Phy 310 chapter 2

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Dr Ahmad Taufek Abdul RahmanSchool of Physics & Material Studies

Faculty of Applied Sciences

Universiti Teknologi MARA Malaysia

Campus of Negeri Sembilan

72000 Kuala Pilah, NS

DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory

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To know the Revolutionary impact of quantum physics

one need first to look at pre-quantum physics:


Max Planck

• 1900 : Max Plank introduced the concept of energy

radiated in discrete quanta.

• Found relationship between the radiation emited by

a blackbody and its temperature.

• E=hѵ quanta of energy is proportional to the

frequency with which the blackbody radiate

assuming that energies of the vibrating electrons

that radiate the light are quantized obtain an

expression that agreed with experiment.

he recognized that the theory was

physically absurd, he described as "an act

of desperation" .

3DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory

Albert Einstein

The photoelectric effect

Not explained by Maxwell's theory since the rate of electrons not

depended on the intensity of light, but in the frequency.

1905: Einstein applied the idea of Plank's constant to the problem

of the photoelectric effect light consists of individual quantum

particles, which later came to be called photons (1926).

Electrons are released from certain materials only when particular

frequencies are reached corresponding to multiples of Plank's

constant .


Niels Bohr

• 1913 : Bohr quantized energy explain how electrons orbit a

nucleus.

• Electrons orbit with momenta, and energies quantized.

• Electrons do not loose energy as they orbit the nucleus, only

change their energy by "jumping" between the stationary states

emitting light whose wavelength depends on the energy difference.

• Explained the Rydberg formula (1888), which correctly modeled

the light emission spectra of atomic hydrogen

• Although Bohr's theory was full of contradictions, it provided a

quantitative description of the spectrum of the hydrogen atom


Two theorist, Niels Bohr and

Max Planck, at the blackboard.


1916 Arnold Sommerfeld :

- To account for the Zeeman effect (1896): atomic absorption or

emission spectral lines change when the light is first shinned

through a magnetic field,

- he suggested ―elliptical orbits‖ in atoms in addition to spherical

orbits.

In 1924, Louis de Broglie:

- theory of matter waves

- particles can exhibit wave characteristics and vice versa, in

analogy to photons.

1924, another precursor Satyendra N. Bose:

- new way to explain the Planck radiation law.

- He treated light as if it were a gas of massless particles (now

called photons).

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By the late 1910s :


Scientific revolution 1925 to January 1928

Wolfang Pauli: the exclusion principle

Werner Heisemberg, with Max Born and Pascual Jordan,

- discovered matrix mechanics first version of quantum mechanics.

Erwin Schrödinger:

- invented wave mechanics, a second form of quantum mechanics in which

the state of a system is described by a wave function,

- Electrons were shown to obey a new type of statistical law, Fermi- Dirac

statistics

Heisenberg :Uncertainty Principle.

Dirac :contributions to quantum mechanics and quantum electrodynamics


Many physicists have also contributed to the

quantum theory:

• Max Planck : Light quanta

• Einstein ―photon‖: photoelectric

• Louis de Broglie: Matter waves

• Erwin Schrödinger: waves equations

• Max Born: probability waves

• Heisenberg: uncertainty

• Paul Dirac: Spin electron equation

• Niels Bohr: Copenhagen

• Feynman: Quantum-electrodynamics

• John Bell: EPR Inequality locality

• David Bohm: Pilot wave (de Broglie)

• ...

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Paul Dirac and Werner

Heisemberg in Cambrige,1930.


The first Solvay Congress in 1911 assembled the pioneers of

quantum theory.


Old faces and new at 1927 Solvay Congress



Werner Karl Heisenberg : Brief chronology

• 1901 - 5Dec: He was born in Würzburg, Germany

• 1914 :Outbreak of World War I.

• 1920 he entered at the University of Munich

Arnold Sommerfeld admitted him to his advanced seminar.

• 1925. 29 June Receipt of Heisenberg's paper providing breakthrough to quantum

mechanics

• 1927. 23 Mar. Receipt of Heisenberg's paper on the uncertainty principle.

• 1932. 7 June Receipt of his first paper on the neutron-proton model of nuclei.

• 1933 .11 Dec. Heisenberg receives Nobel Prize for Physics (for 1932).

• 1976. 1 Feb. Dies because of cancer at his home in Munich.


http://www.aip.org/history/heisenberg/p14.htm

Influences- Studied with three of the world‘s leading atomic

theorists: Sommerfeld, Max Born and Niels

Bohr.

