sk nature of matter waves [1 of 3]

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MATERIAL PARTICLESWave-Particle Nature of

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Wave Particle Duality of Light

After fighting over hundreds of years, the issue of weather light is a particle or a wave is finally settled, or rather, compromised. It is now generally accepted that light is a particle as well as a wave. It is a hybrid called a wave-particle.Some clever people coined the word ‘wavicle’ for a particle that is wave and at the same time as particle. They are like the mythical hybrids of ancient Greece.


Light imitates material particles and become wavicles. But what about the material particles themselves? There are many micro-particles such as electrons, neutrons, and protons and even quarks and gluons. Do they behave like waves as well?

Material Particles?


Matter WavesPhotons are different from matter particles in that they are singularly characterized.

Matter particles are traditionally believed to be solid, localized and unwavy– even when they are in motion.The idea of matter particles that waves like a photon was not known until 1924.

Photon = Particle + Wave

Matter particles and objects


de Broglie Hypothesis

In 1924, a young French nobleman named Louis de Broglie submitted his doctoral thesis to the University of Paris under the title “Research in the Quantum Theory”[1]. It contained a new message that seemed to have the potential to completely revolutionize the classical view of matter.

[1] Louis de Broglie: ‘Recherches sur la théorie des quanta (Researches on the quantum theory)’ Thesis (Paris), 1924; L. de Broglie, Ann. Phys. (Paris) 3, 22 (1925).

Louis de Broglie (1892-1987) ⦊


Particle Waves

Based on the work of Max Planck (1858-1947)[2] and Albert Einstein (1879-1955)[3], de Broglie proposed in his thesis that material particles such as electrons should have both wave and particle properties, just like photons.

[2] Max Planck: Energy of radiation [Link][3] Albert Einstein: Relativistic energy [Link]

Photon = Particle + Wave

Matter particles


Early Days of the Theory

His thesis was so original that it was easily recognized. But there was not any experimental evidence of such a kind of wave at the time. So this new idea of matter-wave was not considered to have any physical reality. However, Paul Langevin (1872-1946) drew the attention of Albert Einstein (1879-1955) to the matter.Einstein immediately recognized its significance and promoted it to the attention of other physicists.

Good stuff!

[Because it came from my

formula!]


Nobel Prize AwardThree years later in 1927, de Broglie’s idea was confirmed by experiments and he received the Nobel Prize for his discovery of the wave nature of electrons in 1929. This made him the first person to receive a Nobel Prize on a PhD thesis.

Nobel Prize Medal


Debut of Matter Waves

de Broglie said in 1929:

“We thus find that in order to describe the properties of Matter, as well as those of Light, we must employ waves and corpuscles simultaneously. We can no longer imagine the electron as being just a minute corpuscle of electricity: we must associate a wave with it.” [4].

[4] Louis de Broglie: Nobel prize speech 1929.

Classical picture of an electron in

motion

Matter wave picture of an electron particle in

motion


Significance

With this new discovery, de Broglie opened the gateway to the development of wave and quantum mechanics in the early 19th century. Although in the later stage of development, the precision descriptions of a particle in motion no longer prevailed and gave way to probabilistic interpretations, the discovery of de Broglie remained a historic landmark in physics.

Picture source: Wikipedia


de Broglie’s

WAVE EQUATION


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 MomentumIn 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 𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 of 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 𝑝 by 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 𝜆𝜆by:

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

𝑝𝑝𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑝𝑝𝑟𝑟𝑜𝑜𝑝𝑝

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.


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 =𝑝𝑚𝑚𝑣𝑣

𝜆𝜆 =6.63 × 10−34𝐽𝐽 � 𝑀𝑀

9.11 × 10−31𝑘𝑘𝑤𝑤 × 2 × 106𝑚𝑚/𝑀𝑀

= 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 =𝑝𝑚𝑚𝑣𝑣

=6.63 × 10−34𝐽𝐽 � 𝑀𝑀0.15𝑘𝑘𝑤𝑤 × 45𝑚𝑚/𝑀𝑀

= 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.

sk nature of matter waves [1 of 3]

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idea of matter particles

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kind of wave

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