March 30, 2020

HOW ARE DISCOVERIES MADE - Class Discussion

Explain experiments:

luck, chance:

imagination and deep thinking:

Using some very basic principles of nature..: Anti-matter and spin as an intrinsic property

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Dirac Equation – Piece of Magic

Classical Mechanics

Newton’s equationnon-Relativistic

v << c

Einstein’s TheoryRelativistic0 < v < c

QuantumMechanics

Schrodinger’s equation

non-Relativisticv << c

Dirac TheoryRelativistic 0 < v < c

No Absolute Time

Probabilisticdescription

Electrons have spin,have anti-particle

and vacuum is complicated

Particle is describedby position & velocity

Particle is describedby wave function

Quantization, Zero point energyposition & momentum obey Heisenberg’

uncertainity rule

Schodinger equation for a particle of mass m is:

i~∂

∂tΨ =

[−~2

2m∇2 + V

]Ψ (1)

Let us write the equation as,

i~∂tΨ =

[−~2

2m∂2x + V

]Ψ (2)

If you solve the equation, you determine Ψ , that determines everything you want to know

a particle of mass m. It determines the probability of where the particle is, it then determines

average position, velocity, energy etc. etc...

Schrodiner equation is not the correct equation when particles move close to speed of light.

Note that electrons inside an atom move with speed that is about one percent the speed of light.

Therefore, Schrodinger equation is successful in describing most of the properties of atomic

electrons.

Schrodiner knew that his equation will break down if electrons were moving close to speed

of light. He tried to generalize his equation and so did Pauli without success

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(1)Why do we need equations describing particles close to speed of light ??

(a) Although both in classical and quantum world, most particles ( including particles

like electrons inside the atom ) move very slowly compared to speed of light, particles in

cosmic rays that come from outer space move close to speed of light. Also, there are experiments

in the laboratory where scientist study collision between particles that move close to speed of light.

(b) Very high precision measurements require talking relativistic effects – small corrections.

Note that even though electrons in the atom move with speed that is one percent the speed of

light, there are small corrections to Bohr energy levels ( called fine structure of levels ) due to this

and high precision measurement can measure this.

Dirac wanted to find equation for an electron that marries quantum mechanics and relativity.

It is like finding the generalization of Newton’s equation by Einstein.

Dirac did not proceed from experimental facts. He was not trying to explain any laboratory data....

He guided his search using Simplicity and Principle of Mathematical beauty....

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He looked for equation which satisfies some general symmetries:

• Equations should be same at all points in space

• Space and Time should be treated on equal footing

• Equation should respect mirror symmetry ( left-right) and time reversal symmetry ( future

and past)

Dirac believed that both the correct equation must have both space and time derivatives to

be first order ( unlike Schrodinger equation which contains first derivative in time and second

derivative in space) partly due to simplicity and partly due to the same reason why the equation is

first order in time in Schrodinger theory.

Dirac equation is,

i~∂tΨ =[−i~cα · ∂x +mc2 + V

])Ψ (3)

Compare it with Schodinger equation :

i~∂tΨ =

[−~2

2m∂2x + V

]Ψ (4)

Some “Important” differences between the Dirac equation and Schodinger equation are

• In Dirac equation, we have ∂x ( first derivative of space coordinate ) while in Schrodinger

equation, we have ∂2x - that is second derivative of space coordinate. In other words, in Dirac

equation, both space and time derivatives are treated on equal footing.

• There is this strange term α in the Dirac equation. To make equation consistent with

relativity, it turns out that “α” are matrices: they are 4 × 4 matrices. This implies that

Ψ has four components.

Ψ =

ψ1

ψ2

ψ3

ψ4

(5)

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Dirac wave function has four components.

What does it Mean ???

Electron can exist in four different forms

(1) Electron with spin up,

(2) Electron with spin down,

(3) Electron with positive charge ( now called positron) with spin up and,

(4) Positron with spin down.

Here positron is an anti-particle of electron, that is a particle of same mass as electron, but

positive charge. In other words, Dirac theory predicts that for every particle, there should exists

anti-particle ( opposite charge ).

