Artículos de revistas
Brackets de transformación para funciones de oscilador armónico
Autor
BRODY , TOMÁS
Institución
Resumen
THE USEFULNESS OF THE SHELL MODEL IS GREATLY BY THE POSSIBILITY, RECENTLY DEVELOPED BY MOSHINSKY, OF EXPRESSING TWO-NUCLEON MATRIX ELEMENTS DIRECTLY IN TERMS OF TALMI INTEGRALS BY MEANS OF TRANSFORMATION BRACKETS CONNECTING THE TWO-NUCLEON WAVE-FUNCTIONS WITH THE CENTRE-OF-MASS AN RELATIVE COORDINATE WAVE-FUNCTIONS. I- N1,N2; I1, I2 ARE THE RADIAL AND ANGULAR MOMENTUM QUANTUM NUMBERS OF THE TWO PARTICLE, MOVING IN A COMMON POTENTIAL TAKEN AS THAT OF THE HARMONIC OSCILLATOR, AND N, L; N, I ARE THE CORRESPONDING QUANTUM NUMBERS IN THE CENTRE-OF-MASS AND RELATIVE COORDINATE REPRESENTATION, WHILE - IS THE TOTAL ANGULAR MOMENTUM, THE TRANSFORMATION BRACKET IS WRITTEN AS ______________ . FOR N1 = N2 = 0 A CLOSED EXPRESSION IS GIVEN BY (3) AND (4) OF THE TEXT, AND IN TABLE 2 THE NUMERICAL VALUES OF THESE PARTICULAR TRANSFORMATION BRACKETS ARE GIVEN FOR ________________ . THE VALUES ARE GROUPED IN 140 GROUPS ACCORDING TO THE VALUES OF _____ AND __, WICH ARE GIVEN AT THE BEGINNING OF EACH GROUP IN THE FIRST COLUMNS IN THAT ORDER. THE NEXT FOUR COLUMNS GIVE THE VALUE OF __________________ FOR THE GROUP, AND THE FOLLOWING CULUMN SHOWS THE NUMERICAL VALUE OF THE TRANSFORMATION BRACKET _______________ . IN THE LAST COLUMN THE NUMBER __ OF TRANSFORMATION BRACKETS IN THE GROUP IS GIVEN AT THE END. __ IS GIVEN BY THE RELATIONS (20) OF THE TEXT. THE TRANSFORMATION BRACKETS OBEY THE ORTHONORMALITY RELATION (17), SO THAT THE SUN OF THE SQUARES OF THE VALUES IN EACH GROUP SHOULD BE I. THIS HAS BEEN CHECKED FOR ALL 140 GROUPS, AS HAS THE ORTHOGONALITY OF A NUMBER OF PAIRS OF GROUPS WITH EQUAL __ AND __ BUT DIFFERENT __ AND __. DUE TO THE COMULATION OF ROUNDING-OFF ERRORS, THE LAST DECIMAL OF THE NUMERICAL VALUES IN THE TABLES MAY BE IN ERROR BY SEVERAL UNITS. THE ROOT-MEAN-SQUARE DERIVATION FROM THE CORRECT VALUES, OBTAINED FROM THE NORMALITY CHECK, IS 2.9 UNITS IN THE LAST DECIMAL, SO THAT THE SEVENTH DECIMAL MAY BE TAKEN AS CORRECT ALMOST ALL CASES. A LIST IS GIVEN (TABLE 1) OF THOSE GROUPS FOR WICH THE SUN OF THE SQUARES DIFFERS BY MORE THAN 5 X 10- FROM1. NO ROUNDING OFF BY HAND WAS ATTEMPTED. EXPLICIT FORMULAE FOR THE USE OF THESE TRANSFORMATION BRACKETS IN CALCULATING SHELL-MODEL MATRIX ELEMENTS, AS WELL AS SOME SUPPLEMENTARY TABLES, WILL BE FOUND IN ANOTHER PAPER. THE USEFULNESS OF THE SHELL MODEL IS GREATLY BY THE POSSIBILITY, RECENTLY DEVELOPED BY MOSHINSKY, OF EXPRESSING TWO-NUCLEON MATRIX ELEMENTS DIRECTLY IN TERMS OF TALMI INTEGRALS BY MEANS OF TRANSFORMATION BRACKETS CONNECTING THE TWO-NUCLEON WAVE-FUNCTIONS WITH THE CENTRE-OF-MASS AN RELATIVE COORDINATE WAVE-FUNCTIONS. I- N1,N2; I1, I2 ARE THE RADIAL AND ANGULAR MOMENTUM QUANTUM NUMBERS OF THE TWO PARTICLE, MOVING IN A COMMON POTENTIAL TAKEN AS THAT OF THE HARMONIC OSCILLATOR, AND N, L; N, I ARE THE CORRESPONDING QUANTUM NUMBERS IN THE CENTRE-OF-MASS AND RELATIVE COORDINATE REPRESENTATION, WHILE - IS THE TOTAL ANGULAR MOMENTUM, THE TRANSFORMATION BRACKET IS WRITTEN AS ______________ . FOR N1 = N2 = 0 A CLOSED EXPRESSION IS GIVEN BY (3) AND (4) OF THE TEXT, AND IN TABLE 2 THE NUMERICAL VALUES OF THESE PARTICULAR TRANSFORMATION BRACKETS ARE GIVEN FOR ________________ . THE VALUES ARE GROUPED IN 140 GROUPS ACCORDING TO THE VALUES OF _____ AND __, WICH ARE GIVEN AT THE BEGINNING OF EACH GROUP IN THE FIRST COLUMNS IN THAT ORDER. THE NEXT FOUR COLUMNS GIVE THE VALUE OF __________________ FOR THE GROUP, AND THE FOLLOWING CULUMN SHOWS THE NUMERICAL VALUE OF THE TRANSFORMATION BRACKET _______________ . IN THE LAST COLUMN THE NUMBER __ OF TRANSFORMATION BRACKETS IN THE GROUP IS GIVEN AT THE END. __ IS GIVEN BY THE RELATIONS (20) OF THE TEXT. THE TRANSFORMATION BRACKETS OBEY THE ORTHONORMALITY RELATION (17), SO THAT THE SUN OF THE SQUARES OF THE VALUES IN EACH GROUP SHOULD BE I. THIS HAS BEEN CHECKED FOR ALL 140 GROUPS, AS HAS THE ORTHOGONALITY OF A NUMBER OF PAIRS OF GROUPS WITH EQUAL __ AND __ BUT DIFFERENT __ AND __. DUE TO THE COMULATION OF ROUNDING-OFF ERRORS, THE LAST DECIMAL OF THE NUMERICAL VALUES IN THE TABLES MAY BE IN ERROR BY SEVERAL UNITS. THE ROOT-MEAN-SQUARE DERIVATION FROM THE CORRECT VALUES, OBTAINED FROM THE NORMALITY CHECK, IS 2.9 UNITS IN THE LAST DECIMAL, SO THAT THE SEVENTH DECIMAL MAY BE TAKEN AS CORRECT ALMOST ALL CASES. A LIST IS GIVEN (TABLE 1) OF THOSE GROUPS FOR WICH THE SUN OF THE SQUARES DIFFERS BY MORE THAN 5 X 10- FROM1. NO ROUNDING OFF BY HAND WAS ATTEMPTED. EXPLICIT FORMULAE FOR THE USE OF THESE TRANSFORMATION BRACKETS IN CALCULATING SHELL-MODEL MATRIX ELEMENTS, AS WELL AS SOME SUPPLEMENTARY TABLES, WILL BE FOUND IN ANOTHER PAPER.