Download Historical Review of Quantum Mechanics

An Introduction to Physical Chemistry
Konstantinos Petridis1,2, Nikolaos Lydakis3 and Minas
Stilianakis1
1Nanomaterials
and Organic Electronics Laboratory
2Department
of Electronic Engineering
3Department of Natural Sources & Environment
Technological Educational Institute of Crete, Greece
Heraklion
September 2014
An Introduction to Physical Chemistry: Part I
Konstantinos Petridis1,2
1Nanomaterials
and Organic Electronics Laboratory
2Department
of Electronic Engineering
Technological Educational Institute of Crete, Greece
Heraklion
September 2014
Course Description
Course Description (5 ECTS)
This module refers presents an introduction to quantum theory and spectroscopy. It
begins with a short historical review of quantum mechanics. Then examines the
properties of particles and waves, the Schrodinger equation, systems like the particle in
a box, the harmonic oscillator. The lectures continue with a discussion of atomic
structure & orbitals and Periodic Table. Continuously the course presents the molecular
bonding including valence bond and molecular orbital theory and structure. The final
lectures are devoted spectroscopy: rotational & vibrational spectra, electronic
transitions and magnetic resonance. The course consists from theoretical part and
laboratory part.
Grading
Activities
Lab Reports
Participation
Final Exam & Evaluation Tests
Percentage
20
20
50 (40 + 10)
K. Petridis, N. Lydakis and M. Stilianakis
Content – Part I
• A Historical Review of Quantum Mechanics
• The Dual Nature of Particles and Waves
•
The one dimensional Schrodinger equations
•
The one dimensional Schrodinger equation’s Applications
•
The 3D Schrodinger equation and the Hydrogen Atom
•
The basic postulates of quantum mechanics
•
Angular Momentum I (operators, angular eigenvalues, angular eigenfunctions)
•
Angular Momentum II (Pauli spin Matrices, Dirac Notation, Zeeman effect)
•
Time dependence
•
Many particle systems
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
• Quantum Mechanics (QM) is a response to the inability of the classical theories of
mechanics and electromagnetism to explain some of the properties of
electromagnetic radiation and atomic structure
• QM basic principles are used to explain the structure & properties of molecules,
solids but also those of nuclei and of elementary particles such as the proton and
neutron
• The development of QM can be splitted into two periods:
- Period #1: (1900 – 1923) the development of the old QM theory
- Period #2: (1924 – 1927) the introduction to the modern QM theory
• Old QM theory: the dual nature of light & matter
…until then
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Quantum Mechanics
• Describes rules that apply to electrons in atoms and molecules
• Non-deterministic, probalistic! A new philosophy of nature
• QM managed to explain unsolved problems of late 19th century physics
• Explains bonding, structure and reactivity in chemistry
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
• Classical Physics in the late 19th century:
Atoms are the basic constituents of matter
Newton’s Laws apply universally
The world is deterministic
• According to the C.M.: Given the positions and velocities and given all forces 
all the future can be predicted
• Physics was complete since:
Newtonian mechanics explained macroscopic behavior of matter –planetary
motion, fluid flow, elasticity, etc
Thermodynamics had its first two laws and most of their consequences
Light was explained as an electromagnetic wave
Basic statistical mechanics has been applied to chemical systems
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
• …however there were several experiments that could not be explained by C.P. and
the accepted dogma!
Blackbody radiation
Photoelectric effect
Discrete atomic spectra
The electron as a subatomic particle
• The study of the above experiments shows:
Atoms are not the most microscopic objects
Newton’s law do not apply to the microscopic world of electron
Outcome: Quantum Mechanics !!!!
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
• …however there were several experiments that could not be explained by C.P. and
the accepted dogma!
Blackbody radiation
Photoelectric effect
Discrete atomic spectra
The electron as a subatomic particle
• The study of the above experiments shows:
Atoms are not the most microscopic objects
Newton’s law do not apply to the microscopic world of electron
Outcome: Quantum Mechanics !!!!