- In 3 of the world‘s leading centres for theoretical

atomic physics: Munich, Göttingen and

Copenhagen.

- “From Sommerfeld I

learn optimism, from

the Göttigen people

mathematics and

from Bohr physics” –

Heisemberg

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- In Munich he began a life-long friendship with Wolfgang Pauli.

Arnold Sommerfeld (left)

and Niels Bohr

Wolfgang Pauli

Max Born


Heisenberg‘s travels and teachers during help him to become

one of the leading physicists of his time.

Goal fortune of entering in the ―world atomic physics‖ just in

the right moment for breakthrough.

Found that properties of the atoms predicted from the

calculations did not agree with existing experimental data.

―The old quantum theory‖, worked well in simple cases, but

experimental and theoretical study was revealing many

problems crisis in quantum theory.

The old quantum theory had failed but Heisenberg and his

colleagues saw exactly where it failed.

15

During 1920


Quantum mechanics 1925-1927

The leading theory of the atom when Heisenberg entered at University was quantum theory of Bohr.

Although it had been highly successful, three areas of research indicated that this theory was inadequate:

light emitted and absorbed by atoms

the predicted properties of atoms and molecules

The nature of light, did it act like waves or like a stream of particles?

1924 physicists were agreed old quantum theory had to be replaced by ―quantum mechanics‖.


Heisenberg set the task of finding the new

quantum mechanics:

Since the electron orbits in atoms could not be observed, hetried to develop a quantum mechanics without them.

By 1925 he had an answer, but the mathematics was so unfamiliar that he was not sure if it made any sense.

These unfamiliar mathematics contain arrays of numbers known as ―matrix‖.

Born sent Heisenberg‘s paper off for publication.

―All of my meagre efforts go toward

killing off and suitably replacing the

concept of the orbital path which cannot

observe‖ Heisemberg, letter to Pauli

1925

The breakthrough to quantum mechanics:


The first page of Heisenberg's

break-through paper on

quantum mechanics,

published in the Zeitschrift für

Physik, 33 (1925),

18

“The present paper seeks to

establish a basis for theoretical

quantum mechanics founded

exclusively upon relationships

between quantities which in

principle are observable”.

Heisemberg, summary abstract

of his first paper on quantum

mechanics


The wave-function formulation

1926: Erwin Schrödinger proposed another quantum

mechanics, ―wave mechanics‖.

Appealed to many physicists because it seemed to

do everything that matrix mechanics could do but

much more easily and seemingly without giving up

the visualization of orbits within the atom.

“I knew of [Heisemberg] theory, of course, but I felt discouraged, not to say

repelled, by the methods of transcendental algebra, which appeared difficult to

me, and by the lack of visualizability.”- Schrödinger in 1926.


The Uncertainty Principle

1926: The rout to uncertainty relations lies in a debate

between alternative versions of quantum mechanics:

- Heisenberg and his closest colleagues who espoused

the “matrix form” of quantum mechanics

- Schrödinger and his colleagues who espoused the new

“wave mechanics ‖.

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May 1926, Matrix mechanics and wave mechanics, apparently

incompatible proof that gave equivalent results.

“The more I think about the physical portion of

Schrödinger’s theory, the more repulsive I find it.. What

Schrödinger writes about the visualizability of his theory is

not quite right, in other words it’s crap” Heisenberg, writing

to Pauli, 1926


.

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In 1927 the intensive work led to Heisenberg‘s uncertainty

principle and the ―Copenhagen Interpretation‖

“The more precisely the position is determined, the less

precisely the momentum is known in this instant, and vice versa”

Heisenberg, uncertainty paper, 1927

After that, Born presented a statistical interpretation of the wave

function, Jordan in Göttingen and Dirac in Cambridge, created

unified equations known as ―transformation theory‖. The basis of

what is now regarded as quantum mechanics.

The uncertainty principle was not accepted by everyone. It‘s

most outspoken opponent was Einstein.


Conclusion

The history of Quantum mechanics it‘s not easy, many events

pass simultaneously difficult period.

Quantum mechanics was created to describe an abstract

atomic world far removed from daily experience, its impact on

our daily lives has become very important.

Spectacular advances in chemistry, biology, and medicine…

Quantum information

The creation of quantum physics has transformed our world,

bringing with it all the benefits—and the risks—of a scientific

revolution.


DR.ATAR @ UiTM.NS 23PHY310 - Early Quantum Theory

Ancient Philosophy Who: Aristotle, Democritus

When: More than 2000 years ago

Where: Greece

What: Aristotle believed in 4 elements: Earth,

Air, Fire, and Water. Democritus believed that

matter was made of small particles he named

―atoms‖.

