My back-of-the-envelope examples have been drawn mostly from mechanics (for example, air resistance1 and pendulums2) and from mathematics (for example, the quadratic formula,3 the birthday paradox, and random walks4). Notably absent has been quantum mechanics. I now atone for my sin against Schrödinger by illustrating how back-of-the-envelope methods help us find energy spectra in power-law potentials,

V(x)=C|x|β
(1)

(the absolute value |x| allows odd and negative β values). Our goal is En, the energy of a particle at the nth energy level (for one-dimensional solutions of the Schrödinger equation).

Dimensional analysis, the first refuge of us approximators, helps only slightly. The dimensions of the relevant quantities are shown in Table I. These four quantities are constructed from three independent dimensions. Thus, from the Buckingham Pi theorem, they make one...

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