By Maxim Olshanii
Dimensional and order-of-magnitude estimates are practiced via virtually every person yet taught nearly nowhere. whilst physics scholars interact of their first theoretical study venture, they quickly research that precisely solvable difficulties belong in basic terms to textbooks, that numerical types are lengthy and source eating, and that “something else” is required to fast achieve perception into the method they're going to research. Qualitative equipment are this “something else”, yet in most cases, scholars have by no means heard of them before.
The goal of this publication is to educate the craft of qualitative research utilizing a collection of difficulties, a few with ideas and a few with no, in complicated undergraduate and starting graduate Quantum Mechanics. Examples comprise a dimensional research answer for the spectrum of a quartic oscillator, uncomplicated 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 sequence of dimensional research questions in Integrable Nonlinear Partial Differential Equations with out dimensions latest 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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Additional info for Back-of-the-Envelope Quantum Mechanics: With Extensions to Many-Body Systems and Integrable PDEs
We get En ∼ ω(n − 1/4) n = 1, 2, 3, . . 2 Order-of-magnitude estimate Make an order-of-magnitude estimate of the spectrum: use the BohrSommerfeld quantization rule and approximate the phase space trajectory by a rectangle. Keep the δ correction. 1) can be approximately replaced by the “circumscribed rectangle” of the trajectory, see Fig. 2. The phase space integral then reads: dx 2m(En − mω 2 x2 /2) ∼ 2En /(mω 2 ) xmax ∼ En /ω. 1) gives En ∼ ω(n − 1/4) n = 1, 2, 3, . . August 8, 2013 15:6 World Scientific Book - 9in x 6in B1662-ch03 “Halved” Harmonic Oscillator: A Case Study 45 p p max 0 x x max Fig.
128, 2425 (1962). 2 33 Bohr-Sommerfeld quantization for periodic boundary conditions Derive a Bohr-Sommerfeld quantization rule for a one-dimensional bounded motion with periodic boundary conditions. 3) can be generalized to any number of spatial dimensions. 4 Ground state energy of a logarithmic potential Consider a one-dimensional particle moving in a logarithmic potential: 2 − ∂2 ψ(x) + 2ϵ ln(r/a)ψ(x) = Eψ(x). 13) (a) Estimate the ground state energy using the Heisenberg uncertainty principle.
S. s. (x) is a time-independent ground state of the Hamiltonian H. e. 1. 1 Bohr-Sommerfeld quantization All useful cases of Bohr-Sommerfeld quantization can be concisely expressed in a single formula that reads n + # soft points × ∆soft + # hard points × ∆hard ] dxp(x; En˜ ) = 2π [˜ with n ˜ = 0, 1, 2, 3, . . 15) where π 2 = π, ∆soft = ∆hard # soft points(# hard points) counts the number of the soft(hard) turning points encountered by the classical trajectory during the cycle of motion (see Fig.
Back-of-the-Envelope Quantum Mechanics: With Extensions to Many-Body Systems and Integrable PDEs by Maxim Olshanii