QUANTUM MECHANICS – CONCEPTS AND APPLICATIONS
CONTENTS MATHEMATICAL PRELIMINARIES THE STATE VECTORS THE INNER PRODUCT LINEAR OPERATORS EIGENSTATES AND EIGENVALUES THE DIRAC DELTA FUNCTION THE LAWS (POSTULATES) OF QUANTUM MECHANICS A LESSON FROM CLASSICAL MECHANICS THE POSTULATES OF QUANTUM MECHANICS SOME HISTORY OF THE POSTULATES POPULAR REPRESENTATIONS THE POSITION REPRESENTATION THE MOMENTUM REPRESENTATION SOME SIMPLE EXAMPLES THE HAMILTONIAN, CONSERVED QUANTITIES AND EXPECTATION VALUE FREE PARTICLE IN ONE DIMENSION MOMENTUM ENERGY POSITION THE HARMONIC OSCILLATOR SOLUTION IN POSITION REPRESENTATION A REPRESENTATION FREE SOLUTION LANDAU LEVELS MORE ONE DIMENSIONAL EXAMPLES GENERAL CHARACTERISTICS OF SOLUTIONS E < V (X) FOR ALL X BOUND STATES SCATTERING STATES SOME OVERSIMPLI¯ED EXAMPLES RECTANGULAR POTENTIAL WELL (BOUND STATES) RECTANGULAR POTENTIAL BARRIER (SCATTERING STATES) NUMERICAL TECHNIQUES IN ONE SPACE DIMENSION FINITE DI®ERENCES ONE DIMENSIONAL SCATTERING ONE DIMENSIONAL BOUND STATE PROBLEMS OTHER TECHNIQUES ACCURACY SPEED SYMMETRIES AND CONSERVED QUANTITIES SYMMETRY GROUPS AND THEIR REPRESENTATION SPACE TRANSLATION SYMMETRY TIME TRANSLATION SYMMETRY ROTATION SYMMETRY EIGENVALUES OF ANGULAR MOMENTUM ADDITION OF ANGULAR MOMENTA DISCRETE SYMMETRIES SPACE INVERSION TIME REVERSAL THREE DIMENSIONAL SYSTEMS GENERAL CHARACTERISTICS OF BOUND STATES SPHERICALLY SYMMETRIC POTENTIALS ANGULAR MOMENTUM THE TWO BODY PROBLEM THE HYDROGEN ATOM (BOUND STATES) SCATTERING IN THREE DIMENSIONS CENTER OF MASS FRAME VS LABORATORY FRAME RELATION BETWEEN ASYMPTOTIC WAVEFUNCTION AND CROSS SECTION SCATTERING DUE TO A SPHERICALLY SYMMETRIC POTENTIAL NUMERICAL TECHNIQUES IN THREE SPACE DIMENSIONS BOUND STATES (SPHERICALLY SYMMETRIC POTENTIALS) BOUND STATES (GENERAL POTENTIAL) SCATTERING STATES (SPHERICALLY SYMMETRIC POTENTIALS) SCATTERING STATES (GENERAL POTENTIAL) 10 APPROXIMATION METHODS (BOUND STATES) 10 1 PERTURBATION METHOD (NONDEGENERATE STATES) 10 2 DEGENERATE STATE PERTURBATION ANALYSIS 10 3 TIME DEPENDENT PERTURBATION ANALYSIS 10 4 THE VARIATIONAL METHOD 11 APPROXIMATION METHODS (SCATTERING STATES) 11 1 THE GREEN'S FUNCTION METHOD 11 2 THE SCATTERING MATRIX 11 3 THE STATIONARY CASE 11 4 THE BORN APPROXIMATION 12 SPIN AND ATOMIC SPECTRA 12 1 DEGENERATE POSITION EIGENSTATES 12 2 SPIN-HALF PARTICLES 12 3 SPIN MAGNETIC MOMENT (STERN-GERLACH EXPERIMENT) 12 4 SPIN-ORBIT COUPLING 12 5 ZEEMAN E®ECT REVISITED 13 RELATIVISTIC QUANTUM MECHANICS 13 1 THE KLEIN-GORDON EQUATION 13 2 THE DIRAC EQUATION 13 3 SPIN AND THE DIRAC PARTICLE 13 4 SPIN-ORBIT COUPLING IN THE DIRAC HAMILTONIAN 13 5 THE DIRAC HYDROGEN ATOM 13 6 THE DIRAC PARTICLE IN A MAGNETIC ¯ELD CHAPTER THE STATE VECTORS IN THE NEXT CHAPTER WE SHALL CONSIDER THE COMPLETE DESCRIPTOR OF A SYSTEM TO BE ITS STATE VECTOR HERE I SHALL DE¯NE THE STATE VECTOR THROUGH ITS PROPERTIES SOME PROPERTIES AND DE¯NITIONS THAT ARE TOO OBVIOUS WILL BE OMITTED I SHALL USE A SLIGHTLY MODI¯ED VERSION OF THE CONVENIENT NOTATION GIVEN BY DIRAC [1] A STATE VECTOR MIGHT ALSO BE CALLED A STATE OR A VECTOR FOR SHORT IN THE FOLLOWING, THE READER IS ENCOURAGED TO SEE ANALOGIES FROM COMPLEX MATRIX ALGEBRA A STATE VECTOR FOR SOME STATE S CAN BE REPRESENTED BY THE SO CALLED KET VECTOR JSI THE LABEL S CAN BE CHOSEN CONVENIENTLY FOR SPECI¯C PROBLEMS JSI WILL IN GENERAL DEPEND ON ALL DEGREES OF FREEDOM OF THE SYSTEM AS WELL AS TIME THE SPACE OF ALL POSSIBLE KETS FOR A SYSTEM WILL BE CALLED THE LINEAR VECTOR SPACE V IN THE FOLLOWING, THE TERM LINEAR WILL BE DROPPED AS ALL VECTOR SPACES CONSIDERED HERE WILL BE LINEAR THE FUNDAMENTAL PROPERTY (OR RULE) OF V IS RULE 1 IF JSI; JRI 2 V THEN AJSI + BJRI 2 V; WHERE A; B 2 C (SET OF COMPLEX NUMBERS) THE MEANING OF ADDITION OF KETS AND MULTIPLICATION BY COMPLEX NUMBERS WILL BECOME OBVIOUS IN THE SENSE OF COMPONENTS OF THE VECTOR ONCE COMPONENTS ARE DE¯NED THE PHYSICAL CONTENT OF THE STATE VECTOR IS PURELY IN ITS \DIRECTION", THAT IS RULE 2 THE PHYSICAL CONTENTS OF JSI AND AJSI ARE THE SAME IF A 2 C AND A = 0 AT THIS STAGE THE FOLLOWING COMMONLY USED TERMS CAN BE DE¯NED