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Back-of-the-Envelope Quantum Mechanics: With Extensions to by Maxim Olshanii

By Maxim Olshanii

Dimensional and order-of-magnitude estimates are practiced via virtually every body yet taught nearly nowhere. whilst physics scholars interact of their first theoretical learn venture, they quickly research that precisely solvable difficulties belong basically to textbooks, that numerical versions are lengthy and source eating, and that “something else” is required to quick achieve perception into the procedure they'll research. Qualitative tools are this “something else”, yet more often than not, scholars have by no means heard of them before.

The goal of this ebook is to educate the craft of qualitative research utilizing a collection of difficulties, a few with options and a few with no, in complex undergraduate and starting graduate Quantum Mechanics. Examples contain a dimensional research resolution for the spectrum of a quartic oscillator, basic WKB formulation for the matrix parts of a coordinate in a gravitational good, and a three-line-long estimate for the ionization strength of atoms uniformly legitimate around the entire periodic desk. The pièce de résistance within the assortment is a chain of dimensional research questions in Integrable Nonlinear Partial Differential Equations with out dimensions current a priori. Solved difficulties contain the connection among the scale and the rate of solitons of the Korteweg–de Vries equation and an expression for the oscillation interval of a Nonlinear Schrödinger breather as a functionality of its width.

Readership: complex undergraduate and starting graduate scholars in physics.

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On the other hand, ψ(x) ∝ sin[ 1 σa (x; E)], and thus ψ(x) = 0 everywhere. In this case, WKB would predict an unnormalizable solution, which must be excluded. August 8, 2013 15:6 World Scientific Book - 9in x 6in B1662-ch03 Chapter 3 “Halved” Harmonic Oscillator: A Case Study Introduction Consider a potential for a particle in the field of a hard wall and a harmonic force: ⎧ ⎪ ⎨+∞ V (x) = mω 2 x2 ⎪ ⎩ 2 for x < 0 for x ≥ 0. V (x ) = m ω x2 2 /2 energy In what follows, we will study its spectrum, using the methods described in previous chapters.

3 Ground state energy in power-law potentials Consider the Schr¨ odinger equation for a particle in a “2q-tic” potential: 2 − ∂2 ψ(x) + Kq x2q ψ(x) = Eψ(x) 2m ∂x2 q = 1, 2, 3, . . s. s. ⇒ ∆x ∼ E Kq 1 2q . The ground state minimizes the Heisenberg uncertainty: ∆p ∆x ∼ . s. ∼ q q+1 2 m 1 Kqq+1 (b) The same as the above, but using dimensional analysis. Solution: The input parameters are 2 /m ≡ η and Kq . Their units are given by [η] = EL2 and [Kq ] = E/L2q . s. s. ] = E. s. 4 q q+1 2 m 1 Kqq+1 Spectrum of power-law potentials Consider the Schr¨ odinger equation for a particle in a “2q-tic” potential: 2 ∂2 ψ(x) + Kq x2q ψ(x) = Eψ(x).

4 We deliberately leave δ undefined. The question of a proper WKB quantization of a three-dimensional radially symmetric potential, including Coulomb problems at both non-zero and zero angular momenta, has a long and convoluted history. A rigorous analysis of the question is presented in a comprehensive book on WKB by Heading [M. A. Heading, An Introduction to Phase-Integral Methods (John Wiley, New York (1962))]. The currently most accepted quantization recipe is based on the classic paper by Langer [R.

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