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EQUATION DERIVED BY DE BROGLIEMatter-Waves [002]

𝜆𝜆 =ℎ𝑚𝑚𝑚𝑚


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.

𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑐𝑐𝑉𝑉𝑉𝑉𝑉𝑉 = 𝑐𝑐

𝐹𝐹𝑟𝑟𝑉𝑉𝑟𝑟𝑟𝑟𝑉𝑉𝑟𝑟𝑐𝑐𝑉𝑉 = 𝑓𝑓

𝑃𝑃ℎ𝑉𝑉𝑉𝑉𝑉𝑉𝑟𝑟

𝑃𝑃𝑟𝑟𝑉𝑉𝑝𝑝𝑉𝑉𝑟𝑟𝑉𝑉𝑉𝑉𝑉𝑉𝑃𝑃 𝑉𝑉𝑓𝑓 𝑎𝑎 𝑝𝑝ℎ𝑉𝑉𝑉𝑉𝑉𝑉𝑟𝑟


Classical MomentumIn classical mechanics, the momentum 𝑝𝑝𝑝𝑝 of a particle is equal to the product of its mass 𝑚𝑚𝑝𝑝 and velocity 𝑚𝑚𝑝𝑝, or 𝑝𝑝𝑝𝑝 = 𝑚𝑚𝑝𝑝𝑚𝑚𝑝𝑝. 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.

𝑚𝑚𝑝𝑝 ≪ 𝑐𝑐 𝑚𝑚𝑝𝑝 ≈ 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 ℎby the famous Planck’s equation:

𝐸𝐸γ = ℎ𝑓𝑓γ

where ℎ 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 =𝑃𝑃𝑝𝑝𝑉𝑉𝑉𝑉𝑠𝑠

𝑤𝑤𝑎𝑎𝑚𝑚𝑉𝑉𝑉𝑉𝑉𝑉𝑟𝑟𝑤𝑤𝑉𝑉ℎ =𝑐𝑐λ

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

𝐸𝐸𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 = ℎ𝑓𝑓 = ℎ 𝑐𝑐/λOr:

λ𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 = 𝑐𝑐/𝑓𝑓

𝐸𝐸𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 = ℎ𝑐𝑐/λ

λ

c

λ𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝 = 𝑐𝑐/𝑓𝑓𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝


Planck + Einstein

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

𝐸𝐸 = ℎ𝑓𝑓 = 𝑝𝑝𝑐𝑐

ℎ𝑓𝑓 = 𝑝𝑝𝑐𝑐or:

𝑝𝑝𝑐𝑐 = ℎ𝑓𝑓

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 = ℎ/𝑝𝑝

As seen in previous page 𝑐𝑐/𝑓𝑓 = 𝜆𝜆.

𝑝𝑝 𝑐𝑐 = ℎ𝑓𝑓

𝑐𝑐/𝑓𝑓 = ℎ/𝑝𝑝

𝜆𝜆 = ℎ/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 =ℎ

𝑝𝑝𝑝𝑝𝑝𝑜𝑜𝑝𝑝𝑜𝑜𝑝𝑝

𝜆𝜆𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑝𝑝𝑟𝑟𝑜𝑜𝑝𝑝 =ℎ

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 =ℎ

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.


Exercise 01 - The Wavelength of an Electron

Find the de Broglie wavelength of an electron (𝑚𝑚 = 9.11 × 10−31 𝑘𝑘𝑤𝑤) moving at 2 × 106 m/s.

The de Broglie wave equation is:

𝜆𝜆 =ℎ𝑚𝑚𝑚𝑚

𝜆𝜆 =6.63 × 10−34𝐽𝐽 ∙ 𝑃𝑃

9.11 × 10−31𝑘𝑘𝑤𝑤 × 2 × 106𝑚𝑚/𝑃𝑃

= 3.639 × 10−10𝑚𝑚

Compared with the classical electron radius which is about 2.8179×10−15 m, this is a relatively large wave length.


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?

𝜆𝜆 =ℎ𝑚𝑚𝑚𝑚 =

6.63 × 10−34𝐽𝐽 ∙ 𝑃𝑃0.15𝑘𝑘𝑤𝑤 × 45𝑚𝑚/𝑃𝑃

= 9.8 × 10−35

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.


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

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Download - PM [D02] de Broglie deriving the Equation

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