Numerical Methods in Quantum Mechanics

 Corso di Laurea Magistrale Interateneo (Master) in Physics

Academic Year 2020-2021

Teacher: Paolo Giannozzi, room L1-2-BE, Department of Mathematics, Computer Science and Physics, Via delle Scienze 206, Udine
tel. +39 0432 558216, fax +39 0432 558222
web: http://www.fisica.uniud.it/~giannozz, e-mail: paolo . giannozzi at uniud . it
Office hours: during the semester when possible, I'll be in the Physics building Monday and Wednesday from 9:30 to 11:00. Send me an e-mail or phone me to get an appointment (in Trieste or in Udine) at a different time when not possible.

Course Description

Goals: this course provides an introduction to numerical methods and techniques useful for the numerical solution of quantum mechanical problems, especially in atomic and condensed-matter physics. The course is organized as a series of theoretical lessons in which the physical problems and the numerical concepts needed for their resolution are presented, followed by practical sessions in which examples of implementatation for specific simple problems are presented. The student will learn to use the concepts and to practise scientific programming by modifying and extending the examples presented during the course.

Syllabus: Schroedinger equations in one dimension: techniques for numerical solutions. Solution of the Schroedinger equations for a potential with spherical symmetry. Scattering from a potential. Variational method: expansion on a basis of functions, secular problem, eigenvalues and eigenvectors. Examples: gaussian basis, plane-wave basis. Many-electron systems: Hartree-Fock equations, self-consistent field, exchange interaction. Numerical solution of Hartree-Fock equations in atoms with radial integration and on a gaussian basis set. Introduction to numerical solution of electronic states in molecules. Electronic states in solids: solution of the Schroedinger equation for periodic potentials. Introduction to exact diagonalization of spin systems. Introduction to Density-Functional Theory.

Bibliography
Lecture Notes (updated): Introduction, Ch.1, Ch.2, Ch.3, Ch.4, Ch.5, Ch.6, Ch.7, Ch.8, Ch.9, Ch.10, Ch.11, Appendix, all.
See also: J. M. Thijssen, Computational Physics, Cambridge University Press, Cambridge, 1999, Ch.2-4, 5, 6.1-6.4, 6.7.
A rather detailed introduction to Density-Functional techniques can be found in the first chapters of the lecture notes of my (now defunct) course on Numerical Methods in Electronic Structure.

Requirements: basic knowledge of Quantum Mechanics, of Fortran or C programming, of an operating system (preferrably Linux).

Exam: personal project consisting in the numerical solution of a problem, followed by oral examination (typically consisting in the discussion of a subject, different from the one of the personal project, chosen by the student). Contact me a few weeks before the exam to receive the personal project (if not yet assigned) and to set a date. A short written report on the personal project and the related code(s) should be provided no later than the day before the exam.

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Schedule

The course starts on March 8th. Classes are held online on Teams (Link on , https://corsi.units.it/didattica-a-distanza, send me an email if you cannot find it) until further notice

• Monday 11-13, Aula C
• Wednesday 11-13, Lab. T21

Deviation from the above schedule are marked in boldface in the detailed table of arguments (subject to changes>/i>) below.

N.	Date	Subject
1.	8 March	One-dimensional Schroedinger equation: Reminder: harmonic oscillator, analytical solution. Discretization, Numerov algorithm, numerical stability, eigenvalue search using stable outwards and inwards integrations. (Notes: Ch.1)
2.	10 March practical session	Numerical solution of the one-dimensional Schroedinger equation: examples for the harmonic oscillator (code harmonic0: fortran, C; code harmonic1: fortran, C).
3.	15 March	Three-dimensional Schroedinger equation: Central potentials, variable separation, logarithmic grids, perturbative estimate to accelerate eigenvalue convergence. (Notes: Ch.2). A glimpse on true three-dimensional problems on a grid. (Notes: appendix A)
4.	17 March practical session	Numerical solution for spherically symmetric potentials: example for Hydrogen atom (code hydrogen_radial: fortran, C)
5.	22 March	Scattering from a potential: cross section, phase shifts, resonances. (Notes: Ch.3; Thijssen: Ch.2)
6.	24 March pratical session	Calculation of cross sections: numerical solution for Lennard-Jones potential (code crossection: fortran, C).
7.	29 March	Variational method: Schroedinger equation as minimum problem, expansion on a basis of functions, secular problem, introduction to diagonalization algorithms. (Notes: Ch.4; Thijssen: Ch.3)
8.	31 March pratical session	Variational method using an orthonormal basis set: example of a potential well in plane waves (code pwell: fortran, C).
9.	12 April	Non-orthonormal basis sets: gaussian functions. (Notes: Ch.5)
10.	14 April practical session	Variational method with gaussian basis set: solution for Hydrogen atom (code hydrogen_gauss: fortran, C)
11.	19 April	The Hartree-Fock method: Slater determinants, Hartree-Fock equations, self-consistent field. (Notes: Ch.6; Thijssen: Ch.4.1-4.5)
12.	21 April practical session	He atom in Hartree-Fock approximation: solution with radial integration and self-consistency (code helium_hf_radial: fortran, C).
13.	26 April	Molecules: Born-Oppenheimer approximation, potential energy surface, diatomic molecules. introduction to numerical solution for molecules. (Notes: Ch.7; Thijssen: Ch.4.6-4.8)
14.	28 April practical session	Molecules with gaussian basis: solution of Hartree-Fock equations on a gaussian basis for a H2 molecule (code h2_hf_gauss: fortran, C).
15.	3 May	Electronic states in crystals: Bloch theorem, band structure. (Notes: Ch.8; Thijssen: Ch.4.6-4.8)
16.	5 May practical session	Periodic potentials: numerical solution with plane waves of the Kronig-Penney model (code periodicwell: fortran, C; example of usage of FFT: fortran, C).
17.	10 May	Electronic states in crystals II: three-dimensional case, methods of solution, plane wave basis set, introduction to the concept of pseudopotential. (Notes: Ch.9; Thijssen: Ch.6.1-6.4, 6.7)
18.	12 May pratical session	Pseudopotentials: solution of the Cohen-Bergstresser model for Silicon (code cohenbergstresser: fortran, C).
19.	17 May	Spin systems Introduction to spin systems: Heisenberg model, exact diagonalization, iterative methods for diagonalization, sparseness. (Notes: Ch.10)
20.	19 May practical session	Exact Diagonalization Solution of the Heisenberg model with Lanczos chains (code heisenberg_exact: fortran, C).
21.	24 May	Density-Functional Theory Hohenberg-Kohn theorem, Kohn-Sham equations (Notes: Ch.11)
22.	26 May semi-practical session	DFT with plane waves and pseudopotentials Fast Fourier-Trasform and iterative techniques; sample code ah (fortran, C) solving Si with Appelbaum-Hamann pseudopotentials).
23.	31 May	Density-Functional Theory II
24.	7 June	Assignment of exam problems

