0. Prelude -- Development of Classical Physics and Dark Clouds

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0. Prelude -- Development of Classical Physics
and Dark Clouds

• (before 20th century)

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Classical Mechanics
Newton, Sir Isaac, PRS, (1643 1727), English
physicist and mathematician
Euler, Leonhard (1707 -- 1783), Swiss
mathematician.
Lagrange, Joseph Louis (1736 -- 1813),
Italian-French mathematician, astronomer and
physicist.
Hamilton, William Rowan (1805 -- 1865), Irish
mathematician and astronomer.
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Classical Electrodynamics
Coulomb, Charles Augustin (1736 1806), French
physicist
Biot, Jean Baptiste (1774 --1862), French
Physicist Savart, Félix (1791 -- 1841), French
Physicist
Ampere, Andre Marie (1775 -- 1836), French
Physicist
Faraday, Michael (1791 -- 1867), English Physicist
Lorentz, Hendrik Antoon (1853 -- 1928), Dutch
Physicist
Maxwell, James Clerk (1831 1879), Scottish
physicist
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Classical Thermodynamics
Clausius, Rudolf Julius Emanuel (1822 -- 1888) ,
German mathematical physicist.
Dalton, John (1766 -- 1844), British chemist and
physicist.
Carnot, Nicolas Léonard Sadi (1796 -- 1832),
French physicist.
Joule, James Prescott (1818 -- 1889), British
physicist.
Helmholtz, Hermann Ludwig Ferdinand von (1821 --
1894), German physicist and physician.
Boltzmann, Ludwig, (1844 1906), Austrian
physicist.
Maxwell, James Clerk (1831 1879), Scottish
physicist
Thomson, William (Baron Kelvin) (1824 - 1907),
British physicist and mathematician.
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Classical Statistical Mechanics
Equal a priori probability postulate (Boltzmann)
Given an isolated system in equilibrium, it is
found with equal probability in each of its
accessible microstates.
Canonical ensemble (isolated system)
Grandcanonical ensemble (opened system)
Boltzmann, Ludwig, (1844 1906), Austrian
physicist.
Microcanonical ensemble (independent system)
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Dark Clouds
Lord and Lady Kelvin at the coronation of King
Edward VII in 1902.
Sir William Thomson working on a problem of
science in 1890.
William Thomson produced 70 patents in the U.K.
from 1854 to 1907.
There is nothing new to be discovered in physics
now. All that remains is more and more precise
measurement.
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Dark Clouds
"Beauty and clearness of theory... Overshadowed
by two clouds..."
Nineteenth-Century Clouds over the Dynamical
Theory of Heat and Light (27th April 1900, Lord
Kelvin)
Michelson, Albert
Morley, Edward
Einstein, Albert
Planck, Max
Michelson-Morley Experiment (1887)
Ultraviolet catastrophy in blackbody radiation
(before October, 1900)
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I. Experiments and Ideas Prior to Quantum Theory

• (Before 1913)

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Radiation Blackbody Radiation and Quanta of
Energy
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Planck (1858 -- 1947), German physicist.
Planck's law of black body radiation (1900)
Plancks assumption (1900) radiation of a given
frequency ? could only be emitted and absorbed
in quanta of energy Eh?
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Radiation interaction with matter Photoelectric
Effect and Quanta of Light
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1. In 1839, Alexandre Edmond Becquerel observed the
   photoelectric effect via an electrode in a
   conductive solution exposed to light.
2. In 1873, Willoughby Smith found that selenium is
   photoconductive.
3. In 1887, Heinrich Hertz made observations of the
   photoelectric effect and of the production and
   reception of electromagnetic (EM) waves.
4. In 1899, Joseph John Thomson (N) investigated
   ultraviolet light in Crookes tubes.
5. In 1901, Nikola Tesla received the U.S. Patent
   685957 (Apparatus for the Utilization of Radiant
   Energy) that describes radiation charging and
   discharging conductors by "radiant energy".
6. In 1902, Philipp von Lenard (N) observed the
   variation in electron energy with light
   frequency.

