Physics 635: Physics on the Back of an Envelope (Estimation in Physics)
TextBook: Maxim Olchanyi, Quantum Mechanics on the Back of an Envelope (World Scientific
(2013)) [in preparation].
Course outline: The goal of the course is to introduce the students to a comprehensive
array of qualitative methods practiced by researchers but rarely taught. The focus will be
on occurrences in a wide range of areas, including Quantum Mechanics and Differential
Equations. As the course title indicates, these methods allow for the quick, and relatively
painless, estimation of magnitude of effects prior to subjecting them to more rigors. The
ability to do this is widely regarded as the defining characteristic of physicists.
Topics covered are:
1. Overview of the methods
i) Order-of-magnitude estimate vs. dimensional analysis
ii) Determined, underdetermined, and overdetermined cases in dimensional analysis
iii) Symmetry considerations
iv) Quantum-Classical Correspondence
v) Equations with no a priori set dimensionality
vi) Virial and Hellmann-Feynman theorems
2. WKB
i) State index becomes a continuous parameter: one input parameter less
ii) Estimates for the ground state energy
iii) Spectra of one-dimensional systems from WKB
iv) Spectra of one-dimensional systems from dimensional analysis
v) Feingold-Peres theorem and the selection rules
vi) Spectrum and wavefunctions of spherical harmonics
3. Perturbation Theory
i) The power of the perturbation theory parameter is fixed: one input parameter less
ii) Power-law perturbations applied to a harmonic oscillator
iii) No-go theorems from the perturbation theory: the “p-wave scatterer problem”
4. Thomas-Fermi Theory of Atoms
i) Number of electrons becomes a continuous parameter: one input parameter less
ii) A “naïve” model with mutually noninteracting electrons
iii) Dimensional estimate for the size and ground state energy of an atom
iv) Discussion of the exact solution and the empirical data
v) Verifying the virial theorem
5. Thomas-Fermi Theory of Bose-Condensates
i) Number of bosons becomes a continuous parameter: one input parameter less
ii) Speed of sound
iii) Healing length and the vortex core size
iv) Frequencies of the elementary excitations in traps
6. Solitons in Integrable Nonlinear Partial Differential Equations
i) Assigning a “bogus” dimensionality to the parameters
ii) Sine-Gordon equation
iii) Nonlinear Schrodinger equation
iv) Korteweg-de-Vries equation
v) Kadomtsev-Petviashvilli equation
Pre-requisites: Physics 613: Quantum Mechanics, Atomic and Molecular Physics
Physics 616: Mathematical Methods for Physicists
Homework: There will be four homework assignments.
Exams: There will be one mid-term exam, and one final exam.
Grading: The total grade will consist of: assignments 30%, mid-term exam 20%, final exam
50%.
Additional texts/references:
Vladimir P. Krainov and Kevin Hendzel, Qualitative Methods in Physical Kinetics and
Hydrodynamics (American Institute of Physics (1992))
A. B. Migdal, Qualitative Methods in Quantum Theory (Westview Press (2000))
NEQwiki, the nonlinear equations encyclopedia
< http://www.primat.mephi.ru/wiki/ow.asp?EquationsList>
STATEMENT ON DISABILITY ACCMMODATIONS:
Section 504 of the Americans with Disabilities Act of 1990 offers guidelines for
curriculum modifications and adaptations for students with documented disabilities. If
applicable, students may obtain adaptation recommendations from the Ross Center for
Disability Services, M-1-401, (617-287-7430). The student must present these
recommendations and discuss them with each professor within a reasonable period,
preferably by the end of Drop/Add period.
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http://www.umb.edu/life_on_campus/policies/code/