Why: Aristotle and Democritus used

observation and inferrence to explain the

existence of everything.


Aristotle

Democritus


Alchemists Who: European Scientists

When: 800 – 900 years ago

Where: Europe

What: Their work developed into what is now

modern chemistry.

Why: Trying to change ordinary materials into

gold.


Alchemic Symbols


Particle Theory Who: John Dalton

When: 1808

Where: England

What: Described atoms as tiny particles that

could not be divided. Thought each element

was made of its own kind of atom.

Why: Building on the ideas of Democritus in

ancient Greece.


John Dalton


Discovery of Electrons Who: J. J. Thompson

When: 1897

Where: England

What: Thompson discovered that electrons

were smaller particles of an atom and were

negatively charged.

Why: Thompson knew atoms were neutrally

charged, but couldn‘t find the positive particle.


J. J. Thompson


Atomic Structure I Who: Ernest Rutherford

When: 1911

Where: England

What: Conducted an experiment to isolate the

positive particles in an atom. Decided that the

atoms were mostly empty space, but had a

dense central core.

Why: He knew that atoms had positive and

negative particles, but could not decide how

they were arranged.


Ernest Rutherford


Atomic Structure II Who: Niels Bohr

When: 1913

Where: England

What: Proposed that electrons traveled in fixed

paths around the nucleus. Scientists still use

the Bohr model to show the number of

electrons in each orbit around the nucleus.

Why: Bohr was trying to show why the negative

electrons were not sucked into the nucleus of

the atom.


Niels Bohr


Electron Cloud Model

Electrons travel around the nucleus in random

orbits.

Scientists cannot predict where they will be at

any given moment.

Electrons travel so fast, they appear to form a

―cloud‖ around the nucleus.


Electron Cloud Model


Defining the Atom

OBJECTIVES:

Describe Democritus‘s

ideas about atoms.


Defining the Atom

OBJECTIVES:

Explain Dalton‘s atomic

theory.


Defining the Atom

OBJECTIVES:

Identify what instrument is

used to observe individual

atoms.


Defining the Atom

The Greek philosopher Democritus (460

B.C. – 370 B.C.) was among the first to

suggest the existence of atoms (from

the Greek word ―atomos‖)

He believed that atoms were indivisible and

indestructible

His ideas did agree with later scientific

theory, but did not explain chemical

behavior, and was not based on the

scientific method – but just philosophyDR.ATAR @ UiTM.NS 41PHY310 - Early Quantum Theory

Dalton‘s Atomic Theory (experiment based!)

3) Atoms of different elements combine in simple whole-

number ratios to form chemical compounds

4) In chemical reactions, atoms are combined, separated,

or rearranged – but never changed into atoms of

another element.

1) All elements are composed of tiny

indivisible particles called atoms

2) Atoms of the same element are

identical. Atoms of any one element

are different from those of any other

element.John Dalton

(1766 – 1844)


Sizing up the Atom

Elements are able to be subdivided into smaller

and smaller particles – these are the atoms, and

they still have properties of that element

If you could line up 100,000,000 copper atoms

in a single file, they would be approximately 1

cm long

Despite their small size, individual atoms are

observable with instruments such as scanning

tunneling (electron) microscopes


Structure of the Nuclear Atom

OBJECTIVES:

Identify three types of

subatomic particles.



OBJECTIVES:

Describe the structure of

atoms, according to the

Rutherford atomic model.



One change to Dalton‘s atomic

theory is that atoms are divisible

into subatomic particles:

Electrons, protons, and neutrons are

examples of these fundamental

particles

There are many other types of

particles, but we will study these threeDR.ATAR @ UiTM.NS 46PHY310 - Early Quantum Theory

Discovery of the ElectronIn 1897, J.J. Thomson used a cathode ray

tube to deduce the presence of a negatively

charged particle: the electron


Modern Cathode Ray Tubes

Cathode ray tubes pass electricity through a gas

that is contained at a very low pressure.