In 1905, Albert Einstein (N) proposed the
well-known Einstein's equation for photoelectric
effect.
In 1916, Robert Andrews Millikan (N) finished a
decade-long experiment to confirm Einsteins
explanation of photoelectric effect.
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Atomic Structure
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Nuclear atom model (1911) Ernest Rutherford
Rutherford, Ernest, 1st Baron Rutherford of
Nelson, OM, PC, FRS (1871 -- 1937), New
Zealand-English nuclear physicist.
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Classical physics atoms should collapse!
This means an electron should fall into the
nucleus.
New mechanics is needed!
Classical Electrodynamics charged particles
radiate EM energy (photons) when their velocity
vector changes (e.g. they accelerate).
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Spectroscopy
Balmer, Johann Jakob (1825 -- 1898), Swiss
mathematician and an honorary physicist.
from n 3 to n 2
visible spectrum
Balmer series (1885)
Balmer's formula (1885)
Rydberg formula for hydrogen (1888)
Rydberg formula for all hydrogen-like atom (1888)
Rydberg, Johannes Robert (1854 -- 1919), Swedish
physicist.
Bohr's formula (1913)
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II. Old Quantum Theory

• (1913 -- 1924)

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Bohr's model of atomic structure, 1913
The electron's orbital angular momentum is
quantized
Bohr, Niels Henrik David (1885 -- 1962), Danish
physicist.
The theory that electrons travel in discrete
orbits around the atom's nucleus, with the
chemical properties of the element being largely
determined by the number of electrons in each of
the outer orbits
The idea that an electron could drop from a
higher-energy orbit to a lower one, emitting a
photon (light quantum) of discrete energy (this
became the basis for quantum theory).
Much work on the Copenhagen interpretation of
quantum mechanics.
The principle of complementarity that items
could be separately analyzed as having several
contradictory properties.
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Bohrs theory in 1 page
Quantum predictions must match classical results
for large n
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Summary

• Electron Transitions

Failures of the Bohr Model
It fails to provide any understanding of why
certain spectral lines are brighter than others.
There is no mechanism for the calculation of
transition probabilities.
The Bohr model treats the electron as if it were
a miniature planet, with definite radius and
momentum. This is in direct violation of the
uncertainty principle which dictates that
position and momentum cannot be simultaneously
determined.
The Bohr model gives us a basic conceptual model
of electrons orbits and energies. The precise
details of spectra and charge distribution must
be left to quantum mechanical calculations, as
with the Schrödinger equation.
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Prince de Broglie gets his Ph.D.
de Broglie matter wave hypothesis (1923) All
matter has a wave-like nature (wave-particle
duality) and that the wavelength and momentum of
a particle are related by a simple equation.
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Davisson-Germer Experiment (1927)
Davisson, Clinton Joseph (1881 -- 1958), American
physicist
Germer, Lester Halbert (1896 1971), American
physicist
Electron has wave nature (diffraction)!
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Later developments

• Borns statistical interpretation of wavefunction
• Matrix mechanics (Heisenberg, Born, Jordan)
• Wave mechanics (Schroedinger)
• Uncertainty principle (Heisenberg)
• Relativistic QM (Dirac)
• Exclusion principle (Pauli)

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Birth of QM

• The necessity for quantum mechanics was thrust
  upon us by a series of observations.
• The theory of QM developed over a period of 30
  years, culminating in 1925-27 with a set of
  postulates.
• QM cannot be deduced from pure mathematical or
  logical reasoning.
• QM is not intuitive, because we dont live in the
  world of electrons and atoms.
• QM is based on observation. Like all science, it
  is subject to change if inconsistencies with
  further observation are revealed.

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Fundamental postulates of QM

• How is the physical state described?
• How are physical observables represented?
• What are the results of measurement?
• How does the physical state evolve in time?

These postulates are fundamental, i.e., their
explanation is beyond the scope of the theory.
The theory is rather concerned with the
consequences of these postulates.
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Goal of PHYS521 and 522

• We will focus on non-relativistic QM.
• Our goal is to understand the meaning of the
  postulates the theory is based on, and how to
  operationally use the theory to calculate
  properties of systems.
• The first semester will lay out the ground work
  and mathematical structure, while the second will
  deal more with computation of real problems.

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Linear Algebra of Quantum Mechanics
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The mathematical structure QM describes is a
linear algebra of operators acting on a vector
space.
Under Dirac notation, we denote a vector using a
ket
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A vector space is n-demensional if the maximum
number of linearly independent vectors in the
space is n.
A set of n linearly independent vectors in
n-dimensional space is a basis --- any vector
can be written in a unique way as a sum over a
basis
Once the basis is chosen, a vector can be
represented by a column vector
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Usually we require the basis to be orthonormal
A linearly independent set of basis vectors can
be made orthonormal using the Gram-Schmidt
procedure.
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Unitary operator possesses the following
properties
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Eigenkets and Eigenvalues
Eigenvalues are roots to the characteristic
polynomial
The set of eigenvalues of an operator satisfy
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Eigenkets and Eigenvalues of Hermitian Operators
All the eigenvalues are real. Eigenkets belonging
to different eigenvalues are orthogonal. The
complete eigenkets can form an orthonormal
basis. The operator can be written as