Early Pioneers of Quantum Physics

Dr Ahmad Taufek Abdul Rahman
School of Physics & Material Studies
Faculty of Applied Sciences
Universiti Teknologi MARA Malaysia
Campus of Negeri Sembilan
72000 Kuala Pilah, NS

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PHY310 - Early Quantum Theory

1

To know the Revolutionary impact of quantum physics
one need first to look at pre-quantum physics:

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2

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" .
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3

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 .

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4

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

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5

Two theorist, Niels Bohr and
Max Planck, at the blackboard.

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6

By the late 1910s :


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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7

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

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8

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.

9

The first Solvay Congress in 1911 assembled the pioneers of
quantum theory.

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Old faces and new at 1927 Solvay Congress

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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.
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13

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.

-

Max Born

“From Sommerfeld I
learn optimism, from
the Göttigen people
mathematics and
from Bohr physics” –
Heisemberg
Arnold Sommerfeld (left)
and Niels Bohr
Wolfgang Pauli

- In Munich he began a life-long friendship with Wolfgang Pauli.
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During 1920


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.

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15

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‖.

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The breakthrough to quantum mechanics:

Heisenberg set the task of finding the new
quantum mechanics:


Since the electron orbits in atoms could not be observed, he
tried 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

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17

The first page of Heisenberg's
break-through paper on
quantum mechanics,
published in the Zeitschrift für
Physik, 33 (1925),
“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

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18

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.

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19

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 ‖.
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.

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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.

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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.


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Democritus

Aristotle

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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.


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Alchemic Symbols

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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.


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John Dalton
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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.


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J. J. Thompson
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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.


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32

Ernest Rutherford
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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.


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Niels Bohr
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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.


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Electron Cloud Model

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Defining the Atom
OBJECTIVES:

Describe Democritus‘s

ideas about atoms.

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Defining the Atom
OBJECTIVES:

Explain Dalton‘s atomic

theory.

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Defining the Atom
OBJECTIVES:

Identify what instrument is

used to observe individual
atoms.

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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 philosophy
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Dalton‘s Atomic Theory (experiment based!)

John Dalton
(1766 – 1844)

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.

3) Atoms of different elements combine in simple wholenumber ratios to form chemical compounds
4) In chemical reactions, atoms are combined, separated,
or rearranged – but never changed into atoms of
another element.
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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

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Structure of the Nuclear Atom
OBJECTIVES:

Identify three types of

subatomic particles.

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OBJECTIVES:

Describe the structure of

atoms, according to the
Rutherford atomic model.

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 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 three
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Discovery of the Electron
In 1897, J.J. Thomson used a cathode ray
tube to deduce the presence of a negatively
charged particle: the electron

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Modern Cathode Ray Tubes

Television

Computer Monitor

Cathode ray tubes pass electricity through a gas
that is contained at a very low pressure.
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Mass of the Electron
Mass of the
electron is
9.11 x 10-28 g

The oil drop apparatus
1916 – Robert Millikan determines the mass of the
electron: 1/1840 the mass of a hydrogen atom;
has one unit of negative charge
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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
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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
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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

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Thomson‘s Atomic Model

J. J. Thomson
Thomson believed that the electrons were like
plums embedded in a positively charged
“pudding,” thus it was called the “plum
pudding” model.
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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
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Rutherford’s Findings
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:
a) The nucleus is small
b) The nucleus is dense
c) The nucleus is positively

charged
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The Rutherford Atomic Model


Based on his experimental evidence:
 The atom is mostly empty space
 All the positive charge, and almost all the
mass is concentrated in a small area in the
center. He called this a ―nucleus‖
 The nucleus is composed of protons and
neutrons (they make the nucleus!)
 The electrons distributed around the
nucleus, and occupy most of the volume
 His model was called a ―nuclear model‖

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Distinguishing Among Atoms
OBJECTIVES:

Explain what makes

elements and isotopes
different from each other.

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OBJECTIVES:

Calculate the number of

neutrons in an atom.

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OBJECTIVES:

Calculate the atomic

mass of an element.

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OBJECTIVES:

Explain why chemists use

the periodic table.

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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
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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

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Mass Number
Mass number is the number of protons and
neutrons in the nucleus of an isotope:

Mass # = p+ + n0

p+

n0

e- Mass #

8

10

8

18

Arsenic - 75

33

42

33

75

Phosphorus - 31

15

16

15

31

Nuclide
Oxygen - 18

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Complete Symbols
 Contain

the symbol of the element,
the mass number and the atomic
number.
Mass
Superscript →
number
Subscript →

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Atomic
number

X

64

Symbols


Find each of these:
a) number of protons
b) number of
neutrons
c) number of
electrons
d) Atomic number
e) Mass Number

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35

Br

65

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

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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

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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

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Isotopes
 Dalton

was wrong about all
elements of the same type being
identical
 Atoms of the same element can
have
different
numbers
of
neutrons.
 Thus, different mass numbers.
 These are called isotopes.
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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.