Television Computer Monitor


Mass of the Electron

1916 – Robert Millikan determines the mass of the

electron: 1/1840 the mass of a hydrogen atom;

has one unit of negative charge

The oil drop apparatus

Mass of the

electron is

9.11 x 10-28 g


Conclusions from the Study of

the Electron:

a) Cathode rays have identical properties

regardless of the element used to produce

them. All elements must contain identically

charged electrons.

b) Atoms are neutral, so there must be positive

particles in the atom to balance the negative

charge of the electrons

c) Electrons have so little mass that atoms

must contain other particles that account for

most of the mass


Conclusions from the Study

of the Electron:

Eugen Goldstein in 1886 observed what

is now called the “proton” - particles

with a positive charge, and a relative

mass of 1 (or 1840 times that of an

electron)

1932 – James Chadwick confirmed the

existence of the “neutron” – a particle

with no charge, but a mass nearly

equal to a proton


Subatomic Particles

Particle Charge Mass (g) Location

Electron

(e-) -1 9.11 x 10-28 Electron

cloud

Proton

(p+) +1 1.67 x 10-24 Nucleus

Neutron

(no) 0 1.67 x 10-24 Nucleus


Thomson‘s Atomic Model

Thomson believed that the electrons were like

plums embedded in a positively charged

“pudding,” thus it was called the “plum

pudding” model.

J. J. Thomson


Ernest Rutherford’s

Gold Foil Experiment - 1911

• Alpha particles are helium nuclei - The alpha

particles were fired at a thin sheet of gold

foil

• Particles that hit on the detecting screen

(film) are recorded


Rutherford’s Findings

a) The nucleus is small

b) The nucleus is dense

c) The nucleus is positively

charged

Most of the particles passed right through

A few particles were deflected

VERY FEW were greatly deflected

“Like howitzer shells bouncing

off of tissue paper!”

Conclusions:


The Rutherford Atomic Model

Based on his experimental evidence:

The atom is mostly empty space

All the positive charge, and almost all themass is concentrated in a small area in thecenter. He called this a ―nucleus‖

The nucleus is composed of protons andneutrons (they make the nucleus!)

The electrons distributed around thenucleus, and occupy most of the volume

His model was called a ―nuclear model‖


Distinguishing Among Atoms

OBJECTIVES:

Explain what makes

elements and isotopes

different from each other.



OBJECTIVES:

Calculate the number of

neutrons in an atom.



OBJECTIVES:

Calculate the atomic

mass of an element.



OBJECTIVES:

Explain why chemists use

the periodic table.


Atomic Number

Atoms are composed of identical

protons, neutrons, and electrons

How then are atoms of one element

different from another element?

Elements are different because they

contain different numbers of PROTONS

The ―atomic number‖ of an element is

the number of protons in the nucleus

# protons in an atom = # electrons


Atomic Number

Atomic number (Z) of an element is

the number of protons in the nucleus

of each atom of that element.

Element # of protons Atomic # (Z)

Carbon 6 6

Phosphorus 15 15

Gold 79 79


Mass NumberMass number is the number of protons and

neutrons in the nucleus of an isotope:

Mass # = p+ + n0

Nuclide p+ n0 e- Mass #

Oxygen - 10

- 33 42

- 31 15

8 8 1818

Arsenic 75 33 75

Phosphorus 15 3116


Complete Symbols

Contain the symbol of the element,

the mass number and the atomic

number.

XMass

number

Atomic

numberSubscript →

Superscript →


Symbols

Find each of these:

a) number of protons

b) number of

neutrons

c) number of

electrons

d) Atomic number

e) Mass Number

Br80

35


Symbols

If an element has an atomic

number of 34 and a mass

number of 78, what is the:

a) number of protons

b) number of neutrons

c) number of electrons

d) complete symbol


Symbols

If an element has 91 protons

and 140 neutrons what is the

a) Atomic number

b) Mass number

c) number of electrons

d) complete symbol


Symbols

If an element has 78

electrons and 117 neutrons

what is the

a) Atomic number

b) Mass number

c) number of protons

d) complete symbol


Isotopes

Dalton was wrong about allelements of the same type beingidentical

Atoms of the same element canhave different numbers ofneutrons.

Thus, different mass numbers.

These are called isotopes.


Isotopes

Frederick Soddy (1877-1956) proposed the idea

of isotopes in 1912

Isotopes are atoms of the same element

having different masses, due to varying

numbers of neutrons.

Soddy won the Nobel Prize in Chemistry in 1921

for his work with isotopes and radioactive

materials.


Naming Isotopes

We can also put the mass

number after the name of the

element:

carbon-12

carbon-14

uranium-235


Isotopes are atoms of the same element

having different masses, due to varying

numbers of neutrons.

Isotope Protons Electrons Neutrons Nucleus

Hydrogen–1

(protium) 1 1 0

Hydrogen-2

(deuterium) 1 1 1

Hydrogen-3

(tritium)

1 1 2


IsotopesElements

occur in

nature as

mixtures of

isotopes.