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
The discovery of the electron
Faraday in 1870 had already shown using electrochemistry that amounts of electric
current proportional to amounts of some substances could be liberated in an electrolytic
cell
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
The discovery of the electron (continue)
JJ Thomson in 1897 discovers the electron and measures (e/me). Inadvertently invents
the cathode ray (TV) tube
JJ Thomson experiment’s conclusions: The results are independent of the cathode
material and gas composition. The electron is a distinct particle, present in all materials
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
1909 Milliken oil drop experiment
Milliken oil drop experiment determines e, me separately
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
1909 Milliken oil drop experiment
Milliken oil drop experiment conclusions: e = 1.59 × 10-19 Cb
Combing the Thomson’s experiment result and the above value we manage to measure
the electron mass, me = 9.1 × 10-31 kgr
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
Where are the electrons ?
1st atomic model “the jelly” model
This model was failed after Rutherford backscattering experiments demonstrated that
the atoms are mostly empty !!!
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
Where are the electrons ?
1st atomic model “the jelly” model
This model was failed after Rutherford backscattering experiments demonstrated that
the atoms are mostly empty !!!
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
Where are the electrons ?
Rutherford planetary model:
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
Where are the electrons ?
Rutherford planetary model failed because was not consistent with the classical
electrodynamic theory:
 Accelerating charge emits radiation (1)
 And since light has energy, E must be getting more negative with time (2)
Due to (1) and (2) we should be able (but we do not) to observe the following:
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
Where are the electrons ?
Rutherford planetary model failure to be consistent with the classical electrodynamics was
also due to: As the radius r of an electron reduces with time its speed should increase. This
means that the frequency of the supposed emitted light (due to electron acceleration) should
change accordingly:
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
……….BUT the emission from the atoms was known to be discrete which is another
failure of the planetary Rutherford model!!!
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Summary: Rutherford’s model of the atom:
① Is not stable relative to collapse of electron into nucleus
② Does not yield discrete emission lines
③ Does not explain the Rydberg formula
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
The Atom of Niels Bohr
• Niles Bohr a Danish Physicist who established the Copenhagen School
• Assumptions that underlying the Bohr atom:
 Atoms can exist in stable “states” without radiating. The states have
quantized energy levels En, n = 1, 2, 3, …….where n = 1 corresponds to
the fundamental state, n = 2 corresponds to the 1st excited state
 Number n is called a quantum number
 Transitions between states can be occurred with the absorption or the emission of a photon
with a frequency f where f =
DE
h
Conclusion: These assumptions are further evidence of the discrete spectra of atoms
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
The Atom of Niels Bohr
• Assumptions that underlying the Bohr atom: (continue)
 The electrons in the various states have an angular momentum that is an
integer multiple of the Planck’s constant:
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
The Atom of Niels Bohr
• Assumptions that underlying the Bohr atom: (continue)
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
The Atom of Niels Bohr
• Assumptions that underlying the Bohr atom: (continue)
 The application of the Bohr’s ideas in the Hydrogen atom (the results are valid
also in hydrogen type atoms):
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
The Atom of Niels Bohr
• Assumptions that underlying the Bohr atom: (continue)
 The application of the Bohr’s ideas in the Hydrogen atom (the results are valid
also in hydrogen type atoms): (continue)
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
The Atom of Niels Bohr
• Assumptions that underlying the Bohr atom: (continue)
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
A blackbody is an ideal body which allows the whole of the
incident radiation to pass into itself (without reflecting the
energy) and absorbs the total of the incident radiation (without
passing on the energy). This property is valid for radiation
corresponding to all wavelengths and to all angles of incidence.
Therefore, the black body is an ideal absorber of incident
radiation. When heated all objects, independent of their
constituents, emit light!!! This radiation is called blackbody
radiation
The blackbody is used as a standard with which the absorption of real bodies is compared
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
• According to the classical mechanics:
 Blackbody radiation is the result of electrons oscillating with frequency ν
 The electrons oscillate (& radiate) equally well at any frequency
 Rayleigh-Jeans Law describes the spectral density ρ(ν) of the blackbody radiation
and predicts that the intensity, within a spectral region, I(v) ~ ν2 (UV catastrophe)!!