Last modified 23 May 2021

PAOLO GIANNOZZI

 

 

PAOLO GIANNOZZI
Professore Associato, struttura della materia

Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, Polo dei Rizzi
Via delle Scienze 206, I-33100 Udine, Italy
tel. (+39)-0432-558216, fax -558222
e-mail: paolo.giannozzi at uniud.it

Ricevimento studenti: sempre, se libero. Se volete essere sicuri di trovarmi, speditemi un messaggio

 

 

TEACHING/DIDATTICA

• Fisica Generale Modulo I per Matematica
• Fisica Moderna per Matematica
• Numerical Methods in Quantum Mechanics for Physics Master (Trieste-Udine)
• (old) Introduzione alla fisica dello stato solido per la Scuola Superiore di Udine
• (old) Metodi Numerici per la Struttura Elettronica

SHORT CV

1958  Born in Poggibonsi (Siena, Italy)
1982  Laurea (Degree) in Physics, Università di Pisa (Italy)
1988  Docteur ès Sciences (Ph.D.), Université de Lausanne (Switzerland)
1992  Ricercatore (assistant professor) at Scuola Normale Superiore di Pisa
2006  Professor ("associato") at Università di Udine
2013  Fellow of the American Physical Society, Division of Computational Physics
CURRENT SCIENTIFIC INTERESTS

• Structure and electronic transport in nanostructures
• New methods for the calculation of hybrid functionals in a plane-wave basis set
• Verification and validation in electronic-structure calculations
• Development of software for electronic-structure calculations

Slides of recent talks

RECENT PUBLICATIONS

• Unit cell restricted Bloch functions basis for first-principle transport models: Theory and application, M. G. Pala, P. Giannozzi, D. Esseni, Phys. Rev. B 102, 045410 (2020).
• Quantum ESPRESSO toward the exascale, P. Giannozzi, O. Baseggio, P. Bonfà, D. Brunato, R. Car, I. Carnimeo, C. Cavazzoni, S. de Gironcoli, P. Delugas, F. Ferrari Ruffino, A. Ferretti, N. Marzari, I. Timrov, A. Urru, S. Baroni, J. Chem. Phys. 152, 154105 (2020).
• Improved understanding of metal-graphene contacts, F. Driussi, S. Venica, A. Gahoi, A. Gambi, P. Giannozzi, S. Kataria, M.C. Lemme, P. Palestri, D. Esseni, Microelectronic Engineering, 216, 111035 (2019).
• Fast hybrid density-functional computations using plane-wave basis sets, I. Carnimeo, S. Baroni, P. Giannozzi, Electron. Struct. 1, 015009 (2019)
• Quantum Crystallography: Current Developments and Future Perspectives, A. Genoni et al., Chem. Eur. J. 24, 10881-10905 (2018)

Complete list of publications, with links to pdf files if available

IN REAL LIFE

 http://www.fisica.uniud.it/~giannozz/

 

Quantum Mechanics for Engineers

 

Leon van Dommelen

08/26/18 Version 5.63 alpha

Copy­right and Dis­claimer

Copy­right © 2004, 2007, 2008, 2010, 2011, and on, Leon van Dom­me­len. You are al­lowed to copy and/or print out this work for your per­sonal use. How­ever, do not dis­trib­ute any parts of this work to oth­ers or post it pub­licly with­out writ­ten per­mis­sion of the au­thor. In­stead please link to this work. I want to be able to cor­rect er­rors and im­prove ex­pla­na­tions and ac­tu­ally have them cor­rected and im­proved. Every at­tempt is made to make links as per­ma­nent as pos­si­ble.

Read­ers are ad­vised that text, fig­ures, and data may be in­ad­ver­tently in­ac­cu­rate. All re­spon­si­bil­ity for the use of any of the ma­te­r­ial in this book rests with the reader. (I took these sen­tences straight out of a printed com­mer­cial book. How­ever, in this web book, I do try to cor­rect in­ac­cu­ra­cies, OK, blun­ders, pretty quickly if pointed out to me by help­ful read­ers, or if I hap­pen to think twice.)

As far as search en­gines are con­cerned, con­ver­sions to html of the pdf ver­sion of this doc­u­ment are stu­pid, since there is a much bet­ter na­tive html ver­sion al­ready avail­able. So try not to do it.

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• Ded­i­ca­tion
• Con­tents
• List of Fig­ures
• List of Ta­bles
• Pref­ace
  • To the Stu­dent
  • Ac­knowl­edg­ments
  • Com­ments and Feed­back
  
  
• I. Spe­cial Rel­a­tiv­ity
  • 1. Spe­cial Rel­a­tiv­ity [Draft]
    • 1.1 Overview of Rel­a­tiv­ity
      • 1.1.1 A note on the his­tory of the the­ory
      • 1.1.2 The mass-en­ergy re­la­tion
      • 1.1.3 The uni­ver­sal speed of light
      • 1.1.4 Dis­agree­ments about space and time
    • 1.2 The Lorentz Trans­for­ma­tion
      • 1.2.1 The trans­for­ma­tion for­mu­lae
      • 1.2.2 Proper time and dis­tance
      • 1.2.3 Sub­lu­mi­nal and su­per­lu­mi­nal ef­fects
      • 1.2.4 Four-vec­tors
      • 1.2.5 In­dex no­ta­tion
      • 1.2.6 Group prop­erty
    • 1.3 Rel­a­tivis­tic Me­chan­ics
      • 1.3.1 In­tro to rel­a­tivis­tic me­chan­ics
      • 1.3.2 La­grangian me­chan­ics
  