Isotopes are

atoms of the

same element

that differ in the

number of

neutrons.


Atomic Mass How heavy is an atom of oxygen?

It depends, because there are different

kinds of oxygen atoms.

We are more concerned with the average

atomic mass.

This is based on the abundance

(percentage) of each variety of that element

in nature.

We don‘t use grams for this mass because

the numbers would be too small.


Measuring Atomic Mass

Instead of grams, the unit we use is theAtomic Mass Unit (amu)

It is defined as one-twelfth the mass ofa carbon-12 atom.

Carbon-12 chosen because of itsisotope purity.

Each isotope has its own atomic mass,thus we determine the average frompercent abundance.


To calculate the average:

Multiply the atomic mass of eachisotope by it‘s abundance (expressedas a decimal), then add the results.

If not told otherwise, the mass of theisotope is expressed in atomic mass

units (amu)


Atomic Masses

Isotope Symbol Composition of

the nucleus

% in nature

Carbon-12 12C 6 protons

6 neutrons

98.89%


7 neutrons

1.11%


8 neutrons

<0.01%

Atomic mass is the average of all the naturally

occurring isotopes of that element.

Carbon = 12.011DR.ATAR @ UiTM.NS 77PHY310 - Early Quantum Theory

- Page 117

Question

Solution

Answer

Knowns

and

Unknown


The Periodic Table: A Preview

A “periodic table” is an arrangement

of elements in which the elements are

separated into groups based on a set

of repeating properties

The periodic table allows you to

easily compare the properties of one

element to another


The Periodic Table: A Preview

Each horizontal row (there are 7 of them)

is called a period

Each vertical column is called a group, or

family

Elements in a group have similar

chemical and physical properties

Identified with a number and either an

“A” or “B”

More presented in Chapter 6



Louis de Broglie

Louis, 7th duc de Broglie was born on August 15, 1892, in Dieppe,

France. He was the son of Victor, 5th duc de Broglie. Although he

originally wanted a career as a humanist (and even received his

first degree in history), he later turned his attention to physics and

mathematics. During the First World War, he helped the French

army with radio communications.

In 1924, after deciding a career in physics and mathematics, he

wrote his doctoral thesis entitled Research on the Quantum Theory.

In this very seminal work he explains his hypothesis about

electrons: that electrons, like photons, can act like a particle and a

wave. With this new discovery, he introduced a new field of study

in the new science of quantum physics: Wave Mechanics!


Fundamentals of Wave Mechanics

First a little basics about waves. Waves are disturbances

through a medium (air, water, empty vacuum), that usually

transfer energy.

Here is one:


Fundamentals of Wave Mechanics

(Cont’d.) The distance between each bump is called a wavelength (λ),

and how many bumps there are per second is called the frequency (f). The velocity at which the wave crest moves is jointly proportional to λ and f:

V = λ f

Now there are two velocities associated with the wave:

the group velocity (v) and the phase velocity (V).

When dealing with waves going in oscillations (cycles of periodic movements), we use notations of angular frequency (ω) and the wavenumber (k) – which is inversely proportional to the wavelength. The equations for both are:

ω = 2πf and k = 2π/ λ


Fundamentals (Cont’d)

The phase velocity of the wave (V) is directly proportional to the angular frequency, but inversely proportional to the wavenumber, or:

V = ω / kThe phase velocity is the velocity of the oscillation (phase) of the wave.

The group velocity is equal to the derivative of the angular frequency with respect to the wavenumber, or:

v = d ω / d k

The group velocity is the velocity at which the energy of the wave propagates. Since the group velocity is the derivative of the phase velocity, it is often the case that the phase velocity will be greater than the group velocity. Indeed, for any waves that are not electromagnetic, the phase velocity will be greater than ‗c‘ – or the speed of light, 3.0 * 108 m/s.


Derivation for De Broglie Equation

De Broglie, in his research, decided to look at Einstein‘s research

on photons – or particles of light – and how it was possible for light

to be considered both a wave and a particle. Let us look at how

there is a relationship between them.

We get from Einstein (and Planck) two equations for energy:

E = h f (photoelectric effect) & E = mc2 (Einstein‘s Special

Relativity)

Now let us join the two equations:

E = h f = m c2


Derivation (Cont’d.)