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In its simplest form ( use units where c = 1, ~ = 1 ), the equation can also be written as,

(iγ · ∂ −m)Ψ = 0 (6)

In 1928, the idea of spin was around and Dirac equation implies the existence of spin as some

intrinsic degree of freedom that emerges if we impose basic symmetries of Nature on equation.

However, the idea that for every electron, there exists a positron was just a theoretical

prediction from Dirac equation. At that time as NO such particle was known.

POSITRON DISCOVERY

Carl David Anderson was an American physicist who discovered positron in 1932, an

achievement for which he received the 1936 Nobel Prize in Physics. At the time of his discovery,

he did not know Dirac’s prediction.

Anti-Matter

It is believed that in the early part of the universe, there were equal number of particles and

their antiparticles. Why, no antiparticles left in the universe is a mystery.

What about neutral particles ?? It turns out that they also have an anti-particle, which for

fermion is different from particle in other properties..like there is anti-neutron and anti-neutrino

which are different from neutron and neutrino.

What happens when particle and anti-particle meet ??

They annihilate each other and two photons are emitted. The total energy is conserved,

mass gets covered into light according to E = mc2. Why two photons and not just one... –

Conservation of momentum

WHERE IS THE ANTI-MATTER ???

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It turns out that all particles have their partner anti-particles ?? Do they exist in nature ???

Can we have anti-H-atom made up of anti-proton plus positron ???

Why NO ???

PET SCAN

A positron emission tomography (PET) scan is an imaging test that helps reveal how your

tissues and organs are functioning. A PET scan uses a radioactive drug (tracer) to show this

activity. This scan can sometimes detect disease before it shows up on other imaging tests

A PET scan is an effective way to examine the chemical activity in parts of your body. It may

help identify a variety of conditions, including many cancers, heart disease and brain disorders.

The pictures from a PET scan provide information different from that uncovered by other types

of scans, such as computerized tomography (CT) or magnetic resonance imaging (MRI). A PET

scan or a combined CT-PET scan enables your doctor to better diagnose illness and assess your

condition.

Positron-Electron tomography... Radio active atoms that decay by emitting positron. These

positrons do not travel very far, before being annihilated by an electron, producing two photons,

which escape our skull and can be detected. From there one can trace out the path of the atom and

obtain image of brain tissue.

Where does the positron come from ???

Neutron→ Proton + electron + neutrino

Proton→ Neutron + positron + neutrino

PET scan represents one of the most sophisticated nuclear medicine techniques of the last

years. It was initially used to study the brain and the heart, but today it is used mainly in oncology.

PET scanning is a non-invasive imaging method that differs from others because it observes ”in

vivo” metabolic activity using radio-isotopes with specific tissutal tropism. More in detail PET

scans necessitate the injection of a small quantity of biologically important material like glucose

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or oxygen which have labelled with radio-nuclides such as carbon-11, nitrogen-13, oxygen-15 and

fluoride-18.

All the used isotopes are radioactive with a rapid time of decaying by positron emission:

carbon, or 11C is a radioactive isotope of carbon with a half-life in the order of twenty minutes.

Decay of Carbon into Boron is given by:

C116 → B11

5 + e+ + ν + 0.96Mev.

Nitrogen-13 or 13N is an isotope of nitrogen with a half life of approximately ten minutes.

Oxygen-15 or 15O is an isotope of oxygen having a half life of about two minutes:

Nuclear Reaction

Note that emission of positron is a nuclear reaction, where particles inside the nucleus decay.

Compare nuclear and chemical reaction

In chemical reactions, electrons participate while in nuclear reactions, particles inside the

nuclear participate. Nuclear reaction involve lot more energy than chemical reaction.

Why do some Elements emit Positrons ?

Because positron emission decreases proton number relative to neutron number, positron

decay happens typically in large “proton-rich” radionuclides. Positron decay results in nuclear

transmutation, changing an atom of one chemical element into an atom of an element with an

atomic number that is less by one unit.

Majorana Fermions

A fermion that is its own anti-particle is called Majorana fermion, predicted by Ettore

Majorana in 1937. They have not been found and there search constitutes one of the most active

line of research in physics.

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