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
• Planck (~1900), 1st “quantum” ideas:
 The energy of the oscillator ~ frequency,
E µn
 The total energy of an electromagnetic wave is an integral multiple of ν,
 The analogy constant is the Planck’s constant h = 6.626 × 10-34 J – sec
 E = hν is known as “quantum” of energy
• Stefan – Boltzmann law:
• Wien’s law:
K. Petridis, N. Lydakis and M. Stilianakis
E µ nn
Part I: Historical Review of QM
• Planck combined the Stefan – Boltzmann and the Wien law and described the
blackbody radiation according to the following formula:
where J(f,T) expresses the intensity per frequency at temperature T
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Summary: Planck’s theory:
① Solves the problem of the infinite energy: UV catastrophe
② Blackbody radiation emits it’s energy in quanta of energy
③ The minimum energy is this of E = hν
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K = eVo
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Φ represents how hard it is to remove an electron…
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Summary: Einstein’s theory about photoelectric effect:
① Structure of atom can not be explained classically
② Discrete atomic spectra and Rydberg’s formula can not be explained
③ Blackbody radiation can be “explained” by quantifying energy of oscillators, E = hν
④ Photoelectric effect can be “explained” by quantifying energy of light
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Classroom Solved Example:
The maximum energy of photoelectrons emitted from potassium is 2.1 eV when
illuminated by light of wavelength 300 nm and 0.5 eV when the light wavelength is 500
nm. Use these results to obtain values for Planck’s constant and the minimum energy
needed to free an electron from potassium
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
The Compton Effect
• The existence of photons also demonstrated by A.H. Compton
• A.H. Compton experiments involved the scattering of x-rays by electrons
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
The Compton Effect
• Compton scattering experiment demonstrated that a photon apart of its energy carries
momentum, p
• The photon momentum expression can be shown using Classical Mechanics
Terminology:
• The Compton experiment noticed the incident photon scattering from an electron of
mass m. After the collision the scattered photon has different momentum p’ and
direction k’
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Conclusion
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
The Compton Effect
• Compton & Photoelectric effects provide conclusive evidence
for the photon nature of electromagnetic waves
• Both experiments described the interaction of electromagnetic
radiation with electrons
• Question: Why in Photoelectric effect the photon (optical)
transfers all its’ energy to photoelectrons whereas in Compton
effect the X-rays are scattered by the electrons?
Classroom Solved Example:
An x-ray photon of wavelength 1× 10-12 m is incident on a stationary electron.
Calculate the wavelength of the scattered photon if it is detected at an angle 60o to the
incident radiation.
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
De Broglie Waves
• Compton experiment demonstrated that the photons have
particle properties
• L. De Broglie suggested that particles such as electrons might
also have wave properties (matter waves)
• The matter properties, energy & momentum, related with wave
properties, frequency & wavenumber: (De Broglie relations)
• The matter waves are exploited in electron microscopes that are
used to display diffraction patterns created by the objects under
investigation. Neutrons can be used as matter waves to
investigate the structural properties of matter
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
•
De Broglie principle states that all the objects have and a wave nature in parallel with
their material nature. They are particles and waves at the same time.
•
De Broglie principle explains the 2nd Bohr condition.
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Classroom Solved Example:
Calculate the wavelength and speed of the neutrons in a double slit diffraction
experiment assuming the following: slit separation equals to 0.12 mm, neutron mass
equals to 1,675 * 10-21 kg, separation of diffraction peaks about 75 μm, distance from
the slit 5 m
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Wave – Particle Duality
• QM predicts that both the wave and the particle models apply to all objects whatever
the size
• Depends on the energy (and thus the wavelength) if for an example an electron
behaves as a wave or as a particle
• Depends on with a photon interacts in order to know how to handle it: when a photon
interacts with a photon has wave properties and when a photon interacts with matter
has a particle properties
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Wave – Particle Duality
• Light as a wave
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Wave – Particle Duality
• Light as a wave (continue)
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Wave – Particle Duality
• Light as a wave (continue)
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Wave – Particle Duality
• Light as a particle
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Wave – Particle Duality
• Matter as wave
K. Petridis, N. Lydakis and M. Stilianakis
Part I: Historical Review of QM
Wave – Particle Duality
• Matter as a wave
K. Petridis, N. Lydakis and M. Stilianakis
Bibliography
• Quantum Mechanics by Alastair I.M. Rae (Taylor and Francis Group, 2008)
• Quantum Mechanics by Yoav Peleg, Schaum’s outlines
• Quantum Mechanics I and II by Stepahnos Trachanas (University of Crete Press, 2012)
• Physical Chemistry by Atkins
• Physical Chemistry notes in MIT Opencourseware
K. Petridis, N. Lydakis and M. Stilianakis