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.