PHYSICAL PROBLEMS SOLVED BY THE PHASE-INTEGRAL METHOD
CONTENTS 1 HISTORICAL SURVEY 1 1 DEVELOPMENT FROM 1817 TO 1 1 1 CARLINI’S PIONEERING WORK 1 1 2 THE WORK BY LIOUVILLE AND GREEN JACOBI’S CONTRIBUTION TOWARDS MAKING CARLINI’S WORK KNOWN SCHEIBNER’S ALTERNATIVE TO CARLINI’S TREATMENT OF PLANETARY MOTION 1 1 6 FIRST TRACES OF A CONNECTION FORMULA 1 1 8 BOTH CONNECTION FORMULAS ARE DERIVED IN EXPLICIT FORM 1 1 9 THE METHOD IS REDISCOVERED IN QUANTUM MECHANICS 1 2 DEVELOPMENT AFTER 2 DESCRIPTION OF THE PHASE-INTEGRAL METHOD 2 1 FORM OF THE WAVE FUNCTION AND THE Q-EQUATION 2 2 PHASE-INTEGRAL APPROXIMATION GENERATED FROM AN UNSPECI?ED BASE FUNCTION 2 3 F-MATRIX METHOD 2 3 1 EXACT SOLUTION EXPRESSED IN TERMS OF THE F-MATRIX 2 3 2 GENERAL RELATIONS SATIS?ED BY THE F-MATRIX 2 3 3 F-MATRIX CORRESPONDING TO THE ENCIRCLING OF A SIMPLE ZERO OF Q2(Z) 2 3 4 BASIC ESTIMATES 2 3 5 STOKES AND ANTI-STOKES LINES SYMBOLS FACILITATING THE TRACING OF A WAVE FUNCTION IN THE COMPLEX Z-PLANE REMOVAL OF A BOUNDARY CONDITION FROM THE REAL Z-AXIS TO AN ANTI-STOKES LINE DEPENDENCE OF THE F-MATRIX ON THE LOWER LIMIT OF INTEGRATION IN THE PHASE INTEGRAL F-MATRIX EXPRESSED IN TERMS OF TWO LINEARLY INDEPENDENT SOLUTIONS OF THE DIFFERENTIAL EQUATION F-MATRIX CONNECTING POINTS ON OPPOSITE SIDES OF A WELL-ISOLATED TURNING POINT, AND EXPRESSIONS FOR THE WAVE FUNCTION IN THESE REGIONS SYMMETRY RELATIONS AND ESTIMATES OF THE F-MATRIX ELEMENTS 2 4 2 PARAMETERIZATION OF THE MATRIX F(X1, X2) 2 4 2 1 CHANGES OF ?, ? AND ? WHEN X1 MOVES IN THE CLASSICALLY FORBIDDEN REGION CHANGES OF ?, ? AND ? WHEN X2 MOVES IN THE CLASSICALLY ALLOWED REGION 2 4 2 3 LIMITING VALUES OF ?, ? AND ? 2 4 3 WAVE FUNCTION ON OPPOSITE SIDES OF A WELL-ISOLATED TURNING POINT 2 4 4 POWER AND LIMITATION OF THE PARAMETERIZATION METHOD 2 5 PHASE-INTEGRAL CONNECTION FORMULAS FOR A REAL, SMOOTH, SINGLE-HUMP POTENTIAL BARRIER 2 5 1 EXACT EXPRESSIONS FOR THE WAVE FUNCTION ON BOTH SIDES OF THE BARRIER 2 5 2 PHASE-INTEGRAL CONNECTION FORMULAS FOR A REAL BARRIER 2 5 2 1 WAVE FUNCTION GIVEN AS AN OUTGOING WAVE TO THE LEFT OF THE BARRIER 2 5 2 2 WAVE FUNCTION GIVEN AS A STANDING WAVE TO THE LEFT OF THE BARRIER 3 PROBLEMS WITH SOLUTIONS BASE FUNCTION FOR THE RADIAL SCHR¨ODINGER EQUATION WHEN THE PHYSICAL POTENTIAL HAS