Dimensional and order-of-magnitude estimates are practiced by
almost everybody but taught almost nowhere. When physics students
engage in their first theoretical research project, they soon learn
that exactly solvable problems belong only to textbooks, that
numerical models are long and resource consuming, and that
"something else" is needed to quickly gain insight into the system
they are going to study. Qualitative methods are this "something
else", but typically, students have never heard of them before.The
aim of this book is to teach the craft of qualitative analysis
using a set of problems, some with solutions and some without, in
advanced undergraduate and beginning graduate Quantum Mechanics.
Examples include a dimensional analysis solution for the spectrum
of a quartic oscillator, simple WKB formulas for the matrix
elements of a coordinate in a gravitational well, and a
three-line-long estimate for the ionization energy of atoms
uniformly valid across the whole periodic table. The piece de
resistance in the collection is a series of dimensional analysis
questions in Integrable Nonlinear Partial Differential Equations
with no dimensions existing a priori. Solved problems include the
relationship between the size and the speed of solitons of the
Korteweg-de Vries equation and an expression for the oscillation
period of a Nonlinear Schroedinger breather as a function of its
width.
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