  
• II. Ba­sic Quan­tum Me­chan­ics
  • 2. Math­e­mat­i­cal Pre­req­ui­sites
    • 2.1 Com­plex Num­bers
    • 2.2 Func­tions as Vec­tors
    • 2.3 The Dot, oops, IN­NER Prod­uct
    • 2.4 Op­er­a­tors
    • 2.5 Eigen­value Prob­lems
    • 2.6 Her­mit­ian Op­er­a­tors
    • 2.7 Ad­di­tional Points
      • 2.7.1 Dirac no­ta­tion
      • 2.7.2 Ad­di­tional in­de­pen­dent vari­ables
  • 3. Ba­sic Ideas of Quan­tum Me­chan­ics
    • 3.1 The Re­vised Pic­ture of Na­ture
    • 3.2 The Heisen­berg Un­cer­tainty Prin­ci­ple
    • 3.3 The Op­er­a­tors of Quan­tum Me­chan­ics
    • 3.4 The Or­tho­dox Sta­tis­ti­cal In­ter­pre­ta­tion
      • 3.4.1 Only eigen­val­ues
      • 3.4.2 Sta­tis­ti­cal se­lec­tion
    • 3.5 A Par­ti­cle Con­fined In­side a Pipe
      • 3.5.1 The phys­i­cal sys­tem
      • 3.5.2 Math­e­mat­i­cal no­ta­tions
      • 3.5.3 The Hamil­ton­ian
      • 3.5.4 The Hamil­ton­ian eigen­value prob­lem
      • 3.5.5 All so­lu­tions of the eigen­value prob­lem
      • 3.5.6 Dis­cus­sion of the en­ergy val­ues
      • 3.5.7 Dis­cus­sion of the eigen­func­tions
      • 3.5.8 Three-di­men­sional so­lu­tion
      • 3.5.9 Quan­tum con­fine­ment
  • 4. Sin­gle-Par­ti­cle Sys­tems
    • 4.1 The Har­monic Os­cil­la­tor
      • 4.1.1 The Hamil­ton­ian
      • 4.1.2 So­lu­tion us­ing sep­a­ra­tion of vari­ables
      • 4.1.3 Dis­cus­sion of the eigen­val­ues
      • 4.1.4 Dis­cus­sion of the eigen­func­tions
      • 4.1.5 De­gen­er­acy
      • 4.1.6 Noneigen­states
    • 4.2 An­gu­lar Mo­men­tum
      • 4.2.1 De­f­i­n­i­tion of an­gu­lar mo­men­tum
      • 4.2.2 An­gu­lar mo­men­tum in an ar­bi­trary di­rec­tion
      • 4.2.3 Square an­gu­lar mo­men­tum
      • 4.2.4 An­gu­lar mo­men­tum un­cer­tainty
    • 4.3 The Hy­dro­gen Atom
      • 4.3.1 The Hamil­ton­ian
      • 4.3.2 So­lu­tion us­ing sep­a­ra­tion of vari­ables
      • 4.3.3 Dis­cus­sion of the eigen­val­ues
      • 4.3.4 Dis­cus­sion of the eigen­func­tions
    • 4.4 Ex­pec­ta­tion Value and Stan­dard De­vi­a­tion
      • 4.4.1 Sta­tis­tics of a die
      • 4.4.2 Sta­tis­tics of quan­tum op­er­a­tors
      • 4.4.3 Sim­pli­fied ex­pres­sions
      • 4.4.4 Some ex­am­ples
    • 4.5 The Com­mu­ta­tor
      • 4.5.1 Com­mut­ing op­er­a­tors
      • 4.5.2 Non­com­mut­ing op­er­a­tors and their com­mu­ta­tor
      • 4.5.3 The Heisen­berg un­cer­tainty re­la­tion­ship
      • 4.5.4 Com­mu­ta­tor ref­er­ence
    • 4.6 The Hy­dro­gen Mol­e­c­u­lar Ion
      • 4.6.1 The Hamil­ton­ian
      • 4.6.2 En­ergy when fully dis­so­ci­ated
      • 4.6.3 En­ergy when closer to­gether
      • 4.6.4 States that share the elec­tron
      • 4.6.5 Com­par­a­tive en­er­gies of the states
      • 4.6.6 Vari­a­tional ap­prox­i­ma­tion of the ground state
      • 4.6.7 Com­par­i­son with the ex­act ground state
  • 5. Mul­ti­ple-Par­ti­cle Sys­tems
    • 5.1 Wave Func­tion for Mul­ti­ple Par­ti­cles
    • 5.2 The Hy­dro­gen Mol­e­cule
      • 5.2.1 The Hamil­ton­ian
      • 5.2.2 Ini­tial ap­prox­i­ma­tion to the low­est en­ergy state
      • 5.2.3 The prob­a­bil­ity den­sity
      • 5.2.4 States that share the elec­trons
      • 5.2.5 Vari­a­tional ap­prox­i­ma­tion of the ground state
      • 5.2.6 Com­par­i­son with the ex­act ground state
    • 5.3 Two-State Sys­tems
    • 5.4 Spin
    • 5.5 Mul­ti­ple-Par­ti­cle Sys­tems In­clud­ing Spin
      • 5.5.1 Wave func­tion for a sin­gle par­ti­cle with spin
      • 5.5.2 In­ner prod­ucts in­clud­ing spin
      • 5.5.3 Com­mu­ta­tors in­clud­ing spin
      • 5.5.4 Wave func­tion for mul­ti­ple par­ti­cles with spin
      • 5.5.5 Ex­am­ple: the hy­dro­gen mol­e­cule
      • 5.5.6 Triplet and sin­glet states
    • 5.6 Iden­ti­cal Par­ti­cles
    • 5.7 Ways to Sym­metrize the Wave Func­tion
    • 5.8 Ma­trix For­mu­la­tion
    • 5.9 Heav­ier Atoms
      • 5.9.1 The Hamil­ton­ian eigen­value prob­lem
      • 5.9.2 Ap­prox­i­mate so­lu­tion us­ing sep­a­ra­tion of vari­ables
      • 5.9.3 Hy­dro­gen and he­lium
      • 5.9.4 Lithium to neon
      • 5.9.5 Sodium to ar­gon
      • 5.9.6 Potas­sium to kryp­ton
      • 5.9.7 Full pe­ri­odic ta­ble
    • 5.10 Pauli Re­pul­sion
    • 5.11 Chem­i­cal Bonds
      • 5.11.1 Co­va­lent sigma bonds
      • 5.11.2 Co­va­lent pi bonds
