Back-of-the-Envelope Quantum Mechanics

With Extensions to Many-Body Systems and Integrable PDEs

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 pièce de résistance 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 Schrödinger breather as a function of its width.

Contents:

• Ground State Energy of a Hybrid Harmonic-Quartic Oscillator: A Case Study
• Bohr-Sommerfeld Quantization
• “Halved” Harmonic Oscillator: A Case Study
• Semi-Classical Matrix Elements of Observables and Perturbation Theory
• Variational Problems
• Gravitational Well: A Case Study
• Miscellaneous
• The Hellmann-Feynman Theorem
• Local Density Approximation Theories
• Integrable Partial Differential Equations

Readership: Advanced undergraduate and beginning graduate students in physics.
Key Features:

• This book is the only existing title on qualitative methods in Quantum Mechanics at the advanced undergraduate / beginning graduate level. A B Migdal's Qualitative Methods in Quantum Theory (Westview Press (2000)) is far too advanced. M Gitterman and V Halpern's Qualitative Analysis of Physical Problems (Academic Press (1981)) is too broad. [Sanjoy Mahajan's, Street-Fighting Mathematics (The MIT Press (2010)) is on mathematics. Vladimir P Krainov's Qualitative Methods in Physical Kinetics and Hydrodynamics (American Institute of Physics (1992)) is on physical kinetics and hydrodynamics. The above list is likely to exhaust all the textbooks in qualitative methods in physics and mathematics ever published
• The book is structured as a coherent sequence of problems aimed at addressing the whole spectrum of dimensional and order-of-magnitude methods
• This book can be used as a secondary text and a source for homework assignments in any advanced undergraduate / beginning graduate course in Quantum Mechanics or in Partial Differential Equations