Modern Physics
Highlights of this Course
This course is designed to help the students acquire a situated and contextualized knowledge framework in the domain of quantum mechanics. Embedded problem solving sessions will help them apply the framework to actual physical systems by developing necessary analytical skills.
Course Description
This is an introductory course of quantum mechanics. After a brief discussion of the breakdown of classical theories in the microscopic world, we introduce the notion of quantization and wavematter duality. The Schrödinger Equation is “derived” from scratch in order to understand how one exploits old theories to create a new theory. Solutions of the Schrödinger Equation are considered for various standard systems such as an infinite square well potential, a barrier potential, and the simple harmonic oscillator. The course culminates in a full solution for the hydrogen atom. Lectures and problem sessions alternate to provide an opportunity to apply the concept to real physical systems. By seeing in action what they have just learned, the students can further digest the content of the lecture and solidify their understanding of the material. During the course of the semester, mathematical tools and techniques are introduced as necessary. This includes spherical coordinates and techniques to solve linear partial differential equations, so that the course is accessible to a broad audience including freshmen.

 #1 How discrepancies arose between experimental findings and the old theories at the beginning of the 20th century

 #2 How Schrödinger “cooked up” his famous differential equation guided by Newtonian Mechanics and the emerging idea of matter wave

 #3 Probabilistic interpretation of wavefunctions a la Max Born; Expectation values for the position, the momentum, and the total energy

 #4 Derivation of the TimeIndependent Schrödinger Equation using separation of variables; The relation between the timeindependent solutions and the timedependent counterparts

 #5 The zero potential and the step potential when the energy is less than the step height; Reflection and transmission

 #6 The step potential when the energy is greater than the step height and the barrier potential

 #7 The infinite square well potential and the simple harmonic oscillator potential; Power series solution for Hermite polynomials

 #8 The TimeIndependent Schrödinger Equation in three dimensions; Spherical coordinates and separation of variables

 #9 The TimeIndependent Schrödinger Equation in three dimensions in spherical coordinates; The radial, the angular, and the azimuthal differential equations

 #10 Solutions to the azimuthal differential equation and the angular differential equation; The associated Legendre functions

 #11 Solutions to the radial equation; The associated Laguerre polynomials; Energy levels and degeneracy; Agreement with spectroscopy

 #12 Final examination

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Details
 Year/Term
 2005 / First semester
 Faculty/
Graduate School  General Education
 Language
 English
 Instructor name
 Masayasu AOTANI（Associate Professor）
Syllabus
Outline and Purpose of the Course  This course starts with a brief description of Newtonian Mechanics and Maxwell’s Theory of Electricity and Magnetism as contrasted with Quantum Mechanics and Einstein’s Special Theory of Relativity. We will first discuss how old theories break down in the realm of atoms and molecules and at high speeds. The significant roles played by Quantum Mechanics and Relativity in modern physics are explained to put the old and the new physics in a proper perspective. Relativity is briefly explained as a theory to describe physics at speeds comparable to that of light. Quantum Mechanics is introduced as a device to solve the problems classical theories faced in explaining microscopic phenomena. Having given a grand framework of the current state of affairs in physics, we then specialize to Quantum Mechanics. Planck’s idea of quantum, Einstein’s explanation of photoelectric effect, Bohr model, and de Broglie’s matter wave all set the stage for an introduction of the Schrödinger Equation. The meandering path to the equation is presented in full to give the students a taste of how actual dirty work is done in physics. We then apply the theory to simple systems involving the infinite square well potential, a barrier potential, and the simple harmonic oscillator. In the process, analytical skills to solve second order partial differential equations like the Schrödinger Equation are developed. The course ends with a full solution for the hydrogen atom, and the predicted results are compared with the outcomes of actual experiments. This provides a justification for the nebulous and potentially even dubious process of the development of quantum mechanics. By highlighting the difference between purely theoretical science like mathematics and a basically experimental science like physics, students can acquire an understanding of the role of physics in a larger context of the human pursuit of knowledge and truth. 


Schedule and Contents  #1 Classical Physics and Modern Physics #2 Schrödinger’s Theory of Quantum Mechanics #3 Max Born’s Interpretation of Wavefunctions #4 TimeIndependent Schrödinger Equation #5 Solutions of TimeIndependent Schrödinger Equation #6 Solutions of TimeIndependent Schrödinger Equation #7 Solutions of TimeIndependent Schrödinger Equation #8 The Hydrogen Atom #9 The Hydrogen Atom #10 The Hydrogen Atom #11 The Hydrogen Atom #12 Final Examination 

References, etc.  [Resources] Robert Eisberg, Robert Resnick : Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles 2nd edition, Wiley, 1985 
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