Quantum mechanical c a l c u l a t i o n s , based on t h e Lennard-Jones (12,6) p o t e n t i a l , a r e presented showing t h e dependence of t h e reduced phase s h i f t , 7* , on t h e quantum parameter, h * , f o r a f i x e d reduced e f f e c t i v e p o t e n t i a l and v a r i o u s reduced e n e r g i e s , E". The
observed o s c i l l a t o r y behavior of ?*[.') i s due p r i m a r i l y t o t h e i n c l u s i o n of t h e p h y s i c a l l y unimportant c o n t r i b u t i o n of wv t o t h e phase s h i f t , where i s t h e number of quasi-bound ( v i r t u a l ) s t a t e sof energy l e s s than E*. A modified reduced phase s h i f t , ?* 9
d e f i n e d by e x c l u d i n g t h i s c o n t r i b u t i o n , d i s p l a y s only t h e s h a r p i n f l e c t i o n s a s s o c i a t e d w i t h b a r r i e r p e n e t r a t i o n under resonance c o n d i t i o n s . Except f o r t h e resonance c o n t r i b u t i o n , the phase s h i f t s may be a c c u r a t e l y reproduced by a second-order JWKB procedure. T h i s method a l s o a c c u r a t e l y p r e d i c t s t h e r e s o n a n t e n e r g i e s ( i . e . t h e e n e r g i e s of t h e quasi-bound s t a t e s ) . The f i r s t -o r d e r JWKB treatmentof t h e b a r r i e r p e n e t r a t i o n problem by Ford, H i l l , Wakano, and Wheeler s u f f i c e s f o r t h e purpose of e s t i m a t i n g t h e l e v e l widths and l i f e t i m e s of t h e v i r t u a l s t a t e s a s w e l l a s t h e main f e a t u r e s of t h e r e s o n a n t phase s h i f t s , b u t does n o t a c c u r a t e l y reproduce t h e q u a n t a l c a l c u l a t i o n s .W T h i s r e s e a r c h r e c e i v e d f i n a n c i a l support from t h e N a t i o n a l Aeronautics fi t t i h f r-----e and Space A d m i n i s t r a t i o n Grant . W P r e s e n t a d d r e s s : N a t i o n a l Bureau of Standard https://ntrs.nasa.gov/search.jsp?R=19670008526 2018-05-09T17:30:18+00:00Z I n t r o d u c t i o n I n t h e p r e s e n t paper, we p r e s e n t t h e r e s u l t s of a numerical s t u d y of phase s h i f t s which a r i s e i n c o l l i s i o n s between molecules which i n t e r a c t according t o a Lennard-Jones (12,6) p o t e n t i a l . W e c o n s i d e r , i n p a r t i c u l a r , t h e behavior of t h e phase s h i f t s under c o n d i t i o n s such t h a t t h r e e c l a s s i c a l t u r n i n g p o i n t s e x i s t . Although t h e q u a n t i t a t i v e r e s u l t s apply t o t h e p r e s e n t model p o t e n t i a l , most of t h e q u a l i t a t i v e f e a t u r e s a r e c h a r a c t e r i s t i c of any r e a l i s t i c i n t e r m o l e c u l a r p o t e n t i a 1.The L. -J. (12,6) p o t e n t i a l f u n c t i o n v m a y be w r i t t e n where c p Y ( v * ) = 4['" -1 2 -p -6 3 i s t h e depth of t h e p o t e n t i a l minimum, Ui s t h e s e p a r a t i o n a t which t h e p o t e n t i a l i s zero, and s e p a r a t i o n . The phase s h i f t s , *lp , are defined by t h e asymptotic form o f t...