de droglie particle-wave equation - derivation by de broglie

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WAVE EQUATIONde Broglie’s derivation of his:


Determining the Matter Wave Equation

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

To start with, de Broglie first employed Einstein’s relativistic energy equation.

Light 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑐𝑐𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑐𝑐

Light f𝑟𝑟𝑉𝑉𝑟𝑟𝑟𝑟𝑉𝑉𝑟𝑟𝑐𝑐𝑉𝑉 = 𝑓𝑓

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑚𝑚

𝐷𝐷𝑀𝑀𝑉𝑉𝑀𝑀 𝑉𝑉𝑓𝑓 𝑀𝑀 𝑝𝑝𝑝𝑉𝑉𝑉𝑉𝑉𝑉𝑟𝑟


Classical Momentum

In classical mechanics, the momentum 𝑝𝑝𝑝𝑝 of a particle is equal to the product of its mass 𝑚𝑚𝑝𝑝 and velocity 𝑣𝑣𝑝𝑝, or 𝑝𝑝𝑝𝑝 = 𝑚𝑚𝑝𝑝𝑣𝑣𝑝𝑝. If the speed is so high as close to the speed of light 𝑐𝑐 (relativistic speed), its momentum will be governed by Einstein’s relativistic equation.

𝑣𝑣𝑝𝑝 ≪ 𝑐𝑐 𝑣𝑣𝑝𝑝 ≈ 𝑐𝑐Classical Newtonian Einsteinan

Your need to use my equations

Velocity of particle Velocity of light


Einstein’s Energy Equation

Einstein’s equation for the energy 𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟of a particle at high speed is written as:

𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟2 = 𝑝𝑝2𝑐𝑐2 + (𝑚𝑚𝑜𝑜𝑐𝑐2)2

Taking the square roots on both sides, we have:

𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑝𝑝2𝑐𝑐2 + (𝑚𝑚𝑜𝑜𝑐𝑐2)2

At the same time, Einstein's theory of relativity pointed out that for a particle like a photon of zero rest mass 𝑚𝑚𝑜𝑜 = 0.So we can neglect the (𝑚𝑚𝑜𝑜𝑐𝑐2)2 term and the relativistic energy becomes:

𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑝𝑝2𝑐𝑐2 + (𝑚𝑚𝑜𝑜𝑐𝑐2)2

= 𝑝𝑝2𝑐𝑐2 = 𝑝𝑝𝑐𝑐


Planck’s Equation

On the other hand, according to Planck, the energy 𝐸𝐸γ of a photon is related to its frequency 𝑓𝑓𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 and Planck’s constant 𝑝by the famous Planck’s equation:

𝐸𝐸γ = 𝑝𝑓𝑓γ

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

𝑓𝑓γ

Photon frequency

gamma - symbol for photon h – Planck’s constant


Speed & Wavelength

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

𝑓𝑓𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 =𝑀𝑀𝑝𝑝𝑉𝑉𝑉𝑉𝑠𝑠

𝑤𝑤𝑀𝑀𝑣𝑣𝑉𝑉𝑉𝑉𝑉𝑉𝑟𝑟𝑤𝑤𝑉𝑉𝑝 =𝑐𝑐λ

So in terms of λ, the Planck’s energy relationship can be written as:

𝐸𝐸𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 = 𝑝𝑓𝑓 = 𝑝 𝑐𝑐/λOr:

λ𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 = 𝑐𝑐/𝑓𝑓

𝐸𝐸𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 = 𝑝𝑐𝑐/λ

λ

c

λ𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 = 𝑐𝑐/𝑓𝑓𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝


Planck + Einstein

Linking up Planck’s formulae with Einstein’s energy equation, de Broglie had:

𝐸𝐸 = 𝑝𝑓𝑓 = 𝑝𝑝𝑐𝑐

𝑝𝑓𝑓 = 𝑝𝑝𝑐𝑐or:

𝑝𝑝𝑐𝑐 = 𝑝𝑓𝑓

That is: Planck’s frequency energy= Einstein’s relativistic energy

Kinetic energy of photon

Frequency energy of photon


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:

𝜆𝜆 = 𝑝/𝑝𝑝

As seen in previous page 𝑐𝑐/𝑓𝑓 = 𝜆𝜆.

𝑝𝑝 𝑐𝑐 = 𝑝𝑓𝑓

𝑐𝑐/𝑓𝑓 = 𝑝/𝑝𝑝

𝜆𝜆 = 𝑝/𝑝𝑝

Swap side

Swap side


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:

𝜆𝜆𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 =𝑝

𝑝𝑝𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝

𝜆𝜆𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑝𝑝𝑟𝑟𝑜𝑜𝑝𝑝 =𝑝

𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑝𝑝𝑟𝑟𝑜𝑜𝑝𝑝

Photonwave

Electronwave


de Broglie equation

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

𝜆𝜆𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑝𝑝𝑟𝑟𝑜𝑜𝑝𝑝 =𝑝

𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑝𝑝𝑟𝑟𝑜𝑜𝑝𝑝So 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.


DERIVATION BY SKTo be continued on:

ABCC

The physical origin of de Broglie’s particle-wave equation

de droglie particle-wave equation - derivation by de broglie

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