationPlancks EquationSpeed & WavelengthPlanck + EinsteinWavelength and MomentumDe Broglie Hypothesisde Broglie equationDerivation by sk