Lecture II: Jan 27, 2021

The “Quantum World: Particle & Waves”

Last lecture, in Brian Greene’s “two-hole” or ” double-slit” video, we saw that electrons have

wave like properties. That is, in quantum world, particles—tiny points moving through space

like tiny pebbles flying through the sky—do not act like pebbles in the sky but like ripples on

water. This though experiment which has now been done in many laboratories around the world,

illustrates many weird properties of electrons such as

(1) Although electrons exhibit wave like pattern on the screen after passing through the two holes,

they are detected as full particles, that is they do not split. In that sense they act like particles.

(2) It is impossible to know which slit an electron passes.

The weird thing is that although each electron wears its “wave hat” while propagating through

space (that is, while moving away from point A, passing through the slits, and approaching the

screen), it doesn’t keep that hat on at the very end. Instead, when it finally lands on the screen,

it doffs its “wave hat”, puts its “particle hat” back on, and deposits a little dot in just one single

point on the screen. Why and how does this weird hat-trick take place? No one can say. This

unfathomable mystery lies at the very heart of quantum mechanics. As Richard Feynman said,

“Nobody can explain quantum mechanics.”

We will be revisiting the two-hole experiment later in more detail. Let us I summarize

classical and quantum worlds and how they are described.

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Classical World

Classical Physics, also known as Newtonian physics – Physics from 17th-19th century

In classical physics, Particle and waves are two forms of matter

Particles have mass, size and shape

Waves are characterized by amplitude, wave length ( usually denoted by symbol Lambda: λ

) and frequency ( denoted by f ). Figure () below illustrates these properties. The velocity of

propagation of wave v is related to its wavelength and frequency as,

v = fλ (1)

NOTE: Generally speaking, in classical physics, there are two kinds of waves

(I) Mechanical Waves such as Sound waves and water waves: They are described by particle

picture, that is by Newton’s equations ( 17th century). They require medium to propagate. For

example, astronauts can not talk on the moon.

Waves in water and sound waves in air are two examples of mechanical waves. Mechanical waves

are caused by a disturbance or vibration in matter, whether solid, gas, liquid. Matter that waves

are traveling through is called a medium. Water waves are formed by vibrations in a liquid and

sound waves are formed by vibrations in a gas (air). These mechanical waves travel through a

medium by causing the molecules to bump into each other, like falling dominoes transferring

energy from one to the next. Sound waves cannot travel in the vacuum of space because there is

no medium to transmit these mechanical waves.

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(II): Electro-magnetic ( EM) waves: light waves . It includes radio waves, microwaves,

infrared, (visible) light, ultraviolet, X-rays, and gamma rays.( visible part ) . They are described

by Maxwell’s equations ( 19th century). These waves can travel in vacuum. Astronauts can

communicate on the moon by send sending EM waves. Electromagnetic waves consist of

oscillating electric and magnetic field, perpendicular to one another.

FIG. 1: Figure shows electro-magnetic wave. What is waving here is the electric and the magnetic field at

right angles to each other and also at right angle to the direction of propagation of the wave.

FIG. 2: Graph of various types of EM waves and their wave length in nano-meters = 10−9 meters

We will see later that “Particle Waves”, known as matter waves or de Broglie waves are

neither mechanical not electromagnetic, but can be though of as the “Probability Waves”.

NOTE: Amplitude determines the loudness of sound waves and frequency determines the

pitch. Human ears can hear only certain range of frequencies. For light waves, wave length

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determines the color and amplitude determines brightness. Figure below shows the EM radiations

from the sun. We can see only visible part of the radiations. Note that only certain certain colors

are emitted !! This puzzle will be solved by quantum physics.

FIG. 3: Figure shows the light emitted by the sun - known as the spectrum. Different colors correspond to

different wavelengths, measured in nano-meters = 10−9 meters. Note that only certain wave lengths appear

and we are seeing wave lengths that our eyes can see - called the visible part of the spectrum.

EXPERIMENTAL test whether we have particle or wave is a two hole experiment.

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Quantum World

In quantum world, we can focus on elementary particles as all particles are made up of those.

Elementary particles that make up our ATOM are electrons and quarks: note protons and neutrons

are made up of quarks. Elementary particles have NO size: they have

(1) Mass

(2) Charge

(3) Spin

(4) In addition, particles also have frequency and wave length.

Planetary world or macroscopic motions are described by Newton’s laws. These

mathematical rules describe all motions in macroscopic world, like trajectory of a tennis ball,

fall of an apple.. That is same rules govern the motion of all objects with mass in a macroscopic

world. ( We will talk about light and micro-waves, radio waves etc later ).

