THE QUANTUM MECHANICS OF MANY-BODY SYSTEMS
CONTENTS 1 GROUND STATE ENERGY OF A HYBRID HARMONIC-QUARTIC OSCILLATOR: A CASE STUDY SOLVED PROBLEMS DIMENSIONAL ANALYSIS AND WHY IT FAILS IN THIS CASE SIDE COMMENT: DIMENSIONAL ANALYSIS AND APPROXIMATIONS SIDE COMMENT: HOW TO RECAST INPUT EQUATIONS IN A DIMENSIONLESS FORM DIMENSIONAL ANALYSIS: THE HARMONIC OSCILLATOR ALONE ORDER-OF-MAGNITUDE ESTIMATE: FULL SOLUTION ORDER-OF-MAGNITUDE ESTIMATES VIS-A-VIS DIMENSIONAL ANALYSIS HARMONIC VS QUARTIC REGIMES THE HARMONIC OSCILLATOR ALONE THE QUARTIC OSCILLATOR ALONE THE BOUNDARY BETWEEN THE REGIMES AND THE ?NAL RESULT AN AFTERTHOUGHT: BOUNDARY BETWEEN REGIMES FROM DIMENSIONAL CONSIDERATIONS A GAUSSIAN VARIATIONAL SOLUTION 2 A DIMENSIONAL ESTIMATE FOR THE PLANCK TEMPERATURE: A CASE STUDY ESTIMATING THE PLANCK TEMPERATURE 3 BOHR-SOMMERFELD QUANTIZATION GROUND STATE ENERGY OF A HARMONIC OSCILLATOR SPECTRUM OF A HARMONIC OSCILLATOR WKB TREATMENT OF A “STRAIGHTENED” HARMONIC OSCILLATOR GROUND STATE ENERGY OF POWER-LAW POTENTIALS SPECTRUM OF POWER-LAW POTENTIALS THE NUMBER OF BOUND STATES OF A DIATOMIC MOLECULE COULOMB PROBLEM AT ZERO ANGULAR MOMENTUM QUANTIZATION OF ANGULAR MOMENTUM FROM WKB FROM WKB QUANTIZATION OF 4D ANGULAR MOMENTUM TO QUANTIZATION OF THE COULOMB PROBLEM GROUND STATE ENERGY OF A LOGARITHMIC POTENTIAL, A WKB ANALYSIS PROBLEMS WITHOUT PROVIDED SOLUTIONS SIZE OF A NEUTRAL MESON IN SCHWINGER’S TOY MODEL OF QUARK CON?NEMENT BOHR-SOMMERFELD QUANTIZATION FOR PERIODIC BOUNDARY CONDITIONS GROUND STATE ENERGY OF MULTI-DIMENSIONAL POWER-LAW POTENTIALS 1D BOX AS A LIMIT OF POWER-LAW POTENTIALS GROUND STATE ENERGY OF A LOGARITHMIC POTENTIAL, AN ESTIMATE SPECTRUM OF A LOGARITHMIC POTENTIAL CLOSEST APPROACH TO A LOGARITHMIC HILL AND TO POWER-LAW HILLS SPIN-1/2 IN THE ?ELD OF A WIRE DIMENSIONAL ANALYSIS OF THE TIME-DEPENDENT SCHRÖDINGER EQUATION FOR A HYBRID HARMONIC-QUARTIC OSCILLATOR BACKGROUND BOHR-SOMMERFELD QUANTIZATION MULTI-DIMENSIONAL WKB PROBLEMS LINKED TO THE “BACKGROUND” BOHR-SOMMERFELD QUANTIZATION FOR ONE SOFT TURNING POINT AND A HARD WALL BOHR-SOMMERFELD QUANTIZATION FOR TWO HARD WALLS 4 “HALVED” HARMONIC OSCILLATOR: A CASE STUDY DIMENSIONAL ANALYSIS ORDER-OF-MAGNITUDE ESTIMATE ANOTHER ORDER-OF-MAGNITUDE ESTIMATE STRAIGHTFORWARD WKB EXACT SOLUTION 5 SEMI-CLASSICAL MATRIX ELEMENTS OF OBSERVABLES AND PERTURBATION THEORY QUANTUM EXPECTATION VALUE OF X6 IN A HARMONIC OSCILLATOR EXPECTATION VALUE OF R2 FOR A CIRCULAR COULOMB ORBIT WKB APPROXIMATION FOR SOME INTEGRALS INVOLVING SPHERICAL HARMONICS GROUND STATE WAVEFUNCTION OF A ONE-DIMENSIONAL BOX EIGENSTATES OF THE HARMONIC OSCILLATOR AT THE ORIGIN: HOW A FACTOR OF TWO CAN RESTORE A QUANTUM-CLASSICAL CORRESPONDENCE PROBABILITY DENSITY DISTRIBUTION IN A “STRAIGHTENED” HARMONIC OSCILLATOR EIGENSTATES OF