Week2: Lecture III: Feb 3, 2020

Summary of Last Lecture:

What is going on in the Two-Hole Experiment ??

• Paradox: electrons come in lumps but show interference . That is N12 6= N1 +N2

• The point in the middle of two holes where we expect very few electrons has in fact large

number of electrons when both holes are open.Math is simple : It is math that shows that

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electrons are acting like waves

• You can determine that if you can see if it goes through either hole 1 or hole 2 or it goes

through both holes simultaneously .

WAITCH the electron by shining light: RESULT: If you shine light to see which hole

electron goes through, the interference pattern on the screen disappears. You can say too that

light affects the behaviour of electrons. Electrons are very delicate. When you are looking

at a-baseball and you shine a light on it, it does not make any difference, the baseball still

goes the same way. But when you shine a light on an electron it knocks him about a bit, and

instead of doing one thing he does another, because you have turned the light on and it is so

strong.

Heisenberg Uncertainty Principle:

’It is impossible to design any apparatus what so ever to determine through which hole the

electron passes that will not at the same time disturb the electron enough to destroy the interference

pattern’

He stated it in another way, but they are exactly equivalent, and you can get from one to the

other. Heisenberg noticed, when he discovered the laws of quantum mechanics, that the new laws

of nature that he had discovered could only be consistent if there were some basic limitation to

our experimental abilities that had not been previously recognized.

No one has found a way around this.

NATURE does not know – It only predicts probability

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There are three school of thoughts....

(a) Nature is unpredictable..

(b) Hidden variable Theory...

(c) Parallel Universes

So far there is NO evidence that supports (b) and (c)....

AND there are NO violations of (a)

WATCH a lively discussion on youtube

<https://www.youtube.com/watch?v=IxRfDtaot5U>

I. PRE-QUANTUM ERA – PARTICLES & WAVES

We had telephone ( Bell 1872..), radio ( Marconi 1895), Airplanes ( Right brothers 1903),

Cars... Scientist knew when solar eclipse occurs... In other words, they fully understood how stars

and planets move.. This is all based on Classical or pre-quantum era that rests on two pillars:

• Newtonian mechanics – Describes how things moves ( 17th century): F = ma. The beauty

of this equation is, it applies to all particles, no matter whether it is on the earth, or in

space or on Mars. It applies to all kinds of forces, such as gravitational force or electric or

magnetic force.

• Maxwell’s’ Equations - describes propagation of electromagnetic waves ( 19th century):

Type I: water waves, sound waves...Require medium to propagate.

Water waves are motion of particles and hence are described by Newton’s equation. Sound

waves are pressure waves describing vibrations ( stretching and compression) of the medium

) in which they travel. They can also be described by Newton’ s equation. Note that there

are no sound waves if there is NO medium like on the moon. Yes, astronauts cannot talk on

the moon.

Type II: Light waves, Micro waves, Radio waves, Infrared, Ultra violet They can travel in

vacuum – no medium. You communicate via electro-magnetic waves on moon.

There is a single theory that describes all these waves , theory by James Clerk Maxwell

( 1873) . Maxwell combined theory of electricity, magnetism and light in four equations,

known as Maxwell’s equations.

You might wonder what light has to do with electric and magnetic phenomena. It turns out

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electromagnetic waves can travel in vacuum ( unlike sound waves that require medium like

air ) and these waves are made up of oscillating electric and magnetic fields....

In electro-magnetic waves, it is the Electric and the magnetic field that are oscillating.

If electrons are waves, what kind of wave they are ???

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II. CHARACTERIZATION OF WAVES

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FIG. 1: Colors of the waves from the sun – Spectrum of radiations from H-atom. Note the unit “nm” means

nano meters, that is 10−9m.

Note the following:

• Humans can hear sound waves with frequencies between about 20 Hz and 20 kHz. Sound

above 20 kHz is ultrasound and below 20 Hz is infrasound. In air at standard temperature

and pressure, the corresponding wavelengths of sound waves range from 17 m to 17 mm.

Other animals have different hearing ranges.

• Loudness ( energy) is associated with the amplitude of the wave and pitch is associated

with the frequency. Compare this with light: for light waves, intensity is determined by

amplitudes and color is determined by frequency. Visible light is usually defined as having

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wavelengths in the range of 400 − 700 nanometres (nm), , between the infrared (with

longer wavelengths) and the ultraviolet (with shorter wavelengths). This wavelength means

a frequency range of roughly 430− 750 terahertz (THz) THz = 1012.

