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Phy 310 chapter 2

This document provides an overview of early developments in quantum theory from Max Planck's work on blackbody radiation in 1900 to the formulation of quantum mechanics between 1925-1927. It describes key contributions from Planck, Einstein, Bohr, de Broglie, Heisenberg, Schrodinger, Born, and others that revolutionized physics. Their work established that energy is quantized, light has particle-like properties, matter exhibits wave-particle duality, and measurements in quantum systems are subject to uncertainty. This laid the foundations for modern quantum mechanics.

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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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 1
To know the Revolutionary impact of quantum physics one need first to look at pre-quantum physics: DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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" . DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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 . DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 5
Two theorist, Niels Bohr and Max Planck, at the blackboard. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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). DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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) ... DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory Paul Dirac and Werner Heisemberg in Cambrige,1930. 9
The first Solvay Congress in 1911 assembled the pioneers of quantum theory. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 10
Old faces and new at 1927 Solvay Congress DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 11
DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 12
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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 14
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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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‖. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 16
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 20
 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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 21
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 PHY310 - Early Quantum Theory 22
DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 23
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.  DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 24
Democritus Aristotle DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 25
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.  DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 26
Alchemic Symbols DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 27
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.  DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 28
John Dalton DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 29
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.  DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 30
J. J. Thompson DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 31
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.  DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 32
Ernest Rutherford DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 33
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.  DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 34
Niels Bohr DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 35
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.  DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 36
Electron Cloud Model DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 37
Defining the Atom OBJECTIVES: Describe Democritus‘s ideas about atoms. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 38
Defining the Atom OBJECTIVES: Explain Dalton‘s atomic theory. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 39
Defining the Atom OBJECTIVES: Identify what instrument is used to observe individual atoms. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 40
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 41
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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 42
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 43
Structure of the Nuclear Atom OBJECTIVES: Identify three types of subatomic particles. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 44
Structure of the Nuclear Atom OBJECTIVES: Describe the structure of atoms, according to the Rutherford atomic model. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 45
Structure of the Nuclear Atom  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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 46
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 47
Modern Cathode Ray Tubes Television Computer Monitor Cathode ray tubes pass electricity through a gas that is contained at a very low pressure. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 48
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 49
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 50
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 51
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 52
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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 53
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 54
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 55
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‖ DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 56
Distinguishing Among Atoms OBJECTIVES: Explain what makes elements and isotopes different from each other. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 57
Distinguishing Among Atoms OBJECTIVES: Calculate the number of neutrons in an atom. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 58
Distinguishing Among Atoms OBJECTIVES: Calculate the atomic mass of an element. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 59
Distinguishing Among Atoms OBJECTIVES: Explain why chemists use the periodic table. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 60
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 61
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 62
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 63
Complete Symbols  Contain the symbol of the element, the mass number and the atomic number. Mass Superscript → number Subscript → DR.ATAR @ UiTM.NS Atomic number PHY310 - Early Quantum Theory 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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 80 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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 66
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 67
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 68
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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 69
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.  DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 70
Naming Isotopes We can also put the mass number after the name of the element: carbon-12 carbon-14 uranium-235 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 71
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) DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 72
Isotopes Elements occur in nature as mixtures of isotopes. Isotopes are atoms of the same element that differ in the number of neutrons. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 73
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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 74
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. DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 75
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) DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 76
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 77
- Page 117 Question Knowns and Unknown Solution DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 79
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 80
DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 81
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! DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 82
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: DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 83
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 DR.ATAR @ UiTM.NS PHY310 - Early Quantum Theory 84
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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 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 PHY310 - Early Quantum Theory 100