A QUARTIC POTENTIAL AT THE ORIGIN PERTURBATION THEORY WITH EXACT AND SEMI-CLASSICAL MATRIX ELEMENTS FOR A HARMONIC OSCILLATOR PERTURBED BY A QUARTIC CORRECTION OR OR BY A CUBIC CORRECTION SHIFT OF THE ENERGY OF THE ?RST EXCITED STATE IMPOSSIBLE POTENTIALS CORRECTION TO THE FREQUENCY OF A HARMONIC OSCILLATOR AS A PERTURBATION OUTER ORBITAL OF SODIUM ATOM RELATIVE CONTRIBUTIONS OF THE EXPECTATION VALUES OF THE UNPERTURBED HAMILTONIAN AND THE PERTURBATION TO THE ?RST AND THE SECOND ORDER PERTURBATION THEORY CORRECTION TO ENERGY A PERTURBATION THEORY ESTIMATE EIGENSTATES OF A TWO-DIMENSIONAL HARMONIC OSCILLATOR AT THE ORIGIN APPROXIMATE WKB EXPRESSIONS FOR MATRIX ELEMENTS OF OBSERVABLES IN A HARMONIC OSCILLATOR O?-DIAGONAL MATRIX ELEMENTS OF THE SPATIAL COORDINATE FOR A PARTICLE IN A BOX HARMONIC OSCILLATOR PERTURBED BY A ?-POTENTIAL, AND BY A UNIFORM ?ELD PERTURBATIVE EXPANSION OF THE EXPECTATION VALUE OF THE PERTURBATION ITSELF AND THE VIRIAL THEOREM A LITTLE THEOREM MATRIX ELEMENTS OF OPERATORS IN THE WKB APPROXIMATION PERTURBATION THEORY: A BRIEF SUMMARY NON-POSITIVITY OF THE SECOND ORDER PERTURBATION THEORY SHIFT OF THE GROUND STATE ENERGY 6 VARIATIONAL PROBLEMS INSERTING A WALL PARITY OF THE EIGENSTATES SIMPLE VARIATIONAL ESTIMATE FOR THE GROUND STATE ENERGY OF A HARMONIC OSCILLATOR A PROPERTY OF VARIATIONAL ESTIMATES ABSENCE OF NODES IN THE GROUND STATE ABSENCE OF DEGENERACY OF THE GROUND STATE ENERGY LEVEL DO STRONGER POTENTIALS ALWAYS LEAD TO HIGHER GROUND STATE ENERGIES? VARIATIONAL ANALYSIS MEETS PERTURBATION THEORY ANOTHER VARIATIONAL ESTIMATE FOR THE GROUND STATE ENERGY OF A HARMONIC OSCILLATOR AND YET ANOTHER GAUSSIAN- AND WEDGE- VARIATIONAL GROUND STATE ENERGY OF A QUARTIC OSCILLATOR VARIATIONAL ANALYSIS COMPLEX VS REAL VARIATIONAL SPACES A PROOF THAT THE (??)2 ENERGY FUNCTIONAL DOES NOT HAVE MINIMA WITH DISCONTINUOUS DERIVATIVES 7 GRAVITATIONAL WELL: A CASE STUDY A WKB-BASED ORDER-OF-MAGNITUDE ESTIMATE FOR THE SPECTRUM A WKB-BASED DIMENSIONAL ESTIMATE FOR THE SPECTRUM A PERTURBATIVE CALCULATION OF THE SHIFT OF THE ENERGY LEVELS UNDER A SMALL CHANGE IN THE COUPLING CONSTANT THE ?RST ORDER A DIMENSIONAL ESTIMATE FOR THE PERTURBATIVE CORRECTION TO THE SPECTRUM A PERTURBATIVE CALCULATION OF THE SHIFT OF THE ENERGY LEVELS UNDER A SMALL CHANGE IN THE COUPLING CONSTANT THE SECOND ORDER A SIMPLE VARIATIONAL TREATMENT OF THE GROUND STATE OF A GRAVITATIONAL WELL 8 MISCELLANEOUS A DIMENSIONAL APPROACH TO THE QUESTION OF THE NUMBER OF BOUND STATES IN ?