• Speed of sound is approximately 300 m/sec. Compare this with the speed of light which is

3× 108 m/sec.

Important characteristics of particles are their masses, their charge, their velocity.

Waves are described by wave length, frequency and speed of propagation.

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III. EINSTEIN’S THEORY OF RELATIVITY

You may enjoy seeing the video about Theory of Relativity: <https://youtu.be/

ERgwVm9qWKA>

Newton’s equation fail when particles travel close to speed of light c = 108 m/sec.

Proposed in 1905 ( called special theory of relativity ) and General theory of Relativity

(1916) were not revolutionary theories.. There were considered extension of Newton’s theory to

particles moving close to speed of light.... It can be viewed as a kind of mini-revolution, as speed

of light emerged as the largest speed possible in Newtonian world, in some sense linking particles

and light.

Brief Introduction of Einstein’s theory

(1) The speed of light in a vacuum is the same for all observers, regardless of their relative

motion or of the motion of the light source. That means, two individuals, traveling on two trains

moving in same direction will not feel that they are not moving...

(2) Relativity of simultaneity: Two events, simultaneous for one observer, may not be

simultaneous for another observer if the observers are in relative motion.

(3) Time dilation: Moving clocks are measured to tick more slowly than an observer’s

”stationary” clock.

(4) Length contraction: Objects are measured to be shortened in the direction that they are

moving with respect to the observer. Mass-energy equivalence: E = mc2, energy and mass are

equivalent and transmutable. Maximum speed is finite: No physical object, message or field line

can travel faster than the speed of light in a vacuum.

General Theory of Relativity

General relativity is a theory of gravitation developed by Einstein in the years 1907-1915.

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Einstein first proposed that space-time is curved. In 1915, he devised the Einstein field equations

which relate the curvature of space-time with matter and energy. General relativity generalizes

special relativity and Newton’s law of universal gravitation, providing a unified description of

gravity as a geometric property of space and time, or space-time. In particular, the curvature of

space-time is directly related to the energy and momentum of whatever matter and radiation are

present. The relation is specified by the Einstein field equations.

Rab −1

2Rgab = −8πGTab (1)

The above equation means that ”space-time curvature = energy density”. R and g describe

the structure of space-time and T matter or energy that determines this structure.

NOTE: YOU DO NOT NEED TO UNDERSTAND THIS EQUATION. However, you may

find it fascinating that these few weird symbols encode all the complexity of the theory and that is

what physicist call ”Mathematical Beauty”.

Some of the consequences of general relativity are

(1) Orbits precess in a way unexpected in Newton’s theory of gravity. (This has been observed in

the orbit of Mercury and in binary pulsars).

(2) Rays of light bend in the presence of a gravitational field.

General theory of Relativity is the key to GPS... something Einstein did not envision when he

discovered his theory..

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CAN something be wrong with classical physics ???

Classical physics explained many things and almost NO one doubted them as the inventions

( like radio, telephones, airplanes, cars... radars...) based on these theories had become part of

daily life.

However, near the end of 10th century, there were some noises being made about certain

phenomena that contradicted classical theory.

I: The radiations coming from hot objects were puzzling physicists around 1900.

As we will see, study of radiations from toaster or fire place that gave birth to quantum physics.

Puzzle near the end of the 19th century – Radiations from hot objects

• All macroscopic objects at any finite temperature radiate as well as absorb energy. These

radiations are electromagnetic waves.

• Radiations coming from these objects consists of all wave lengths, that is, the the entire

radiation can be broken into various wave lengths and the total intensity is the sum of the

intensity for each wavelength. Some wave lengths have more intensity than the others.

Interesting thing for scientists is to measure this radiation and make a graph of intensity vs

wave length, known as the spectrum of the radiation.

• Of course different objects will have different spectrum, with almost similar characteristics.

Scientists consider an ideal body, called Black Body, which keeps the common feature

of all the bodies. A black body is an idealized physical body that absorbs all incident

electromagnetic radiation, regardless of frequency or angle of incidence. An excellent

approximation to black body is an old fashion boiler furnace.