From there we get:

h f = p c (where p = mc, for the momentum of a photon)

h / p = c / f

Substituting what we know for wavelengths (λ = v / f, or in this case c / f ):

h / mc = λ

De Broglie saw that this works perfectly for light waves, but does it work for particles other than photons, also?


Derivation (Cont’d.)

In order to explain his hypothesis, he would have to associate two wave velocities with the particle. De Broglie hypothesized that the particle itself was not a wave, but always had with it a pilot wave, or a wave that helps guide the particle through space and time. This wave always accompanies the particle. He postulated that the group velocity of the wave was equal to the actual velocity of the particle.

However, the phase velocity would be very much different. He saw that the phase velocity was equal to the angular frequency divided by the wavenumber. Since he was trying to find a velocity that fit for all particles (not just photons) he associated the phase velocity with that velocity. He equated these two equations:

V = ω / k = E / p (from his earlier equation c = (h f) / p )


Derivation (Cont’d)

From this new equation from the phase velocity we can derive:

V = m c2 / m v = c2 / v

Applied to Einstein‘s energy equation, we have:

E = p V = m v (c2 / v)

This is extremely helpful because if we look at a photon traveling at the velocity c:

V = c2 / c = c

The phase velocity is equal to the group velocity! This allows for the equation to be applied to particles, as well as photons.


Derivation (Cont’d)

Now we can get to an actual derivation of the De Broglie equation:

p = E / V

p = (h f) / V

p = h / λ

With a little algebra, we can switch this to:

λ = h / m v

This is the equation De Broglie discovered in his 1924 doctoral thesis! It accounts for both waves and particles, mentioning the momentum (particle aspect) and the wavelength (wave aspect). This simple equation proves to be one of the most useful, and famous, equations in quantum mechanics!


De Broglie and Bohr

De Broglie‘s equation brought relief to many people, especially

Niels Bohr. Niels Bohr had postulated in his quantum theory that

the angular momentum of an electron in orbit around the nucleus of

the atom is equal to an integer multiplied with h / 2π, or:

n h / 2π = m v r

We get the equation now for standing waves:

n λ = 2π r

Using De Broglie‘s equation, we get:

n h / m v = 2π r

This is exactly in relation to Niels Bohr‘s postulate!


De Broglie and Relativity

Not only is De Broglie‘s equation useful for small particles, such as electrons and protons, but can also be applied to larger particles, such as our everyday objects. Let us try using De Broglie‘s equation in relation to Einstein‘s equations for relativity. Einstein proposed this about Energy:

E = M c2 where M = m / (1 – v2 / c2) ½ and m is rest mass.

Using what we have with De Broglie:

E = p V = (h V) / λ

Another note, we know that mass changes as the velocity of the object goes faster, so:

p = (M v)

Substituting with the other wave equations, we can see:

p = m v / (1 – v / V) ½ = m v / (1 – k x / ω t ) ½

One can see how wave mechanics can be applied to even Einstein‘s theory of relativity. It is much bigger than we all can imagine!


Conclusion

We can see very clearly how helpful De Broglie‘s equation has

been to physics. His research on the wave-particle duality is one of

the biggest paradigms in quantum mechanics, and even physics

itself. In 1929 Louis, 7th duc de Broglie received the Nobel Prize in

Physics for his ―discovery of the wave nature of electrons.‖ It was a

very special moment in history, and for Louis de Broglie himself.

He died in 1987, in Paris, France, having never been married. Let

us pay him tribute as CW Oseen, the Chairman for the Nobel

Committee for Physics, did when he said about de Broglie:

“You have covered in fresh glory a name already crowned for

centuries with honour.”

(On the next two slides contains an appendix on the relation between

wave mechanics and relativity, if it could be of any help to anyone.)


Appendix: Wave Mechanics and

Relativity

We get from Einstein these equations from his Special Theory of Relativity:

t = T / (1 - v2 / c2) ½ , L = l (1 - v2 / c2) ½ , M = m / (1 - v2 / c2) ½

I pointed out earlier that c2 / v2 can be replaced with ω t / k x. One can see

the relationship then that wave mechanics would have on all particles, and

vice versa. Of course, in the case of time, you could replace the k x / ω t

with k v / ω.

Similarly, it is careful to observe this relativity being applied to wave

mechanics. We have, using Einstein‘s equation for Energy, two equations

satisfying Energy:

E = h F = M c2.

Since mass M (which shall be used as m for intent purposes on the early

slides where I derive De Broglie‘s equation) undergoes relativistic changes,

so does the frequency F (which shall be used as f for earlier slides due to

the same reasoning):

E = h f / (1 - v2 / c2) ½ , which gives us the final equation for Energy:

E = h f / (1 - k x / ω t ) ½.


