THEORETICAL AND QUANTUM MECHANICS
CONTENTS MAXWELL AND SCHR¨ODINGER LIGHT AND LINEAR OPERATORS DE BROGLIES IDEA AND SCHR¨ODINGERS EQUATION POTENTIALS AND GAUGE INVARIANCE STATIONARY POTENTIALS, ZEEMAN SHIFTS BOUND STATES SPINLESS HYDROGENLIKE ATOMS LANDAU LEVELS AND HARMONIC OSCILLATOR ORTHOGONALITY AND MEASUREMENTS OPERATOR METHODS, MATRICES 1 10 SCATTERING AND PHASE SHIFTS LORENTZ, PAULI AND DIRAC LORENTZ TRANSFORMATIONS SPINLESS CURRENT, DENSITY OF STATES PAULI’S ELECTRON SPIN THE DIRAC EQUATION ADDITION OF ANGULAR MOMENTA HYDROGEN ATOM AND PARITY BASIS ALTERNATIVE FORM, PERTURBATIONS THE PAULI EQUATION THE ZEEMAN E?ECT 2 10 THE DIRAC CURRENT FREE ELECTRONS QUANTUM FIELDS AND PARTICLES THE PHOTON FIELD C, P AND T FIELD OPERATORS AND WAVE EQUATIONS BREIT OPERATORS TWO-ELECTRON STATES AND PAULI PRINCIPLE ELIMINATION OF COMPONENTS BROWN-RAVENHALL DISEASE, ENERGY PROJECTORS, IMPROVED BREITIAN VARIATIONAL METHOD, SHELL MODEL THE PAULI PRINCIPLE FOR THREE ELECTRONS SCATTERING AND BOUND STATES INTRODUCTION BORN SERIES AND S-MATRIX TWO-BODY SCATTERING AND DECAY CURRENT MATRIX ELEMENTS, FORM FACTORS PARTICLES OF HIGHER SPINS THE EQUATION FOR SPINLESS BINARIES THE LEPTONIUM EQUATION THE INTERACTION IN LEPTONIUM BINARY BOOSTS 4 10 KLEIN-DIRAC EQUATION, HYDROGEN 4 11 DIRAC STRUCTURES OF BINARY BOUND STATES HYPER?NE SHIFTS, RADIATION, QUARKS FIRST-ORDER MAGNETIC HYPER?NE SPLITTING NONRELATIVISTIC MAGNETIC HYPER?NE OPERATORS VACUUM POLARIZATION, DISPERSION RELATIONS ATOMIC RADIATION SOFT PHOTONS, LAMB SHIFT ANTIPROTONIC ATOMS, QUADRUPOLE POTENTIAL THE MAGNETIC MOMENT INTERACTION SU2, SU3, QUARKS BARYON MAGNETIC MOMENTS Z = 0 + ?2, ?0 = ?/?(CT) (1 3) THE FULL USE OF THIS NOMENCLATURE WILL BE POSTPONED TO CHAP 2 FOR THE MO- MENT, T IS EXPRESSED IN TERMS OF X0 MERELY TO SUPPRESS THE CONSTANT C TODAY, C IS IN FACT USED IN THE DE?NITION OF THE LENGTH SCALE, SEE SECT 1 6 DI?ERENTIAL OPERATORS D ARE LINEAR IN THE SENSE D(E1+E2) = DE1+DE2 IF E1 AND E2 ARE TWO DI?ERENT SOLUTIONS OF (1 1), E = E1 + E2 IS A THIRD ONE THIS IS CALLED THE SUPERPOSITION PRINCIPLE THE INTENSITY I OF LIGHT IS NORMALLY MEASURED BY E2, I ?E2 ?SQUARE(E), BUT NONLINEAR OPERA- TORS SUCH AS “SQUARE” ARE NOT USED IN QUANTUM MECHANICS ?AND ?2 ARE RR OF THE LAPLACIAN (1 7), THE ?RST X Y Z ? ? R ? FIG 1 2 SPHERICAL COORDINATES 1 1 LIGHT AND LINEAR OPERATORS Z (1 14) WE SHALL ALSO NEED CYLINDRICAL AND SPHERICAL WAVES, WHERE ?2 IS REQUIRED IN THE FORMS (1 5) AND (1 7) SUCH WAVES CAN ALSO BE MONOCHROMATIC, MEANING THAT THEY CONTAIN ONLY ONE (ANGULAR) FREQUENCY ? THE COMMON WAVE EQUATION FOR ALL MONOCHROMATIC WAVES IN VACUUM IS E(Xµ) = E?I?TE?(R), (?2/C2 + ?2)E?