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

0. Prelude -- Development of Classical Physics and Dark Clouds. (before 20th century). Classical Mechanics. Newton, Sir Isaac, PRS, (1643 – 1727), English physicist and mathematician. Euler, Leonhard (1707 -- 1783), Swiss mathematician. Lagrange, Joseph Louis (1736 -- 1813),

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

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  1. 0. Prelude -- Development of Classical Physics and Dark Clouds (before 20th century)

  2. 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.

  3. Classical Electrodynamics Coulomb, Charles Augustin (1736 – 1806), Frenchphysicist 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

  4. 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.

  5. 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)

  6. 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.”

  7. 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)

  8. I. Experiments and Ideas Prior to Quantum Theory (Before 1913)

  9. Radiation: Blackbody Radiation and Quanta of Energy

  10. Planck (1858 -- 1947), German physicist. Planck's law of black body radiation (1900) Planck’s assumption (1900): radiation of a given frequency ν could only be emitted and absorbed in “quanta” of energy E=hν

  11. Radiation interaction with matter: Photoelectric Effect and Quanta of Light

  12. In 1839, Alexandre Edmond Becquerel observed the photoelectric effect via an electrode in a conductive solution exposed to light. • In 1873, Willoughby Smith found that selenium is photoconductive. • In 1887, Heinrich Hertz made observations of the photoelectric effect and of the production and reception of electromagnetic (EM) waves. • In 1899, Joseph John Thomson (N) investigated ultraviolet light in Crookes tubes. • 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". • 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 Einstein’s explanation of photoelectric effect.

  13. Atomic Structure

  14. Nuclear atom model (1911): Ernest Rutherford Rutherford, Ernest,1st Baron Rutherford of Nelson, OM, PC, FRS (1871 -- 1937), New Zealand-English nuclear physicist.

  15. 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).

  16. Spectroscopy Balmer,Johann Jakob (1825 -- 1898), Swiss mathematician and an honorary physicist. from n ≥ 3 to n = 2 Balmer series (1885) visible spectrum 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)

  17. II. Old Quantum Theory (1913 -- 1924)

  18. 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.

  19. Bohr’s theory in 1 page Quantum predictions must match classical results for large n

  20. Summary Failures of the Bohr Model Electron Transitions 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.

  21. 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.

  22. Davisson-Germer Experiment(1927) Davisson, Clinton Joseph (1881 -- 1958), American physicist Germer,Lester Halbert (1896 – 1971), American physicist Electron has wave nature (diffraction)!

  23. Later developments • Born’s statistical interpretation of wavefunction • Matrix mechanics (Heisenberg, Born, Jordan) • Wave mechanics (Schroedinger) • Uncertainty principle (Heisenberg) • Relativistic QM (Dirac) • Exclusion principle (Pauli)

  24. 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 don’t 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.

  25. 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.

  26. 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.

  27. Linear Algebra of Quantum Mechanics

  28. 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”:

  29. 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:

  30. Usually we require the basis to be orthonormal: A linearly independent set of basis vectors can be made orthonormal using the Gram-Schmidt procedure.

  31. Unitary operator possesses the following properties:

  32. Eigenkets and Eigenvalues: Eigenvalues are roots to the characteristic polynomial The set of eigenvalues of an operator satisfy:

  33. 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

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