PM [D02] de Broglie deriving the Equation

Download PM [D02] de Broglie deriving the Equation

Post on 15-Apr-2017

385 views

Category:

Science

0 download

TRANSCRIPT

  • 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 ?

Recommended

View more >