(R) = 0 (1 15) THIS “HELMHOLTZ EQUATION” IS STILL A PARTIAL DI?ERENTIAL EQUATION IN THREE VARIABLES; WE RECALL A FEW TRICKS FOR THE SOLUTION OF SUCH EQUATIONS THE MAIN TRICK IS TO EXPRESS ?2 IN TERMS OF COMMUTING OPERATORS A, B, C, AND THEN TO CONSTRUCT “EIGENFUNCTIONS” OF THESE OPERATORS WHEN A IS APPLIED TO ANY OF ITS EIGENFUNCTIONS FN, IT MAY BE REPLACED SIMPLY BY A CONSTANT AN, CALLED THE EIGENVALUE: AFN = ANFN (1 16) FOR EXAMPLE, THE SQUARE OF THE OPERATOR ?? OCCURS BOTH IN CYLINDRICAL AND IN SPHERICAL COORDINATES THE NORMALIZED EIGENFUNCTIONS OF ?? ARE ?ML(?) = (2?)?1/2EIML?, ML = 0, ±1, ±2 (1 17) IN QUANTUM MECHANICS, ML IS CALLED THE (ORBITAL) MAGNETIC QUANTUM NUMBER (SECT 1 4) THE NORMALIZATION IS CHOSEN SUCH THAT 2? ?? ML?MLD? = 2? |?ML|2D? = 1 (1 18) L = (L + 2 PL(U) P0 = 1, P1 = U, P2 = 2(3U3 ?1), P3 = 2(5U3 ?3U) (1 23) WHEN APPLIED TO THE SPHERICAL HARMONICS, THE LAPLACIAN (1 6) E?ECTIVELY BE- COMES A RADIAL OPERATOR, I E INDEPENDENT OF ? AND ? THUS E?(R) HAS SOLUTIONS OF THE FORM E?(R) = E0(?)R?,L(R)Y M L (?, ?), (1 24) (?2/C2 + ?2)E? = E0Y M L [?2/C2 + (?R + 1/R)2 ?L(L + 1)/R2]R?,L(R) (1 25) TABLE 1 1 Y M L FOR L < 3 NORMALIZATION (1 186), X± = ?X ?IY Y 2 , Y = (3/4?) 2 COS ? = (3/4?) = ?(3/8?) 2 SIN ?E±I? = (3/8?) 2 X±/R, Y = (5/16?) 2 (3 COS2 ? ?1) = (5/16?) = ?(15/8?) 2 COS ? SIN ?E±I? = (15/8?) = (15/32?) 2 E±2I? SIN2 ? = (15/32?) ±/R2 1 2 DE BROGLIES IDEA AND SCHR¨ODINGERS EQUATION DIVIDING O?THE ?RST TWO FACTORS, ONE ?NDS THE DI?ERENTIAL EQUATION FOR THE RADIAL WAVE FUNCTION R(R), [?2/C2 + (?R + 1/R)2 ?L(L + 1)/R2]R?,L(R) = 0 (1 26) ALSO THIS EQUATION HAS SIMPLE SOLUTIONS, TO BE DISCUSSED IN SECT 1 10 E NEED NOT BE AN EIGENFUNCTION OF ANY OF THESE OPERATORS, BUT IT MAY BE EXPANDED IN TERMS OF THE EIGENFUNCTIONS REAL LIGHT HAS A “SPECTRAL DECOM- POSITION”, E(T, R) = ? ? WE CONCLUDE WITH THE SOLUTION OF (1 26) FOR L = 0, [?2/C2 + (?R + 1/R)2]R?,0(R) = 0 ALSO THIS EQUATION HAS TWO SOLUTIONS, R± = R?1E±IKR, (?R + 1/R)R± = R?1?RE±IKR = ±IKR±, (1 29) WITH K2 = ?2/C2, AS USUAL R+ IS THE SIMPLEST EXAMPLE OF AN OUTGOING SPHERICAL WAVE (IT DOES NOT REPRESENT DIPOLE RADIATION, BECAUSE THE COULOMB GAUGE CONDITION DIVE = 0 HAS BEEN IGNORED ) FOR COMPLEX E, THE INTENSITY IS I ? E?E INSTEAD OF E2 IT DECREASES WITH R AS R?2, AS EXPECTED 1 2 DE BROGLIES IDEA AND SCHR¨ODINGERS EQUATION ALTHOUGH LIGHT DOES PROPAGATE ACCORDING TO THE WAVE EQUATION JUST DIS- CUSSED, IT IS NEVERTHELESS EMITTED AND ABSORBED IN QUANTA CALLED PHOTONS IN MONOCHROMATIC LIGHT OF THE TYPE (1 15), EACH PHOTON HAS THE SAME ENERGY E = H?, AND IN THE CASE OF A PLANE MONOCHROMATIC WAVE (1 11), IT ALSO HAS A ?XED MOMENTUM P = HK/2?: E = H? = ¯H? = HC/?, P = ¯HK, (1 30) ¯H = H/2? = 6 58218 × 10?16 EV S, (1 31)