# PM [D02] de Broglie deriving the Equation

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ABCC Australia 2015 new-physics.com

EQUATION DERIVED BY DE BROGLIEMatter-Waves [002]

=

ABCC Australia 2015 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.

=

=

ABCC Australia 2015 new-physics.com

Classical MomentumIn 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

ABCC Australia 2015 new-physics.com

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 =

ABCC Australia 2015 new-physics.com

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

ABCC Australia 2015 new-physics.com

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

= /

ABCC Australia 2015 new-physics.com

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

ABCC Australia 2015 new-physics.com

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

ABCC Australia 2015 new-physics.com

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

ABCC Australia 2015 new-physics.com

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.

ABCC Australia 2015 new-physics.com

Exercise 01 - The Wavelength of an Electron

Find the de Broglie wavelength of an electron ( = 9.11 1031 ) moving at 2 106 m/s.

The de Broglie wave equation is:

=

=6.63 1034

9.11 1031 2 106/

= 3.639 1010

Compared with the classical electron radius which is about 2.81791015 m, this is a relatively large wave length.

ABCC Australia 2015 new-physics.com

Exercise 02 - The Wavelength of a Baseball

A baseball with a mass of 0.15 kg is pitched at 45 m/s What is its De Broglie wavelength?

= =

6.63 1034 0.15 45/

= 9.8 1035

Diffraction effects of a baseball are negligible.

This is an incredibly small figure compare with the size of the ball. However this is a wrong example, as we shall see later.

ABCC Australia 2015 new-physics.com

WHAT IS THERE WAVING ?To be continued on: Matter-Waves [003]

ABCC

equation derived by de BroglieDetermining the Matter Wave EquationClassical MomentumEinsteins Energy EquationPlancks EquationSpeed & WavelengthPlanck + EinsteinWavelength and MomentumDe Broglie Hypothesisde Broglie equationExercise 01 - The Wavelength of an ElectronExercise 02 - The Wavelength of a BaseballWhat is there waving ?

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