      • 5.11.3 Po­lar co­va­lent bonds and hy­dro­gen bonds
      • 5.11.4 Pro­mo­tion and hy­bridiza­tion
      • 5.11.5 Ionic bonds
      • 5.11.6 Lim­i­ta­tions of va­lence bond the­ory
  • 6. Macro­scopic Sys­tems
    • 6.1 In­tro to Par­ti­cles in a Box
    • 6.2 The Sin­gle-Par­ti­cle States
    • 6.3 Den­sity of States
    • 6.4 Ground State of a Sys­tem of Bosons
    • 6.5 About Tem­per­a­ture
    • 6.6 Bose-Ein­stein Con­den­sa­tion
      • 6.6.1 Rough ex­pla­na­tion of the con­den­sa­tion
    • 6.7 Bose-Ein­stein Dis­tri­b­u­tion
    • 6.8 Black­body Ra­di­a­tion
    • 6.9 Ground State of a Sys­tem of Elec­trons
    • 6.10 Fermi En­ergy of the Free-Elec­tron Gas
    • 6.11 De­gen­er­acy Pres­sure
    • 6.12 Con­fine­ment and the DOS
    • 6.13 Fermi-Dirac Dis­tri­b­u­tion
    • 6.14 Maxwell-Boltz­mann Dis­tri­b­u­tion
    • 6.15 Thermionic Emis­sion
    • 6.16 Chem­i­cal Po­ten­tial and Dif­fu­sion
    • 6.17 In­tro to the Pe­ri­odic Box
    • 6.18 Pe­ri­odic Sin­gle-Par­ti­cle States
    • 6.19 DOS for a Pe­ri­odic Box
    • 6.20 In­tro to Elec­tri­cal Con­duc­tion
    • 6.21 In­tro to Band Struc­ture
      • 6.21.1 Met­als and in­su­la­tors
      • 6.21.2 Typ­i­cal met­als and in­su­la­tors
      • 6.21.3 Semi­con­duc­tors
      • 6.21.4 Semi­met­als
      • 6.21.5 Elec­tronic heat con­duc­tion
      • 6.21.6 Ionic con­duc­tiv­ity
    • 6.22 Elec­trons in Crys­tals
      • 6.22.1 Bloch waves
      • 6.22.2 Ex­am­ple spec­tra
      • 6.22.3 Ef­fec­tive mass
      • 6.22.4 Crys­tal mo­men­tum
      • 6.22.5 Three-di­men­sional crys­tals
    • 6.23 Semi­con­duc­tors
    • 6.24 The P-N Junc­tion
    • 6.25 The Tran­sis­tor
    • 6.26 Zener and Avalanche Diodes
    • 6.27 Op­ti­cal Ap­pli­ca­tions
      • 6.27.1 Atomic spec­tra
      • 6.27.2 Spec­tra of solids
      • 6.27.3 Band gap ef­fects
      • 6.27.4 Ef­fects of crys­tal im­per­fec­tions
      • 6.27.5 Pho­to­con­duc­tiv­ity
      • 6.27.6 Pho­to­voltaic cells
      • 6.27.7 Light-emit­ting diodes
    • 6.28 Ther­mo­elec­tric Ap­pli­ca­tions
      • 6.28.1 Peltier ef­fect
      • 6.28.2 See­beck ef­fect
      • 6.28.3 Thom­son ef­fect
  • 7. Time Evo­lu­tion
    • 7.1 The Schrö­din­ger Equa­tion
      • 7.1.1 The equa­tion
      • 7.1.2 So­lu­tion of the equa­tion
      • 7.1.3 En­ergy con­ser­va­tion
      • 7.1.4 Sta­tion­ary states
      • 7.1.5 The adi­a­batic ap­prox­i­ma­tion
    • 7.2 Time Vari­a­tion of Ex­pec­ta­tion Val­ues
      • 7.2.1 New­ton­ian mo­tion
      • 7.2.2 En­ergy-time un­cer­tainty re­la­tion
    • 7.3 Con­ser­va­tion Laws and Sym­me­tries
    • 7.4 Con­ser­va­tion Laws in Emis­sion
      • 7.4.1 Con­ser­va­tion of en­ergy
      • 7.4.2 Com­bin­ing an­gu­lar mo­menta and par­i­ties
      • 7.4.3 Tran­si­tion types and their pho­tons
      • 7.4.4 Se­lec­tion rules
    • 7.5 Sym­met­ric Two-State Sys­tems
      • 7.5.1 A graph­i­cal ex­am­ple
      • 7.5.2 Par­ti­cle ex­change and forces
      • 7.5.3 Spon­ta­neous emis­sion
    • 7.6 Asym­met­ric Two-State Sys­tems
      • 7.6.1 Spon­ta­neous emis­sion re­vis­ited
    • 7.7 Ab­sorp­tion and Stim­u­lated Emis­sion
      • 7.7.1 The Hamil­ton­ian
      • 7.7.2 The two-state model
    • 7.8 Gen­eral In­ter­ac­tion with Ra­di­a­tion
    • 7.9 Po­si­tion and Lin­ear Mo­men­tum
      • 7.9.1 The po­si­tion eigen­func­tion
      • 7.9.2 The lin­ear mo­men­tum eigen­func­tion
    • 7.10 Wave Pack­ets
      • 7.10.1 So­lu­tion of the Schrö­din­ger equa­tion.
      • 7.10.2 Com­po­nent wave so­lu­tions
      • 7.10.3 Wave pack­ets
      • 7.10.4 Group ve­loc­ity
      • 7.10.5 Elec­tron mo­tion through crys­tals
    • 7.11 Al­most Clas­si­cal Mo­tion
      • 7.11.1 Mo­tion through free space
      • 7.11.2 Ac­cel­er­ated mo­tion
      • 7.11.3 De­cel­er­ated mo­tion
      • 7.11.4 The har­monic os­cil­la­tor
    • 7.12 Scat­ter­ing
      • 7.12.1 Par­tial re­flec­tion
      • 7.12.2 Tun­nel­ing
    • 7.13 Re­flec­tion and Trans­mis­sion Co­ef­fi­cients
  • 8. The Mean­ing of Quan­tum Me­chan­ics
    • 8.1 Schrö­din­ger’s Cat
    • 8.2 In­stan­ta­neous In­ter­ac­tions
    • 8.3 Global Sym­metriza­tion
    • 8.4 A story by Wheeler
    • 8.5 Fail­ure of the Schrö­din­ger Equa­tion?
    • 8.6 The Many-Worlds In­ter­pre­ta­tion
    • 8.7 The Ar­row of Time
  