All normal matter emits electromagnetic radiation when it has a temperature above absolute

zero. The radiation represents a conversion of a body’s thermal energy into electromagnetic

energy, and is therefore called thermal radiation.

• Note, the temperature is measured in absolute unit, called “Kelvin”. Zero degree C, which

is same as 32 degree F which is same as 273 Kelvin.

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FIG. 2: Black Body spectrum at different temperatures. Note that toaster temperature is above 600 degree

C and is not shown in the graph. Note that x-rays ( very short wave lengths) and radio waves ( very long

wave lengths ) form the invisible part of the spectrum.

• Let us see if the graph makes sense, that is consistent with our intuition. At low temperature,

most of the light has long wave lengths. As we increase temperature, visible colors become

stronger...

• NOTE: the graph agrees with the classical theory for very very long wave lengths like radio

waves. So radio waves are well described by classical theory. If there was something wrong

with the classical theory, we will not know from studying radio waves...

• As shown in the graph, Fig. (2), the observed spectrum did not agree with the classical

theory ( Maxwell’s theory) as wave lengths get smaller. Note that classical theory predicts

infinite intensity in the ultraviolet regime ( tiny wave length or very high frequency), known

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as the“Ultraviolet Catastrophe”....

• In Berlin, a theoretical physicist at University of Berlin, a forty year old Max Planck used

some mathematical tricks to explain the black body spectrum

IV. PLANCK’S THEORY – MATHEMATICAL GIMMICK

(1) Planck divided the emitted light at any wavelength into lumps or quanta. The energy of

each such quanta is,

E = hf (2)

That is, energy is proportional to the frequency (E ∝ f ) and h is some constant.

(2) Introduce lump or “discreteness”, the “QUANTA” in the theory. In Planck’s theory,

intensity at certain frequency becomes the number of quanta in the light at that frequency, where

each quantum carries an energy proportional to the frequency, E ∝ f .

(3) With his, he was able to produce the black body curve, provided he chooses

h = 6.6× 10−34 joule-seconds.

(4) Is h related to fundamental constants known to physicist at that time, such as speed of

light and gravitational constant G???

Is Maxwell’s theory broken ??

Note that in Maxwell’s theory, intensity can take any value, independent of frequency, and

depends only on the amplitude of the wave ( for electromagnetic waves, this is the amplitude of

the electric and magnetic field in the wave disturbance of light).

In Planck’s theory, intensity at a certain frequency becomes number of quanta in light at that

frequency, where each quanta carries an energy proportional to the frequency, E = hf . This idea

“feels” more like the concepts of particles. However, light shows interference and should be a

wave according to classical theory.

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Planck did not understand the significance of his theory. He thought of the quanta as as

short bursts or pulses of radiation, coming from hot objects. Once emitted, they travel according

to Maxwell’s theory. This way, he avoided any conflict with the Maxwell’s theory. Note that

Maxwell’s theory was a big success at that time, it has led to many inventions like electric motors,

radio telegraphy... Planck did not want to question Maxwell’s theory... He throught he was just

correcting theory of thermal physics.

He did not realize that the constant “h”, which is now known as Planck’s constant, would

become the cornerstone of quantum revolution.

NOTE that Planck did not say that light is made up of discrete bundle of energy, the quanta.

He somehow believed that light is emitted as ‘quanta” but then continue to propagate as a wave,

as described by Maxwell’s theory.

CULTURE SHOCK

Try to imagine the culture shock in 1900. You are reviewing the radiation data from hot

objects which are described by Maxwell theory, well understood since 1860. But it cannot explain

the Black body curve.. Here comes Planck and explains it using this weird theory that hot objects

emit light as discrete pocket of energy E = hf

The theory threw the “classical” community into a monumental state of confusion....Planck

himself was not sure what his theory meant. In his presentation on Friday, Dec 14, 1900 in Berlin to

German Physical Society, he quietly described his result, without any excitement, without realizing

that he has shaken the Maxwell’s theory and is laying down the foundation of a new theory, a new

era in physics. In his equation E ∝ f , he tried to fit the constant of proportionality to known

fundamental constants of physics, without success. He did not realize that this is a new constant,

the pillar of a new theory.

Most in the scientist regarded his law as a mathematical formula that happened to fit the date.

NO one saw its implication. That was left to a young graduate student in Switzerland, working on

his own.

Important NOTE: There was nothing in Planck’s theory that can be tested in an experiment.

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