Back-of-the-envelope Quantum Mechanics: With Extensions To Many-body Systems And Integrable Pdes

With Extensions to Many-Body Systems and Integrable PDEs

Maxim Olshanii

• Publisher: WSPC
• Year: 2013
• eBook ISBN: 9789814508483
• Topic: Scienze fisiche
• Subtopic: Equazioni differenziali

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

Chapter 1

Ground State Energy of a Hybrid
Harmonic-Quartic Oscillator:
A Case Study

Introduction

Consider the Schrödinger equation for a one-dimensional particle moving in a combination of harmonic potential of frequency ω and a quartic potential of strength β:

(1.1)

where m is the particle’s mass. We will be mainly interested in determining the ground state energy. The Eq. (1.1) does not allow for an exact solution. However, the major features of the dependence of the ground state energy on the system parameters can be determined via elementary methods, such as dimensional analysis, order-of-magnitude estimates, and simple variational bounds. The goal of this chapter is to illustrate the application of these methods using the ground state problem (1.1) as an example.

1.1Solved problems

1.1.1Dimensional analysis and why it fails in this case

The assignment is: perform dimensional analysis of the problem and show that from a dimensional point of view the problem is underdetermined: no estimate for the ground state energy can be produced. However, some information about the structure of the expression for the ground state energy can still be extracted, on purely dimensional grounds.

Solution: The dimensional procedure for finding the ground state energy E_g.s. (or assessing the impossibility of a complete dimensional solution) is as follows:

— Begin by identifying the principal units of measurement for the problem, i.e. the minimal set of units sufficient to describe all input parameters of the problem. For stationary problems in quantum mechanics, the units of length, [

], and energy, [

], have been proven to provide a handy set;

— Identify the input parameters and units used to measure them;

— Determine the maximal set of independent dimensionless parameters: the set will include only the parameters that are generally either much greater or much less than unity. These include both the dimensionless parameters present in the problem a priori (such as the quantum number n), and the dimensionless combinations of the dimensionful input parameters. If the set is empty, the unknown quantities can be determined almost completely, i.e. up to a numerical prefactor of the order of unity. If some dimensionless parameters are present, the class of possible relationships between the unknowns and the input parameters can be narrowed down, but the order of magnitude of the unknown quantities can not be determined.

— For each of the principal units, choose a scale: a combination of the input parameters measured using the unit in question;

— Express the unknown quantities as a multi-power-law of principal scales, times an arbitrary function of all dimensionless parameters, if any. If no dimensionless parameters are present, the arbitrary function is replaced by an arbitrary constant, presumed to be of the order of unity.

In our case, the above procedure gives:

— The principal units—the units of length and the units of energy:

— The input parameters and their units:

where η ≡ ħ²/m and ϒ ≡ mω²;

— The set of independent dimensionless parameters =

(1.2)

It is represented by a single...

Table of contents

• Cover
• Halftitle Page
• Title Page
• Copyright
• Preface
• Contents
• 1. Ground State Energy of a Hybrid Harmonic-Quartic Oscillator: A Case Study
• 2. Bohr-Sommerfeld Quantization
• 3. “Halved” Harmonic Oscillator: A Case Study
• 4. Semi-Classical Matrix Elements of Observables and Perturbation Theory
• 5. Variational Problems
• 6. Gravitational Well: A Case Study
• 7. Miscellaneous
• 8. The Hellmann-Feynman Theorem
• 9. Local Density Approximation Theories
• 10. Integrable Partial Differential Equations
• Further Reading
• Subject Index
• Author Index