  
• III. Gate­way Top­ics
  • 9. Nu­mer­i­cal Pro­ce­dures
    • 9.1 The Vari­a­tional Method
      • 9.1.1 Ba­sic vari­a­tional state­ment
      • 9.1.2 Dif­fer­en­tial form of the state­ment
      • 9.1.3 Us­ing La­grangian mul­ti­pli­ers
    • 9.2 The Born-Op­pen­heimer Ap­prox­i­ma­tion
      • 9.2.1 The Hamil­ton­ian
      • 9.2.2 Ba­sic Born-Op­pen­heimer ap­prox­i­ma­tion
      • 9.2.3 Go­ing one bet­ter
    • 9.3 The Hartree-Fock Ap­prox­i­ma­tion
      • 9.3.1 Wave func­tion ap­prox­i­ma­tion
      • 9.3.2 The Hamil­ton­ian
      • 9.3.3 The ex­pec­ta­tion value of en­ergy
      • 9.3.4 The canon­i­cal Hartree-Fock equa­tions
      • 9.3.5 Ad­di­tional points
  • 10. Solids
    • 10.1 Mol­e­c­u­lar Solids
    • 10.2 Ionic Solids
    • 10.3 Met­als
      • 10.3.1 Lithium
      • 10.3.2 One-di­men­sional crys­tals
      • 10.3.3 Wave func­tions of one-di­men­sional crys­tals
      • 10.3.4 Analy­sis of the wave func­tions
      • 10.3.5 Flo­quet (Bloch) the­ory
      • 10.3.6 Fourier analy­sis
      • 10.3.7 The rec­i­p­ro­cal lat­tice
      • 10.3.8 The en­ergy lev­els
      • 10.3.9 Merg­ing and split­ting bands
      • 10.3.10 Three-di­men­sional met­als
    • 10.4 Co­va­lent Ma­te­ri­als
    • 10.5 Free-Elec­tron Gas
      • 10.5.1 Lat­tice for the free elec­trons
      • 10.5.2 Oc­cu­pied states and Bril­louin zones
    • 10.6 Nearly-Free Elec­trons
      • 10.6.1 En­ergy changes due to a weak lat­tice po­ten­tial
      • 10.6.2 Dis­cus­sion of the en­ergy changes
    • 10.7 Ad­di­tional Points
      • 10.7.1 About fer­ro­mag­net­ism
      • 10.7.2 X-ray dif­frac­tion
  • 11. Ba­sic and Quan­tum Ther­mo­dy­nam­ics
    • 11.1 Tem­per­a­ture
    • 11.2 Sin­gle-Par­ti­cle ver­sus Sys­tem States
    • 11.3 How Many Sys­tem Eigen­func­tions?
    • 11.4 Par­ti­cle-En­ergy Dis­tri­b­u­tion Func­tions
    • 11.5 The Canon­i­cal Prob­a­bil­ity Dis­tri­b­u­tion
    • 11.6 Low Tem­per­a­ture Be­hav­ior
    • 11.7 The Ba­sic Ther­mo­dy­namic Vari­ables
    • 11.8 In­tro to the Sec­ond Law
    • 11.9 The Re­versible Ideal
    • 11.10 En­tropy
    • 11.11 The Big Lie of Dis­tin­guish­able Par­ti­cles
    • 11.12 The New Vari­ables
    • 11.13 Mi­cro­scopic Mean­ing of the Vari­ables
    • 11.14 Ap­pli­ca­tion to Par­ti­cles in a Box
      • 11.14.1 Bose-Ein­stein con­den­sa­tion
      • 11.14.2 Fermi­ons at low tem­per­a­tures
      • 11.14.3 A gen­er­al­ized ideal gas law
      • 11.14.4 The ideal gas
      • 11.14.5 Black­body ra­di­a­tion
      • 11.14.6 The De­bye model
    • 11.15 Spe­cific Heats
  • 12. An­gu­lar mo­men­tum
    • 12.1 In­tro­duc­tion
    • 12.2 The fun­da­men­tal com­mu­ta­tion re­la­tions
    • 12.3 Lad­ders
    • 12.4 Pos­si­ble val­ues of an­gu­lar mo­men­tum
    • 12.5 A warn­ing about an­gu­lar mo­men­tum
    • 12.6 Triplet and sin­glet states
    • 12.7 Cleb­sch-Gor­dan co­ef­fi­cients
    • 12.8 Some im­por­tant re­sults
    • 12.9 Mo­men­tum of par­tially filled shells
    • 12.10 Pauli spin ma­tri­ces
    • 12.11 Gen­eral spin ma­tri­ces
    • 12.12 The Rel­a­tivis­tic Dirac Equa­tion
  • 13. Elec­tro­mag­net­ism
    • 13.1 The Elec­tro­mag­netic Hamil­ton­ian
    • 13.2 Maxwell’s Equa­tions
    • 13.3 Ex­am­ple Sta­tic Elec­tro­mag­netic Fields
      • 13.3.1 Point charge at the ori­gin
      • 13.3.2 Dipoles
      • 13.3.3 Ar­bi­trary charge dis­tri­b­u­tions
      • 13.3.4 So­lu­tion of the Pois­son equa­tion
      • 13.3.5 Cur­rents
      • 13.3.6 Prin­ci­ple of the elec­tric mo­tor
    • 13.4 Par­ti­cles in Mag­netic Fields
    • 13.5 Stern-Ger­lach Ap­pa­ra­tus
    • 13.6 Nu­clear Mag­netic Res­o­nance
      • 13.6.1 De­scrip­tion of the method
      • 13.6.2 The Hamil­ton­ian
      • 13.6.3 The un­per­turbed sys­tem
      • 13.6.4 Ef­fect of the per­tur­ba­tion
  • 14. Nu­clei [Un­fin­ished Draft]
    • 14.1 Fun­da­men­tal Con­cepts
    • 14.2 Draft: The Sim­plest Nu­clei
      • 14.2.1 Draft: The pro­ton
      • 14.2.2 Draft: The neu­tron
      • 14.2.3 Draft: The deuteron
      • 14.2.4 Draft: Prop­erty sum­mary
    • 14.3 Draft: Overview of Nu­clei
    • 14.4 Draft: Magic num­bers
    • 14.5 Draft: Ra­dioac­tiv­ity
      • 14.5.1 Draft: Half-life and de­cay rate
      • 14.5.2 Draft: More than one de­cay process
      • 14.5.3 Draft: Other de­f­i­n­i­tions
    • 14.6 Draft: Mass and en­ergy
    • 14.7 Draft: Bind­ing en­ergy
    • 14.8 Draft: Nu­cleon sep­a­ra­tion en­er­gies
    • 14.9 Draft: Mod­el­ing the Deuteron
    • 14.10 Draft: Liq­uid drop model
      • 14.10.1 Draft: Nu­clear ra­dius
      • 14.10.2 Draft: von Weizsäcker for­mula
      • 14.10.3 Draft: Ex­pla­na­tion of the for­mula
      • 14.10.4 Draft: Ac­cu­racy of the for­mula
    • 14.11 Draft: Al­pha De­cay
      • 14.11.1 Draft: De­cay mech­a­nism
      • 14.11.2 Draft: Com­par­i­son with data
      • 14.11.3 Draft: For­bid­den de­cays
      • 14.11.4 Draft: Why al­pha de­cay?
    • 14.12 Draft: Shell model
      • 14.12.1 Draft: Av­er­age po­ten­tial
      • 14.12.2 Draft: Spin-or­bit in­ter­ac­tion
      • 14.12.3 Draft: Ex­am­ple oc­cu­pa­tion lev­els
      • 14.12.4 Draft: Shell model with pair­ing
      • 14.12.5 Draft: Con­fig­u­ra­tion mix­ing
      • 14.12.6 Draft: Shell model fail­ures
    • 14.13 Draft: Col­lec­tive Struc­ture
      • 14.13.1 Draft: Clas­si­cal liq­uid drop
      • 14.13.2 Draft: Nu­clear vi­bra­tions
      • 14.13.3 Draft: Non­spher­i­cal nu­clei
      • 14.13.4 Draft: Ro­ta­tional bands
    • 14.14 Draft: Fis­sion
      • 14.14.1 Draft: Ba­sic con­cepts
      • 14.14.2 Draft: Some ba­sic fea­tures
    • 14.15 Draft: Spin Data
      • 14.15.1 Draft: Even-even nu­clei
      • 14.15.2 Draft: Odd mass num­ber nu­clei
      • 14.15.3 Draft: Odd-odd nu­clei
    • 14.16 Draft: Par­ity Data
      • 14.16.1 Draft: Even-even nu­clei
      • 14.16.2 Draft: Odd mass num­ber nu­clei
      • 14.16.3 Draft: Odd-odd nu­clei
      • 14.16.4 Draft: Par­ity Sum­mary
    • 14.17 Draft: Elec­tro­mag­netic Mo­ments
      • 14.17.1 Draft: Clas­si­cal de­scrip­tion
      • 14.17.2 Draft: Quan­tum de­scrip­tion
      • 14.17.3 Draft: Mag­netic mo­ment data
      • 14.17.4 Draft: Quadru­pole mo­ment data
    • 14.18 Draft: Isospin
      • 14.18.1 Draft: Ba­sic ideas
      • 14.18.2 Draft: Heav­ier nu­clei
      • 14.18.3 Draft: Ad­di­tional points
      • 14.18.4 Draft: Why does this work?
    • 14.19 Draft: Beta de­cay
      • 14.19.1 Draft: In­tro­duc­tion
      • 14.19.2 Draft: En­er­get­ics Data
      • 14.19.3 Draft: Beta de­cay and magic num­bers
      • 14.19.4 Draft: Von Weizsäcker ap­prox­i­ma­tion
      • 14.19.5 Draft: Ki­netic En­er­gies
      • 14.19.6 Draft: For­bid­den de­cays
      • 14.19.7 Draft: Data and Fermi the­ory
      • 14.19.8 Draft: Par­ity vi­o­la­tion
    • 14.20 Draft: Gamma De­cay
      • 14.20.1 Draft: En­er­get­ics
      • 14.20.2 Draft: For­bid­den de­cays
      • 14.20.3 Draft: Iso­mers
      • 14.20.4 Draft: Weis­skopf es­ti­mates
      • 14.20.5 Draft: Com­par­i­son with data
      • 14.20.6 Draft: In­ter­nal con­ver­sion
  