-POTENTIAL WELL AND IN A PÖSCHL-TELLER POTENTIAL EXISTENCE OF LOSSLESS EIGENSTATES IN THE 1/X2-POTENTIAL ON THE ABSENCE OF THE UNITARY LIMIT IN TWO DIMENSIONS 9 THE HELLMANN-FEYNMAN THEOREM LIEB-LINIGER MODEL EXPECTATION VALUES OF 1/R2 AND 1/R IN THE COULOMB PROBLEM, USING THE HELLMANN-FEYNMAN THEOREM EXPECTATION VALUE OF THE TRAPPING ENERGY IN THE GROUND STATE OF THE CALOGERO SYSTEM VIRIAL THEOREM FROM THE HELLMANN-FEYNMAN THEOREM VIRIAL THEOREM FOR THE LOGARITHMIC POTENTIAL AND ITS COROLLARIES THE HELLMANN-FEYNMAN THEOREM 10 LOCAL DENSITY APPROXIMATION THEORIES A THOMAS-FERMI ESTIMATE FOR THE ATOM SIZE AND TOTAL IONIZATION ENERGY THE SIZE OF AN ION TIME-DEPENDENT THOMAS-FERMI MODEL FOR COLD BOSONS THE QUANTUM DOT DIMENSIONAL ANALYSIS OF AN ATOM BEYOND THE THOMAS-FERMI MODEL 11 INTEGRABLE PARTIAL DI?ERENTIAL EQUATIONS SOLITONS OF THE KORTEWEG-DE VRIES EQUATION BREATHERS OF THE NONLINEAR SCHRÖDINGER EQUATION HEALING LENGTH DIMENSIONAL ANALYSIS OF THE PROJECTILE PROBLEM AS A PRELUDE TO A DISCUSSION ON THE KADOMTSEV-PETVIASHVILI SOLITONS KADOMTSEV-PETVIASHVILI EQUATION THE NONLINEAR TRANSPORT EQUATION BURGERS EQUATION STATIONARY SOLITONS OF THE BURGERS EQUATION STATIONARY SOLITONS OF THE NONLINEAR SCHRÖDINGER EQUATION SOLITONS OF THE SINE-GORDON EQUATION 12 RARE AND EXOTIC METHODS IN ELEMENTARY QUANTUM MECHANICS AND BEYOND QUANTUM-MECHANICAL SUPERSYMMETRY (QM-SUSY): PÖSCHL-TELLER AS AN EXAMPLE WHAT THE SUPERSYMMETRIC STRUCTURE ALONE IMPLIES FOR THE SCATTERING STATES OF THE PÖSCHL-TELLER POTENTIAL POWER-INDEX METHOD EXAMPLE OF THE NONLINEAR SCHRÖDINGER EQUATION, WITH THE PÖSCHL-TELLER PROBLEM AS A BYPRODUCT A STATIONARY-KINK SOLUTION OF THE BURGERS EQUATION THROUGH THE POWER-INDEX METHOD SCALE INVARIANCE: QUANTUM CALOGERO POTENTIAL AS AN EXAMPLE CLASSICAL CALOGERO POTENTIAL: A POSTERIORI MANIFESTATIONS OF THE SCALE INVARIANCE CLASSICAL CALOGERO POTENTIAL: A PRIORI MANIFESTA- TIONS OF SCALE INVARIANCE AT THE MAUPERTUIS-JACOBI LEVEL FINDING THE ZERO-ENERGY ORBIT FROM SYMMETRIES ALONE CIRCLE INVERSION, QUANTUM: ZERO-ENERGY EIGENSTATES IN A 1/R4 POTENTIAL SELF-SIMILAR TILINGS: MOMENT OF INERTIA OF AN EQUILATERAL TRIANGLE AS A PARADIGM 12 1 10 PROPERTIES OF THE SPECTRUM OF AN EQUILATERAL TRIANGULAR QUANTUM BILLIARDS THAT FOLLOW FROM THE TILING SELF-SIMILARITY 12 1 11 CENTER OF MASS OF A ?LLED “GOLDEN B” SHAPE, THROUGH SELF-SIMILAR TILINGS POWER-INDEX METHOD APPLIED TO THE PÖSCHL-TELLER PROBLEM DIRECTLY QUASI-INTEGRABILITY: THE OVERALL IDEA AND THE EXAMPLE OF THE SIXTIC OSCILLATOR A QUASI-INTEGRABLE INSTANCE OF TWO NONLINEARLY COUPLED OSCILLATORS MOMENT OF INERTIA OF A ?LLED “GOLDEN B” SHAPE, THROUGH THE SELF-SIMILAR TILINGS RELATIONSHIP BETWEEN VARIOUS METHODS FURTHER READING: BETHE ANSATZ CHAPTER DIMENSIONAL 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 IN- FORMATION ABOUT THE STRUCTURE OF THE EXPRESSION FOR THE GROUND STATE ENERGY CAN STILL BE EXTRACTED, ON PURELY DIMENSIONAL GROUNDS SOLUTION: THE DIMENSIONAL PROCEDURE FOR ?NDING THE GROUND STATE ENERGY EG S (OR ASSESSING THE IMPOSSIBILITY OF A COMPLETE DIMENSIONAL SOLUTION) IS AS FOLLOWS