AT THE MOST A COULOMB SINGULARITY AT THE ORIGIN BASE FUNCTION AND WAVE FUNCTION CLOSE TO THE ORIGIN WHEN THE PHYSICAL POTENTIAL IS REPULSIVE AND STRONGLY SINGULAR AT THE ORIGIN 3 3 RE?ECTIONLESS POTENTIAL 3 4 STOKES AND ANTI-STOKES LINES PROPERTIES OF THE PHASE-INTEGRAL APPROXIMATION ALONG AN ANTI-STOKES LINE PROPERTIES OF THE PHASE-INTEGRAL APPROXIMATION ALONG A PATH ON WHICH THE ABSOLUTE VALUE OF EXP[IW(Z)] IS MONOTONIC IN THE STRICT SENSE, IN PARTICULAR ALONG A STOKES LINE DETERMINATION OF THE STOKES CONSTANTS ASSOCIATED WITH THE THREE ANTI-STOKES LINES THAT EMERGE FROM A WELL ISOLATED, SIMPLE TRANSITION ZERO CONNECTION FORMULA FOR TRACING A PHASE-INTEGRAL WAVE FUNCTION FROM A STOKES LINE EMERGING FROM A SIMPLE TRANSITION ZERO T TO THE ANTI-STOKES LINE EMERGING FROM T IN THE OPPOSITE DIRECTION CONNECTION FORMULA FOR TRACING A PHASE-INTEGRAL WAVE FUNCTION FROM AN ANTI-STOKES LINE EMERGING FROM A SIMPLE TRANSITION ZERO T TO THE STOKES LINE EMERGING FROM T IN THE OPPOSITE DIRECTION CONNECTION FORMULA FOR TRACING A PHASE-INTEGRAL WAVE FUNCTION FROM A CLASSICALLY FORBIDDEN TO A CLASSICALLY ALLOWED REGION 3 11 ONE-DIRECTIONAL NATURE OF THE CONNECTION FORMULA FOR TRACING A PHASE-INTEGRAL WAVE FUNCTION FROM A CLASSICALLY FORBIDDEN TO A CLASSICALLY ALLOWED REGION 3 12 CONNECTION FORMULAS FOR TRACING A PHASE-INTEGRAL WAVE FUNCTION FROM A CLASSICALLY ALLOWED TO A CLASSICALLY FORBIDDEN REGION 3 13 ONE-DIRECTIONAL NATURE OF THE CONNECTION FORMULAS FOR TRACING A PHASE-INTEGRAL WAVE FUNCTION FROM A CLASSICALLY ALLOWED TO A CLASSICALLY FORBIDDEN REGION VALUE AT THE TURNING POINT OF THE WAVE FUNCTION ASSOCIATED WITH THE CONNECTION FORMULA FOR TRACING A PHASE-INTEGRAL WAVE FUNCTION FROM THE CLASSICALLY FORBIDDEN TO THE CLASSICALLY ALLOWED REGION 3 15 VALUE AT THE TURNING POINT OF THE WAVE FUNCTION ASSOCIATED WITH A CONNECTION FORMULA FOR TRACING THE PHASE-INTEGRAL WAVE FUNCTION FROM THE CLASSICALLY ALLOWED TO THE CLASSICALLY FORBIDDEN REGION 3 16 ILLUSTRATION OF THE ACCURACY OF THE APPROXIMATE FORMULAS FOR THE VALUE OF THE WAVE FUNCTION AT A TURNING POINT EXPRESSIONS FOR THE A-COEF?CIENTS ASSOCIATED WITH THE AIRY FUNCTIONS EXPRESSIONS FOR THE PARAMETERS ?, ? AND ? WHEN Q2(Z) = R(Z) = ?Z SOLUTIONS OF THE AIRY DIFFERENTIAL EQUATION THAT AT A ?XED POINT