Now what about the electrons... the things we cannot see with our eyes..

They do have mass. Are they really particles ????

When they were first discovered by J. J. Thomson in 1897 they were believed to be particles.

Later experiments showed that they act like waves..... In fact, In 1920s, son of J.J Thompson,

George Thompson showed that electrons behave like waves. Both father and son got Nobel prize.

It turns out that electrons sometimes act like particles and sometimes waves. In that sense,

they are neither particle nor wave!!! They are described by laws of quantum physics.

(III) Why bother about the Quantum World

• Firstly, we are curious...and we wonder if the motion of electrons around the nucleus is

similar to the motion of planets around the sun ???..

• Secondly, we need to understand the behavior of these things as they lie at the heart of smart

devices... computers, atomic watches..., lasers... and if we want to build faster and more

powerful computers... the quantum computers.., new ways to generate energy ( other than

coals and hydroelectric power....)., want tiny devices....

• Quantum effects also appear in macroscopic world, like superconducting that can give us

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superfast trains.

• Quantum Revolution - the world beyond our imagination:

Appendix I: Summarizing Wilczek’s 2002 paper

• Twentieth-century physics began around 600 B.C. when Greek mathematician Pythagoras

claimed an awesome vision. By studying the notes from a plucked string, Pythagoras

discovered that the human perception of harmony is connected to numerical ratios. For

example, the ratio 2 : 1 sounds a musical octave, 3 : 2 a musical fifth and so on. The vision

inspired by this discovery is summed up in the maxim “All Things are Whole Number”, that

is integers .

Their most celebrated discovery is the Pythagorean Theorem. Unfortunately, this very

theorem undermined their belief that “ all things are whole numbers” Using the Pythagorean

Theorem, it is not hard to prove that the ratio of the hypotenuse of an isosceles right triangle

to either of its two shorter sides cannot be expressed in whole numbers.

• Kepler famous for this laws describing empirically the motion of the planets, originally

believed in Pythagorean type ideas where the orbits of the planets are related to the spheres

inscribed or circumscribed in the five known platonic solids. The five Platonic solids –

tetrahedron, cube, octahedron, dodecahedron, icosahedron–have faces that are congruent

equilateral polygons and it was shown that only five such solids exist. However, this turned

out to be not valid description of planetary orbits. Further work by Newton and others also

did not show any evidence that whole numbers underlie all there is in this world.

• However, in quantum physics, numbers rule the way things behave. First such example

was the Bohr model of an atom where energy of the electrons were expressed in terms

of whole numbers. The word “quantum ” in quantum physics has its origin that most

physical quantities are quantized, that is some integer multiple of certain constants. These

whole numbers appear in quantum physics because the mathematics describing the vibratory

patterns that define the states of atoms in quantum mechanics is identical to that which

describes the resonance of musical instruments.

• Wilczek argues against the notion that quantum theory is complicated. He says the quantum

theory is simple because very few ingredients are needed for the description. The theory

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describing electrons in an atom, called “QED” ( Quantum Electrodynamics ) require just

two quantities: (a) The mass of the electron, which we will denote as me, and so called fine

structure constant, denoted as α that specifies the strength of electro-magnetic interaction.

The theory of nucleus is called “QCD” ( Quantum chromo dynamics). To describe this

theory, we just need the mass of two quarks ( that make up protons and neutrons) and the

strength of interaction between them, denoted as αs. He further argues that it is sufficient to

have only two quantities: mass of the proton mp and αs.

• Let me summarize. Starting with precisely four numerical ingredients, which must be taken

from experiment, QED and QCD, cook up a concept-world of mathematical objects whose

behavior matches, with remarkable accuracy, the behavior of real-world matter. These

objects are vibratory wave patterns. Stable elements of reality–protons, atomic nuclei,

atoms–correspond, not just metaphorically but with mathematical precision, to pure tones.

Kepler would be pleased.

The key point is that quantum world is simple, needing just few ingredients. In some sense,

this is how mathematicians and philosophers envisioned the world before 17th century when

scientific view points took over.

Appendix II: Platonic Solids

The Platonic solids are prominent in the philosophy of Plato. Plato associated each of the four

classical elements (earth, air, water, and fire) with a regular solid. Earth was associated with the

cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was

intuitive justification for these associations. Of the fifth Platonic solid, the dodecahedron, Plato

obscurely remarked, ”...the god used it for arranging the constellations on the whole heaven”. For

further details, see

<https://en.wikipedia.org/wiki/Platonic_solid#:˜:text=The>

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