Post on 15-Apr-2017
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EQUATION DERIVED BY DE BROGLIEMatter-Waves [002]
ππ =βππππ
Β© 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 πΈπΈ, the velocity of light ππ, the momentum ππ and the frequency ππ of a photon or particle established by Planck and Einstein at the time.
To start with, de Broglie first employed Einsteinβs relativistic energy equation.
ππππππππππππππππ = ππ
πΉπΉππππππππππππππππ = ππ
ππβππππππππ
ππππππππππππππππππππ ππππ ππ ππβππππππππ
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Classical MomentumIn classical mechanics, the momentum ππππ of a particle is equal to the product of its mass ππππ and velocity ππππ, or ππππ = ππππππππ. If the speed is so high as close to the speed of light ππ (relativistic speed), its momentum will be governed by Einsteinβs relativistic equation.
ππππ βͺ ππ ππππ β ππClassical Newtonian Einsteinan
Your need to use my equations
Velocity of particle Velocity of light
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Einsteinβs Energy Equation
Einsteinβs equation for the energy πΈπΈππππππof a particle at high speed is written as:
πΈπΈππππππ2 = ππ2ππ2 + (ππππππ2)2
Taking the square roots on both sides, we have:
πΈπΈππππππ = ππ2ππ2 + (ππππππ2)2
At the same time, Einstein's theory of relativity pointed out that for a particle like a photon of zero rest mass ππππ = 0.So we can neglect the (ππππππ2)2 term and the relativistic energy becomes:
πΈπΈππππππ = ππ2ππ2 + (ππππππ2)2
= ππ2ππ2 = ππππ
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Planckβs Equation
On the other hand, according to Planck, the energy πΈπΈΞ³ of a photon is related to its frequency πππππππππππππ and Planckβs constant βby the famous Planckβs equation:
πΈπΈΞ³ = βππΞ³
where β is Planck's constant; πππππππππππππ is the frequency of the radiation or photon.
ππΞ³
Photon frequency
gamma - symbol for photon h β Planckβs constant
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Speed & Wavelength
In radiation (light), the frequency πππππππππππππ of a photon is related to its velocity ππ and wave length ππ by:
πππππππππππππ =πππππππππ π
π€π€πππππππππππππ€π€ππβ =ππΞ»
So in terms of Ξ», the Planckβs energy relationship can be written as:
πΈπΈπππππππππππ = βππ = β ππ/Ξ»Or:
Ξ»πππππππππππ = ππ/ππ
πΈπΈπππππππππππ = βππ/Ξ»
Ξ»
c
Ξ»πππππππππππ = ππ/πππππππππππππ
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Planck + Einstein
Linking up Planckβs formulae with Einsteinβs energy equation, de Broglie had:
πΈπΈ = βππ = ππππ
βππ = ππππor:
ππππ = βππ
That is: Planckβs frequency energy= Einsteinβs 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:
ππ = β/ππ
As seen in previous page ππ/ππ = ππ.
ππ ππ = βππ
ππ/ππ = β/ππ
ππ = β/ππ
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:
πππππππππππππ =β
πππππππππππππ
ππππππππππππππππππ =β
ππππππππππππππππππ
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. ππππ is called the de Broglie wavelength of the electron:
ππππππππππππππππππ =β
ππππππππππππππππππSo 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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Exercise 01 - The Wavelength of an Electron
Find the de Broglie wavelength of an electron (ππ = 9.11 Γ 10β31 πππ€π€) moving at 2 Γ 106 m/s.
The de Broglie wave equation is:
ππ =βππππ
ππ =6.63 Γ 10β34π½π½ β ππ
9.11 Γ 10β31πππ€π€ Γ 2 Γ 106ππ/ππ
= 3.639 Γ 10β10ππ
Compared with the classical electron radius which is about 2.8179Γ10β15 m, this is a relatively large wave length.
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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?
ππ =βππππ =
6.63 Γ 10β34π½π½ β ππ0.15πππ€π€ Γ 45ππ/ππ
= 9.8 Γ 10β35
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.
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WHAT IS THERE WAVING ?To be continued on: Matter-Waves [003]
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