  
• IV. Sup­ple­men­tary In­for­ma­tion
  • A. Ad­denda
    • A.1 Clas­si­cal La­grangian me­chan­ics
      • A.1.1 In­tro­duc­tion
      • A.1.2 Gen­er­al­ized co­or­di­nates
      • A.1.3 La­grangian equa­tions of mo­tion
      • A.1.4 Hamil­ton­ian dy­nam­ics
      • A.1.5 Fields
    • A.2 An ex­am­ple of vari­a­tional cal­cu­lus
    • A.3 Galilean trans­for­ma­tion
    • A.4 More on in­dex no­ta­tion
    • A.5 The re­duced mass
    • A.6 Con­stant spher­i­cal po­ten­tials
      • A.6.1 The eigen­value prob­lem
      • A.6.2 The eigen­func­tions
      • A.6.3 About free space so­lu­tions
    • A.7 Ac­cu­racy of the vari­a­tional method
    • A.8 Pos­i­tive ground state wave func­tion
    • A.9 Wave func­tion sym­me­tries
    • A.10 Spin in­ner prod­uct
    • A.11 Ther­mo­elec­tric ef­fects
      • A.11.1 Peltier and See­beck co­ef­fi­cient ball­parks
      • A.11.2 Fig­ure of merit
      • A.11.3 Phys­i­cal See­beck mech­a­nism
      • A.11.4 Full ther­mo­elec­tric equa­tions
      • A.11.5 Charge lo­ca­tions in ther­mo­electrics
      • A.11.6 Kelvin re­la­tion­ships
    • A.12 Heisen­berg pic­ture
    • A.13 In­te­gral Schrö­din­ger equa­tion
    • A.14 The Klein-Gor­don equa­tion
    • A.15 Quan­tum Field The­ory in a Nanoshell
      • A.15.1 Oc­cu­pa­tion num­bers
      • A.15.2 Cre­ation and an­ni­hi­la­tion op­er­a­tors
      • A.15.3 The ca­Her­mi­tians
      • A.15.4 Re­cast­ing a Hamil­ton­ian as a quan­tum field one
      • A.15.5 The har­monic os­cil­la­tor as a bo­son sys­tem
      • A.15.6 Canon­i­cal (sec­ond) quan­ti­za­tion
      • A.15.7 Spin as a fermion sys­tem
      • A.15.8 More sin­gle par­ti­cle states
      • A.15.9 Field op­er­a­tors
      • A.15.10 Non­rel­a­tivis­tic quan­tum field the­ory
    • A.16 The adi­a­batic the­o­rem
    • A.17 The vir­ial the­o­rem
    • A.18 The en­ergy-time un­cer­tainty re­la­tion­ship
    • A.19 Con­ser­va­tion Laws and Sym­me­tries
      • A.19.1 An ex­am­ple sym­me­try trans­for­ma­tion
      • A.19.2 Phys­i­cal de­scrip­tion of a sym­me­try
      • A.19.3 De­riva­tion of the con­ser­va­tion law
      • A.19.4 Other sym­me­tries
      • A.19.5 A gauge sym­me­try and con­ser­va­tion of charge
      • A.19.6 Reser­va­tions about time shift sym­me­try
    • A.20 An­gu­lar mo­men­tum of vec­tor par­ti­cles
    • A.21 Pho­ton type 2 wave func­tion
      • A.21.1 The wave func­tion
      • A.21.2 Sim­pli­fy­ing the wave func­tion
      • A.21.3 Pho­ton spin
      • A.21.4 En­ergy eigen­states
      • A.21.5 Nor­mal­iza­tion of the wave func­tion
      • A.21.6 States of def­i­nite lin­ear mo­men­tum
      • A.21.7 States of def­i­nite an­gu­lar mo­men­tum
    • A.22 Forces by par­ti­cle ex­change
      • A.22.1 Clas­si­cal se­lec­to­sta­t­ics
      • A.22.2 Clas­si­cal se­lec­to­dy­nam­ics
      • A.22.3 Quan­tum se­lec­to­sta­t­ics
      • A.22.4 Poin­caré and Ein­stein try to save the uni­verse
      • A.22.5 Lorenz saves the uni­verse
      • A.22.6 Gupta-Bleuler con­di­tion
      • A.22.7 The con­ven­tional La­grangian
      • A.22.8 Quan­ti­za­tion fol­low­ing Fermi
      • A.22.9 The Coulomb po­ten­tial and the speed of light
    • A.23 Quan­ti­za­tion of ra­di­a­tion
      • A.23.1 Prop­er­ties of clas­si­cal elec­tro­mag­netic fields
      • A.23.2 Pho­ton wave func­tions
      • A.23.3 The elec­tro­mag­netic op­er­a­tors
      • A.23.4 Prop­er­ties of the ob­serv­able elec­tro­mag­netic field
    • A.24 Quan­tum spon­ta­neous emis­sion
    • A.25 Mul­ti­pole tran­si­tions
      • A.25.1 Ap­prox­i­mate Hamil­ton­ian
      • A.25.2 Ap­prox­i­mate mul­ti­pole ma­trix el­e­ments
      • A.25.3 Cor­rected mul­ti­pole ma­trix el­e­ments
      • A.25.4 Ma­trix el­e­ment ball­parks
      • A.25.5 Se­lec­tion rules
      • A.25.6 Ball­park de­cay rates
      • A.25.7 Wave func­tions of def­i­nite an­gu­lar mo­men­tum
      • A.25.8 Weis­skopf and Moszkowski es­ti­mates
      • A.25.9 Er­rors in other sources
    • A.26 Fourier in­ver­sion the­o­rem and Par­se­val
    • A.27 De­tails of the an­i­ma­tions
    • A.28 WKB The­ory of Nearly Clas­si­cal Mo­tion
    • A.29 WKB so­lu­tion near the turn­ing points
    • A.30 Three-di­men­sional scat­ter­ing
      • A.30.1 Par­tial wave analy­sis
      • A.30.2 Par­tial wave am­pli­tude
      • A.30.3 The Born ap­prox­i­ma­tion
    • A.31 The Born se­ries
    • A.32 The evo­lu­tion of prob­a­bil­ity
    • A.33 Ex­pla­na­tion of the Lon­don forces
    • A.34 Ex­pla­na­tion of Hund’s first rule