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Naming Isotopes
We

can also put the mass
number after the name of the
element:
carbon-12
carbon-14

uranium-235
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Isotopes are atoms of the same element
having different masses, due to varying
numbers of neutrons.
Isotope

Protons Electrons

Neutrons

Hydrogen–1
(protium)

1

1

0

Hydrogen-2
(deuterium)

1

1

1

1

1

Nucleus

2

Hydrogen-3
(tritium)

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Isotopes
Elements
occur in
nature as
mixtures of
isotopes.
Isotopes are
atoms of the
same element
that differ in the
number of
neutrons.
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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.
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Measuring Atomic Mass
 Instead

of grams, the unit we use is the
Atomic Mass Unit (amu)
 It is defined as one-twelfth the mass of
a carbon-12 atom.
 Carbon-12 chosen because of its
isotope purity.
 Each isotope has its own atomic mass,
thus we determine the average from
percent abundance.
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To calculate the average:
 Multiply

the atomic mass of each
isotope by it‘s abundance (expressed
as a decimal), then add the results.
 If not told otherwise, the mass of the
isotope is expressed in atomic mass
units (amu)

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Atomic Masses
Atomic mass is the average of all the naturally
occurring isotopes of that element.
Isotope

Symbol

Carbon-12

12C

Carbon-13

13C

Carbon-14

14C

Composition of
the nucleus
6 protons
6 neutrons
6 protons
7 neutrons
6 protons
8 neutrons

% in nature

98.89%
1.11%
<0.01%

Carbon = 12.011
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- Page 117

Question

Knowns
and
Unknown

Solution
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Answer

78

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

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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
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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!

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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:

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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=λf
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

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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/k
The 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=dω/dk
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.

DR.ATAR @ UiTM.NS


85

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

DR.ATAR @ UiTM.NS


86

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?

DR.ATAR @ UiTM.NS


87

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)

DR.ATAR @ UiTM.NS


88

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.
DR.ATAR @ UiTM.NS


89

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/mv
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!

DR.ATAR @ UiTM.NS


90

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!

DR.ATAR @ UiTM.NS


91

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!
DR.ATAR @ UiTM.NS


92

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.)

DR.ATAR @ UiTM.NS


93

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.

DR.ATAR @ UiTM.NS


94

Appendix (Cont’d)


With this in mind, it is also worthy to take in mind dealing with suprarelativity (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=Vt
We see when we analyze ω t / k x that we can fiddle with the math:
kx/ωt= x/Vt=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.

DR.ATAR @ UiTM.NS


95

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.

DR.ATAR @ UiTM.NS


96

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

DR.ATAR @ UiTM.NS


97

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

λ

h
h

p mu

DR.ATAR @ UiTM.NS


98

Frequency of a Particle
In an analogy with photons, de Broglie postulated that a particle would
also have a frequency associated with it

ƒ

E
h

These equations present the dual nature of matter:
 Particle nature, p and E
 Wave nature, λ and ƒ

DR.ATAR @ UiTM.NS


99

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.

DR.ATAR @ UiTM.NS


100

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.

DR.ATAR @ UiTM.NS


101

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.

DR.ATAR @ UiTM.NS


102

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.

DR.ATAR @ UiTM.NS


103

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.

DR.ATAR @ UiTM.NS


104

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.

DR.ATAR @ UiTM.NS


105

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.

DR.ATAR @ UiTM.NS


106

Wave Envelope

The dashed line represents the envelope function.
This envelope can travel through space with a different speed than the
individual waves.

DR.ATAR @ UiTM.NS


107

Speeds Associated with Wave
Packet
The phase speed of a wave in a wave packet is given by

v phase  ω

k

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

The group speed is given by

v  dωis the speed of the wave packet itself.
g This dk

DR.ATAR @ UiTM.NS


108

Speeds, cont.
The group speed can also be expressed in terms of energy and
momentum.

dE d  p 2 
1
vg 


 2p   u


dp dp  2m  2m
This indicates that the group speed of the wave packet is identical to
the speed of the particle that it is modeled to represent.

DR.ATAR @ UiTM.NS


109

Electron Diffraction, Set-Up

DR.ATAR @ UiTM.NS


110

Electron Diffraction,
Experiment
Parallel 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.

DR.ATAR @ UiTM.NS


111

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.

DR.ATAR @ UiTM.NS


112

Electron Diffraction,
Equations
A maximum occurs whend sin θ  mλ
 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.

DR.ATAR @ UiTM.NS


113

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.

DR.ATAR @ UiTM.NS


114

Werner Heisenberg
1901 – 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

DR.ATAR @ UiTM.NS


115

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.

DxDpx 
2

DR.ATAR @ UiTM.NS


116

Heisenberg Uncertainty
Principle, Explained
It 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.

DR.ATAR @ UiTM.NS


117

Heisenberg Uncertainty
Principle, Another Form
Another form of the uncertainty principle can be expressed in terms of
energy and time.


2
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.
DE Dt 

DR.ATAR @ UiTM.NS


118

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.

DR.ATAR @ UiTM.NS


119

Max Planck
1858 – 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.

DR.ATAR @ UiTM.NS


120

Planck’s Theory of Blackbody
Radiation
In 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.

DR.ATAR @ UiTM.NS


121

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.

DR.ATAR @ UiTM.NS


122

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.

DR.ATAR @ UiTM.NS


123

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.

DR.ATAR @ UiTM.NS


124

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
e E kBT where E is the energy of the state.

DR.ATAR @ UiTM.NS


125

Planck’s Model, Graph

DR.ATAR @ UiTM.NS


126

Planck’s Wavelength
Distribution Function
Planck generated a theoretical expression for the wavelength
distribution.
2πhc 2
I  λ,T   5 hc λk T
B
λ e
1





 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.

DR.ATAR @ UiTM.NS


127

Thank You

DR.ATAR @ UiTM.NS


128

Early Pioneers of Quantum Physics

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