Appendix (Cont’d)

With this in mind, it is also worthy to take in mind dealing with supra-relativity (my own coined term for events that occur with objects traveling faster than the speed of light). It would be interesting to note that the phase velocity is usually greater than the speed of light. Although no superluminal communication or energy transfer occurs under such a velocity, it would be interesting to see what mechanics could arise from just such a situation.

A person traveling on the phase wave is traveling at velocity V. His position would then be X.

Using classical laws:

X = V t

We see when we analyze ω t / k x that we can fiddle with the math:

k x / ω t = x / V t = X / x

Thus, Einstein‘s equations refined:

t = T / (1 - x / X ) ½ , L = l (1 - x / X ) ½ , M = m / (1 - x / X ) ½

Essentially, if we imagined a particle (or a miniature man) traveling on the phase wave, we could measure his conditions under the particle‘s velocity. Take it as you will.


Photons and Waves Revisited

Some experiments are best explained by the photon model.

Some are best explained by the wave model.

We must accept both models and admit that the true nature of light is

not describable in terms of any single classical model.

The particle model and the wave model of light complement each other.

A complete understanding of the observed behavior of light can be

attained only if the two models are combined in a complementary

matter.


Louis de Broglie

1892 – 1987

French physicist

Originally studied history

Was awarded the Nobel Prize in 1929

for his prediction of the wave nature

of electrons


Wave Properties of Particles

Louis de Broglie postulated that because photons have both wave and

particle characteristics, perhaps all forms of matter have both

properties.

The de Broglie wavelength of a particle is

h hλ

p mu


Frequency of a Particle

In an analogy with photons, de Broglie postulated that a particle would

also have a frequency associated with it

These equations present the dual nature of matter:

Particle nature, p and E

Wave nature, λ and ƒ

ƒE

h


Complementarity

The principle of complementarity states that the wave and particle

models of either matter or radiation complement each other.

Neither model can be used exclusively to describe matter or radiation

adequately.


Davisson-Germer Experiment

If particles have a wave nature, then under the correct conditions, they

should exhibit diffraction effects.

Davisson and Germer measured the wavelength of electrons.

This provided experimental confirmation of the matter waves proposed

by de Broglie.


Wave Properties of Particles

Mechanical waves have materials that are ―waving‖ and can be

described in terms of physical variables.

A string may be vibrating.

Sound waves are produced by molecules of a material vibrating.

Electromagnetic waves are associated with electric and

magnetic fields.

Waves associated with particles cannot be associated with a physical

variable.


Electron Microscope

The electron microscope relies on the

wave characteristics of electrons.

Shown is a transmission electron

microscope

Used for viewing flat, thin

samples

The electron microscope has a high

resolving power because it has a very

short wavelength.

Typically, the wavelengths of the

electrons are about 100 times shorter

than that of visible light.


Quantum Particle

The quantum particle is a new model that is a result of the recognition

of the dual nature of both light and material particles.

Entities have both particle and wave characteristics.

We must choose one appropriate behavior in order to understand a

particular phenomenon.


Ideal Particle vs. Ideal Wave

An ideal particle has zero size.

Therefore, it is localized in space.

An ideal wave has a single frequency and is infinitely long.

Therefore, it is unlocalized in space.

A localized entity can be built from infinitely long waves.


Particle as a Wave Packet

Multiple waves are superimposed so that one of its crests is at x = 0.

The result is that all the waves add constructively at x = 0.

There is destructive interference at every point except x = 0.

The small region of constructive interference is called a wave packet.

The wave packet can be identified as a particle.


Wave Envelope

The dashed line represents the envelope function.

This envelope can travel through space with a different speed than the

individual waves.


Speeds Associated with Wave

PacketThe phase speed of a wave in a wave packet is given by

This is the rate of advance of a crest on a single wave.

The group speed is given by

This is the speed of the wave packet itself.

phaseωv

k

gdωv

dk


Speeds, cont.

The group speed can also be expressed in terms of energy and

momentum.

This indicates that the group speed of the wave packet is identical to

the speed of the particle that it is modeled to represent.

2 1

22 2

g

dE d pv p u

dp dp m m


Electron Diffraction, Set-Up


Electron Diffraction,

ExperimentParallel beams of mono-energetic electrons that are incident on a double slit.

The slit widths are small compared to the electron wavelength.

An electron detector is positioned far from the slits at a distance much greater than the slit separation.