ON ONE SIDE OF THE TURNING POINT ARE REPRESENTED BY A SINGLE, PURE PHASE-INTEGRAL FUNCTION, AND THEIR REPRESENTATION ON THE OTHER SIDE OF THE TURNING POINT CONNECTION FORMULAS AND THEIR ONE-DIRECTIONAL NATURE DEMONSTRATED FOR THE AIRY DIFFERENTIAL EQUATION DEPENDENCE OF THE PHASE OF THE WAVE FUNCTION IN A CLASSICALLY ALLOWED REGION ON THE VALUE OF THE LOGARITHMIC DERIVATIVE OF THE WAVE FUNCTION AT A ?XED POINT X1 IN AN ADJACENT CLASSICALLY FORBIDDEN REGION 3 22 PHASE OF THE WAVE FUNCTION IN THE CLASSICALLY ALLOWED REGIONS ADJACENT TO A REAL, SYMMETRIC POTENTIAL BARRIER, WHEN THE LOGARITHMIC DERIVATIVE OF THE WAVE FUNCTION IS GIVEN AT THE CENTRE OF THE BARRIER EIGENVALUE PROBLEM FOR A QUANTAL PARTICLE IN A BROAD, SYMMETRIC POTENTIAL WELL BETWEEN TWO SYMMETRIC POTENTIAL BARRIERS OF EQUAL SHAPE, WITH BOUNDARY CONDITIONS IMPOSED IN THE MIDDLE OF EACH BARRIER DEPENDENCE OF THE PHASE OF THE WAVE FUNCTION IN A CLASSICALLY ALLOWED REGION ON THE POSITION OF THE POINT X1 IN AN ADJACENT CLASSICALLY FORBIDDEN REGION WHERE THE BOUNDARY CONDITION ?(X1) = 0 IS IMPOSED 3 25 PHASE-SHIFT FORMULA 3 26 DISTANCE BETWEEN NEAR-LYING ENERGY LEVELS IN DIFFERENT TYPES OF PHYSICAL SYSTEMS, EXPRESSED EITHER IN TERMS OF THE FREQUENCY OF CLASSICAL OSCILLATIONS IN A POTENTIAL WELL OR IN TERMS OF THE DERIVATIVE OF THE ENERGY WITH RESPECT TO A QUANTUM NUMBER ARBITRARY-ORDER QUANTIZATION CONDITION FOR A PARTICLE IN A SINGLE-WELL POTENTIAL, DERIVED ON THE ASSUMPTION THAT THE CLASSICALLY ALLOWED REGION IS BROAD ENOUGH TO ALLOW THE USE OF A CONNECTION FORMULA ARBITRARY-ORDER QUANTIZATION CONDITION FOR A PARTICLE IN A SINGLE-WELL POTENTIAL, DERIVED WITHOUT THE ASSUMPTION THAT THE CLASSICALLY ALLOWED REGION IS BROAD DISPLACEMENT OF THE ENERGY LEVELS DUE TO COMPRESSION OF AN ATOM (SIMPLE TREATMENT) 3 30 DISPLACEMENT OF THE ENERGY LEVELS DUE TO COMPRESSION OF AN ATOM (ALTERNATIVE TREATMENT) QUANTIZATION CONDITION FOR A PARTICLE IN A SMOOTH POTENTIAL WELL, LIMITED ON ONE SIDE BY AN IMPENETRABLE WALL AND ON THE OTHER SIDE BY A SMOOTH, IN?NITELY THICK POTENTIAL BARRIER, AND IN PARTICULAR FOR A PARTICLE IN A UNIFORM GRAVITATIONAL ?ELD LIMITED FROM BELOW BY AN IMPENETRABLE PLANE SURFACE IS A QUANTITY WITH THE DIMENSION OF LENGTH, E G THE BOHR RADIUS DETERMINATION OF A