    • A.35 The third law
    • A.36 Al­ter­nate Dirac equa­tions
    • A.37 Maxwell’s wave equa­tions
    • A.38 Per­tur­ba­tion The­ory
      • A.38.1 Ba­sic per­tur­ba­tion the­ory
      • A.38.2 Ion­iza­tion en­ergy of he­lium
      • A.38.3 De­gen­er­ate per­tur­ba­tion the­ory
      • A.38.4 The Zee­man ef­fect
      • A.38.5 The Stark ef­fect
    • A.39 The rel­a­tivis­tic hy­dro­gen atom
      • A.39.1 In­tro­duc­tion
      • A.39.2 Fine struc­ture
      • A.39.3 Weak and in­ter­me­di­ate Zee­man ef­fect
      • A.39.4 Lamb shift
      • A.39.5 Hy­per­fine split­ting
    • A.40 Deuteron wave func­tion
    • A.41 Deuteron model
      • A.41.1 The model
      • A.41.2 The re­pul­sive core
      • A.41.3 Spin de­pen­dence
      • A.41.4 Non­cen­tral force
      • A.41.5 Spin-or­bit in­ter­ac­tion
    • A.42 Nu­clear forces
      • A.42.1 Ba­sic Yukawa po­ten­tial
      • A.42.2 OPEP po­ten­tial
      • A.42.3 Ex­pla­na­tion of the OPEP po­ten­tial
      • A.42.4 Mul­ti­ple pion ex­change and such
    • A.43 Clas­si­cal vi­brat­ing drop
      • A.43.1 Ba­sic de­f­i­n­i­tions
      • A.43.2 Ki­netic en­ergy
      • A.43.3 En­ergy due to sur­face ten­sion
      • A.43.4 En­ergy due to Coulomb re­pul­sion
      • A.43.5 Fre­quency of vi­bra­tion
    • A.44 Rel­a­tivis­tic neu­tri­nos
    • A.45 Fermi the­ory
      • A.45.1 Form of the wave func­tion
      • A.45.2 Source of the de­cay
      • A.45.3 Al­lowed or for­bid­den
      • A.45.4 The nu­clear op­er­a­tor
      • A.45.5 Fermi’s golden rule
      • A.45.6 Mop­ping up
      • A.45.7 Elec­tron cap­ture
  • D. De­riva­tions
    • D.1 Generic vec­tor iden­ti­ties
    • D.2 Some Green’s func­tions
      • D.2.1 The Pois­son equa­tion
      • D.2.2 The screened Pois­son equa­tion
    • D.3 La­grangian me­chan­ics
      • D.3.1 La­grangian equa­tions of mo­tion
      • D.3.2 Hamil­ton­ian dy­nam­ics
      • D.3.3 Fields
    • D.4 Lorentz trans­for­ma­tion de­riva­tion
    • D.5 Lorentz group prop­erty de­riva­tion
    • D.6 Lorentz force de­riva­tion
    • D.7 De­riva­tion of the Euler for­mula
    • D.8 Com­plete­ness of Fourier modes
    • D.9 Mo­men­tum op­er­a­tors are Her­mit­ian
    • D.10 The curl is Her­mit­ian
    • D.11 Ex­ten­sion to three-di­men­sional so­lu­tions
    • D.12 The har­monic os­cil­la­tor so­lu­tion
    • D.13 The har­monic os­cil­la­tor and un­cer­tainty
    • D.14 The spher­i­cal har­mon­ics
      • D.14.1 De­riva­tion from the eigen­value prob­lem
      • D.14.2 Par­ity
      • D.14.3 So­lu­tions of the Laplace equa­tion
      • D.14.4 Or­thog­o­nal in­te­grals
      • D.14.5 An­other way to find the spher­i­cal har­mon­ics
      • D.14.6 Still an­other way to find them
    • D.15 The hy­dro­gen ra­dial wave func­tions
    • D.16 Con­stant spher­i­cal po­ten­tials de­riva­tions
      • D.16.1 The eigen­func­tions
      • D.16.2 The Rayleigh for­mula
    • D.17 In­ner prod­uct for the ex­pec­ta­tion value
    • D.18 Eigen­func­tions of com­mut­ing op­er­a­tors
    • D.19 The gen­er­al­ized un­cer­tainty re­la­tion­ship
    • D.20 De­riva­tion of the com­mu­ta­tor rules
    • D.21 So­lu­tion of the hy­dro­gen mol­e­c­u­lar ion
    • D.22 Unique ground state wave func­tion
    • D.23 So­lu­tion of the hy­dro­gen mol­e­cule
    • D.24 Hy­dro­gen mol­e­cule ground state and spin
    • D.25 Num­ber of bo­son states
    • D.26 Den­sity of states
    • D.27 Ra­di­a­tion from a hole
    • D.28 Kirch­hoff’s law
    • D.29 The thermionic emis­sion equa­tion
    • D.30 Num­ber of con­duc­tion band elec­trons
    • D.31 In­te­gral Schrö­din­ger equa­tion
    • D.32 In­te­gral con­ser­va­tion laws
    • D.33 Quan­tum field de­riva­tions
    • D.34 The adi­a­batic the­o­rem
    • D.35 The evo­lu­tion of ex­pec­ta­tion val­ues
    • D.36 Pho­ton wave func­tion de­riva­tions
      • D.36.1 Rewrit­ing the en­ergy in­te­gral
      • D.36.2 An­gu­lar mo­men­tum states
    • D.37 Forces by par­ti­cle ex­change de­riva­tions
      • D.37.1 Clas­si­cal en­ergy min­i­miza­tion
      • D.37.2 Quan­tum en­ergy min­i­miza­tion
      • D.37.3 Rewrit­ing the La­grangian
      • D.37.4 Coulomb po­ten­tial en­ergy
    • D.38 Time-de­pen­dent per­tur­ba­tion the­ory
    • D.39 Se­lec­tion rules
    • D.40 Quan­ti­za­tion of ra­di­a­tion de­riva­tions
    • D.41 De­riva­tion of the Ein­stein B co­ef­fi­cients
    • D.42 De­riva­tion of the Ein­stein A co­ef­fi­cients
    • D.43 Mul­ti­pole de­riva­tions