Electron Diffraction, cont.

If the detector collects electrons for a long enough time, a typical wave interference pattern is produced.

This is distinct evidence that electrons are interfering, a wave-like behavior.

The interference pattern becomes clearer as the number of electrons reaching the screen increases.


Electron Diffraction,

EquationsA maximum occurs when

This is the same equation that was used for light.

This shows the dual nature of the electron.

The electrons are detected as particles at a localized spot at

some instant of time.

The probability of arrival at that spot is determined by finding the

intensity of two interfering waves.

sin d θ mλ


Electron Diffraction Explained

An electron interacts with both slits simultaneously.

If an attempt is made to determine experimentally which slit the electron goes through, the act of measuring destroys the interference pattern.

It is impossible to determine which slit the electron goes through.

In effect, the electron goes through both slits.

The wave components of the electron are present at both slits at the same time.


Werner Heisenberg1901 – 1976

German physicist

Developed matrix mechanics

Many contributions include:

Uncertainty principle

○ Received Nobel

Prize in 1932 Prediction of two forms of

molecular hydrogen

Theoretical models of the

nucleus


The Uncertainty Principle

In classical mechanics, it is possible, in principle, to make

measurements with arbitrarily small uncertainty.

Quantum theory predicts that it is fundamentally impossible to make

simultaneous measurements of a particle‘s position and momentum

with infinite accuracy.

The Heisenberg uncertainty principle states: if a measurement of the

position of a particle is made with uncertainty Dx and a simultaneous

measurement of its x component of momentum is made with

uncertainty Dpx, the product of the two uncertainties can never be

smaller than /2.

2xx pD D


Heisenberg Uncertainty

Principle, ExplainedIt is physically impossible to measure simultaneously the exact position

and exact momentum of a particle.

The inescapable uncertainties do not arise from imperfections in

practical measuring instruments.

The uncertainties arise from the quantum structure of matter.


Heisenberg Uncertainty

Principle, Another FormAnother form of the uncertainty principle can be expressed in terms of

energy and time.

This suggests that energy conservation can appear to be violated by an

amount DE as long as it is only for a short time interval Dt.

2E tD D


Uncertainty Principle, final

The Uncertainty Principle cannot be interpreted as meaning that a

measurement interferes with the system.

The Uncertainty Principle is independent of the measurement process.

It is based on the wave nature of matter.


Max Planck1858 – 1847

German physicist

Introduced the concept of ―quantum

of action‖

In 1918 he was awarded the Nobel

Prize for the discovery of the

quantized nature of energy.


Planck’s Theory of Blackbody

RadiationIn 1900 Planck developed a theory of blackbody radiation that leads to an equation for the intensity of the radiation.

This equation is in complete agreement with experimental observations.

He assumed the cavity radiation came from atomic oscillations in the cavity walls.

Planck made two assumptions about the nature of the oscillators in the cavity walls.


Planck’s Assumption, 1

The energy of an oscillator can have only certain discrete values En.

En = n h ƒ

○ n is a positive integer called the quantum

number

○ ƒ is the frequency of oscillation

○ h is Planck‘s constant This says the energy is quantized.

Each discrete energy value corresponds to a different quantum

state.

○ Each quantum state is represented by the

quantum number, n.


Planck’s Assumption, 2

The oscillators emit or absorb energy when making a transition from one quantum state to another.

The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation.

An oscillator emits or absorbs energy only when it changes quantum states.

The energy carried by the quantum of radiation is E = h ƒ.


Energy-Level Diagram

An energy-level diagram shows the

quantized energy levels and allowed

transitions.

Energy is on the vertical axis.

Horizontal lines represent the allowed

energy levels.

The double-headed arrows indicate

allowed transitions.


More About Planck’s Model

The average energy of a wave is the average energy difference

between levels of the oscillator, weighted according to the probability of

the wave being emitted.

This weighting is described by the Boltzmann distribution law and gives

the probability of a state being occupied as being proportional to

where E is the energy of the state. BE k Te


Planck’s Model, Graph


Planck’s Wavelength

Distribution FunctionPlanck generated a theoretical expression for the wavelength

distribution.

h = 6.626 x 10-34 J.s

h is a fundamental constant of nature.

At long wavelengths, Planck‘s equation reduces to the Rayleigh-Jeans

expression.

At short wavelengths, it predicts an exponential decrease in intensity

with decreasing wavelength.

This is in agreement with experimental results.

I ,

2

5

2

1Bhc λk T

πhcλ T

λ e


Thank You

DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 128

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