ONE-DIMENSIONAL, SMOOTH, SINGLE-WELL POTENTIAL FROM THE ENERGY SPECTRUM OF THE BOUND STATES DETERMINATION OF A RADIAL, SMOOTH, SINGLE-WELL POTENTIAL FROM THE ENERGY SPECTRUM OF THE BOUND STATES DETERMINATION OF THE RADIAL, SINGLE-WELL POTENTIAL, WHEN THE ENERGY EIGENVALUES ARE ?MZ2E4/[2¯H2(L + S + 1)2], WHERE L IS THE ANGULAR MOMENTUM QUANTUM NUMBER, AND S IS THE RADIAL QUANTUM NUMBER 3 36 EXACT FORMULA FOR THE NORMALIZATION INTEGRAL FOR THE WAVE FUNCTION PERTAINING TO A BOUND STATE OF A PARTICLE IN A RADIAL POTENTIAL 3 37 PHASE-INTEGRAL FORMULA FOR THE NORMALIZED RADIAL WAVE FUNCTION PERTAINING TO A BOUND STATE OF A PARTICLE IN A RADIAL SINGLE-WELL POTENTIAL RADIAL WAVE FUNCTION ?(Z) FOR AN S-ELECTRON IN A CLASSICALLY ALLOWED REGION CONTAINING THE ORIGIN, WHEN THE POTENTIAL NEAR THE ORIGIN IS DOMINATED BY A STRONG, ATTRACTIVE COULOMB SINGULARITY, AND THE NORMALIZATION FACTOR IS CHOSEN SUCH THAT, WHEN THE RADIAL VARIABLE Z IS DIMENSIONLESS, ?(Z)/Z TENDS TO UNITY AS Z TENDS TO ZERO 3 39 QUANTIZATION CONDITION, AND VALUE OF THE NORMALIZED WAVE FUNCTION AT THE ORIGIN EXPRESSED IN TERMS OF THE LEVEL DENSITY, FOR AN S-ELECTRON IN A SINGLE-WELL POTENTIAL WITH A STRONG ATTRACTIVE COULOMB SINGULARITY AT THE ORIGIN 3 40 EXPECTATION VALUE OF AN UNSPECI?ED FUNCTION F (Z) FOR A NON-RELATIVISTIC PARTICLE IN A BOUND STATE 3 41 SOME CASES IN WHICH THE PHASE-INTEGRAL EXPECTATION VALUE FORMULA YIELDS THE EXPECTATION VALUE EXACTLY IN THE ?RST-ORDER APPROXIMATION EXPECTATION VALUE OF THE KINETIC ENERGY OF A NON-RELATIVISTIC PARTICLE IN A BOUND STATE VERI?CATION OF THE VIRIAL THEOREM 3 43 PHASE-INTEGRAL CALCULATION OF QUANTAL MATRIX ELEMENTS 3 44 CONNECTION FORMULA FOR A COMPLEX POTENTIAL BARRIER 3 45 CONNECTION FORMULA FOR A REAL, SINGLE-HUMP POTENTIAL BARRIER ENERGY LEVELS OF A PARTICLE IN A SMOOTH DOUBLE-WELL POTENTIAL, WHEN NO SYMMETRY REQUIREMENT IS IMPOSED ENERGY LEVELS OF A PARTICLE IN A SMOOTH, SYMMETRIC, DOUBLE-WELL POTENTIAL DETERMINATION OF THE QUASI-STATIONARY ENERGY LEVELS OF A PARTICLE IN A RADIAL POTENTIAL WITH A THICK SINGLE-HUMP BARRIER TRANSMISSION COEF?CIENT FOR A PARTICLE PENETRATING A REAL SINGLE-HUMP POTENTIAL BARRIER 3 50 TRANSMISSION COEF?CIENT FOR A PARTICLE PENETRATING A REAL, SYMMETRIC, SUPERDENSE DOUBLE-HUMP POTENTIAL BARRIER REFERENCES