      • D.43.1 Ma­trix el­e­ment for lin­ear mo­men­tum modes
      • D.43.2 Ma­trix el­e­ment for an­gu­lar mo­men­tum modes
      • D.43.3 Weis­skopf and Moszkowski es­ti­mates
    • D.44 De­riva­tion of group ve­loc­ity
    • D.45 Mo­tion through crys­tals
      • D.45.1 Prop­a­ga­tion speed
      • D.45.2 Mo­tion un­der an ex­ter­nal force
      • D.45.3 Free-elec­tron gas with con­stant elec­tric field
    • D.46 De­riva­tion of the WKB ap­prox­i­ma­tion
    • D.47 Born dif­fer­en­tial cross sec­tion
    • D.48 About La­grangian mul­ti­pli­ers
    • D.49 The gen­er­al­ized vari­a­tional prin­ci­ple
    • D.50 Spin de­gen­er­acy
    • D.51 Born-Op­pen­heimer nu­clear mo­tion
    • D.52 Sim­pli­fi­ca­tion of the Hartree-Fock en­ergy
    • D.53 In­te­gral con­straints
    • D.54 De­riva­tion of the Hartree-Fock equa­tions
    • D.55 Why the Fock op­er­a­tor is Her­mit­ian
    • D.56 Num­ber of sys­tem eigen­func­tions
    • D.57 The par­ti­cle en­ergy dis­tri­b­u­tions
    • D.58 The canon­i­cal prob­a­bil­ity dis­tri­b­u­tion
    • D.59 Analy­sis of the ideal gas Carnot cy­cle
    • D.60 Checks on the ex­pres­sion for en­tropy
    • D.61 Chem­i­cal po­ten­tial in the dis­tri­b­u­tions
    • D.62 Fermi-Dirac in­te­grals at low tem­per­a­ture
    • D.63 An­gu­lar mo­men­tum un­cer­tainty
    • D.64 Spher­i­cal har­mon­ics by lad­der op­er­a­tors
    • D.65 How to make Cleb­sch-Gor­dan ta­bles
    • D.66 The tri­an­gle in­equal­ity
    • D.67 Mo­men­tum of shells
    • D.68 Awk­ward ques­tions about spin
    • D.69 More awk­ward­ness about spin
    • D.70 Emer­gence of spin from rel­a­tiv­ity
    • D.71 Elec­tro­mag­netic com­mu­ta­tors
    • D.72 Var­i­ous elec­tro­sta­tic de­riva­tions.
      • D.72.1 Ex­is­tence of a po­ten­tial
      • D.72.2 The Laplace equa­tion
      • D.72.3 Egg-shaped di­pole field lines
      • D.72.4 Ideal charge di­pole delta func­tion
      • D.72.5 In­te­grals of the cur­rent den­sity
      • D.72.6 Lorentz forces on a cur­rent dis­tri­b­u­tion
      • D.72.7 Field of a cur­rent di­pole
      • D.72.8 Biot-Savart law
    • D.73 Or­bital mo­tion in a mag­netic field
    • D.74 Elec­tron spin in a mag­netic field
    • D.75 Solv­ing the NMR equa­tions
    • D.76 Har­monic os­cil­la­tor re­vis­ited
    • D.77 Im­pen­e­tra­ble spher­i­cal shell
    • D.78 Shell model quadru­pole mo­ment
    • D.79 De­riva­tion of per­tur­ba­tion the­ory
    • D.80 Hy­dro­gen ground state Stark ef­fect
    • D.81 Dirac fine struc­ture Hamil­ton­ian
    • D.82 Clas­si­cal spin-or­bit de­riva­tion
    • D.83 Ex­pec­ta­tion pow­ers of r for hy­dro­gen
    • D.84 Band gap ex­pla­na­tion de­riva­tions
  • N. Notes
    • N.1 Why this book?
    • N.2 His­tory and wish list
    • N.3 Na­ture and real eigen­val­ues
    • N.4 Are Her­mit­ian op­er­a­tors re­ally like that?
    • N.5 Why bound­ary con­di­tions are tricky
    • N.6 Is the vari­a­tional ap­prox­i­ma­tion best?
    • N.7 Shield­ing ap­prox­i­ma­tion lim­i­ta­tions
    • N.8 Why the s states have the least en­ergy
    • N.9 Ex­pla­na­tion of the band gaps
    • N.10 A less fishy story
    • N.11 Bet­ter de­scrip­tion of two-state sys­tems
    • N.12 Sec­ond quan­ti­za­tion in other books
    • N.13 Com­bin­ing an­gu­lar mo­men­tum fac­tors
    • N.14 The elec­tric mul­ti­pole prob­lem
    • N.15 A tenth of a googol in uni­verses
    • N.16 A sin­gle Slater de­ter­mi­nant is not ex­act
    • N.17 Gen­er­al­ized or­bitals
    • N.18 Cor­re­la­tion en­ergy
    • N.19 Am­bi­gu­i­ties in elec­tron affin­ity
    • N.20 Why Flo­quet the­ory should be called so
    • N.21 Su­per­flu­id­ity ver­sus BEC
    • N.22 The mech­a­nism of fer­ro­mag­net­ism
    • N.23 Fun­da­men­tal as­sump­tion of sta­tis­tics
    • N.24 A prob­lem if the en­ergy is given
    • N.25 The recipe of life
    • N.26 Physics of the fun­da­men­tal com­mu­ta­tors
    • N.27 Mag­ni­tude of com­po­nents of vec­tors
    • N.28 Adding an­gu­lar mo­men­tum com­po­nents
    • N.29 Cleb­sch-Gor­dan ta­bles are bidi­rec­tional
    • N.30 Ma­chine lan­guage Cleb­sch-Gor­dan ta­bles
    • N.31 Ex­is­tence of mag­netic monopoles
    • N.32 More on Maxwell’s third law
    • N.33 Set­ting the record straight on align­ment
    • N.34 NuDat 2 data se­lec­tion
    • N.35 Auger dis­cov­ery
    • N.36 Draft: Cage-of-Fara­day pro­posal
  
  
• Web Pages
• Ref­er­ences
• No­ta­tions
• In­dex

 

https://web1.eng.famu.fsu.edu/~dommelen/quantum/style_a/index.html