PM [D01] Matter Waves

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DE BROGLIE'S THEORYMatter-Waves [001]


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 in the early twentieth century. It is now generally accepted that light is a particle as well as a wave. It is a hybrid called a wave-particle.


Wavicles

Some clever people coined the word ‘wavicle’ for a particle that is wave and at the same time as particle. The composite particles have dual characteristics, like the mythical hybrids in ancient Greek mythology.


Light imitates material particles and becomes a wavicle. 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 Waves

Photons 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 (1892-1987) 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 théoriedes 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 Award

Three 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.


RELATIVISTIC ENERGYAoppendix : Einstein’s


Relativistic Energy

The momentum 𝑝𝑝 in relativistic expression is:

𝑝𝑝 =𝑚𝑚𝑜𝑜𝑣𝑣

1 − 𝑣𝑣2𝑐𝑐2

𝑚𝑚𝑜𝑜 is the rest mass of the particle; 𝑣𝑣the velocity of the particle; 𝑐𝑐 the speed of light.

Squaring both sides:

(𝑝𝑝)2=𝑚𝑚𝑜𝑜𝑣𝑣


2

We arrive at:

𝑝𝑝2 =𝑚𝑚𝑜𝑜2𝑣𝑣2



Multiplying both sides by 𝑐𝑐2:

𝑝𝑝2𝑐𝑐2 =𝑚𝑚𝑜𝑜2𝑣𝑣2𝑐𝑐2


Or:

𝑝𝑝2𝑐𝑐2 =𝑚𝑚𝑜𝑜2 𝑣𝑣2𝑐𝑐2 𝑐𝑐

4


By adding and subtracting a term it can be put in the form:

𝑝𝑝2𝑐𝑐2 =𝑚𝑚𝑜𝑜2 𝑣𝑣2𝑐𝑐2 𝑐𝑐

4


=𝑚𝑚𝑜𝑜2𝑐𝑐4 𝑣𝑣2

𝑐𝑐2 − 1


+𝑚𝑚𝑜𝑜2𝑐𝑐4


= −𝑚𝑚𝑜𝑜2𝑐𝑐4 + 𝑚𝑚2𝑐𝑐4


The term 𝑚𝑚𝑜𝑜2𝑐𝑐4

1−𝑣𝑣2

𝑐𝑐2

is just the

relativistic mass m. So:

𝑝𝑝2𝑐𝑐2 + 𝑚𝑚𝑜𝑜2𝑐𝑐4 = 𝑚𝑚2𝑐𝑐4

Or

𝑚𝑚2𝑐𝑐4 = 𝑝𝑝2𝑐𝑐2 + 𝑚𝑚𝑜𝑜2𝑐𝑐4

which may be rearranged to give the expression for relativistic energy 𝑚𝑚𝑐𝑐 = 𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟:

𝑚𝑚𝑐𝑐 = 𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑝𝑝2𝑐𝑐2 + 𝑚𝑚𝑜𝑜2𝑐𝑐4

= 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 the relativistic energy becomes:

𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑝𝑝2𝑐𝑐2 + (𝑚𝑚𝑜𝑜𝑐𝑐2)2= 𝑝𝑝𝑐𝑐

𝐸𝐸𝑟𝑟𝑟𝑟𝑟𝑟 = 𝑝𝑝𝑐𝑐


THE EQUATION DERIVED BY DE BROGLIETo be continued in : Matter-Waves [002]

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