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Exact Propagator for Darnped Time-Dependent Harmonic Osciilator

Kang Cheng .

Published Jan 1, 1983• 0 citations
OpenAlex

Abstract

mu1 t i p l i e d bya f u n c t i o n o f x', z and T . Here we have de f ined -r = (tr't r ) / n f o r l a t e r convenience. He then c a r r i e r s o u t c a l c u l a t i o n s as T + O ( o r n +a). H i s method has r e c e n t l y been a p p l i e d f o r e v a l u a t i n g t h e propag a t o r o f t ime-dependent fo rced harmonic occ i 1 l a t o r 5 . J n the p resen t work the sane method has been genera l i zed f u r t h e r t o c a l c u l a t e t h e propaga to r o f time-dependent harmonic o s c i l l a t o r w i t h cons tan t damping. I n Sec t ion 2, we a r e a b l e t o t rans fo rm o u r p a t h i n t e g r a l i n t o a f u n c t i o n o f x', x and T. t h e Gaussian i n t e g r a l (1.2) mul t i p l i e d by

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mu1 t i p l i e d bya f u n c t i o n o f x', z and T . Here we have de f ined -r = (tr't r ) / n f o r l a t e r convenience. He then c a r r i e r s o u t c a l c u l a t i o n s as T + O ( o r n +a). H i s method has r e c e n t l y been a p p l i e d f o r e v a l u a t i n g t h e propag a t o r o f t ime-dependent fo rced harmonic occ i 1 l a t o r 5 . J n the p resen t work the sane method has been genera l i zed f u r t h e r t o c a l c u l a t e t h e propaga to r o f time-dependent harmonic o s c i l l a t o r w i t h cons tan t damping. I n Sec t ion 2, we a r e a b l e t o t rans fo rm o u r p a t h i n t e g r a l i n t o a f u n c t i o n o f x', x and T. t h e Gaussian i n t e g r a l (1.2) mul t i p l i e d by
In recent years, some interes t has been raised by the problem of the quan tum harmonic-osci l la tor wi th a coupling cons tant K(t), which is a funct ion of t ime (1.2). The impor tance of this p roblem has been stressed by the analysis of the mot ion of a charged part ic le in the presence of a magnet ic field H~(t), which has a fixed direct ion bu t changes i ts in tens i ty wi th t ime (1). The reason of this note is tha t , among these recent studies, the most na tura l approach is surprisingly absent. We will describe i t here, and we will analyse some easily solvable potent ials . W e say t h a t i t is a ve ry na tura l approach, because i t consists in a ve ry na ive general ization of Dirae 's method. I n order to in t roduce it, we remark tha t the Dirae creat ion opera tor (in the t imeindependen t case) could be considered as the opera tor t ha t gives a solution once applied to another solution (for ins tance the vacuum). So we in t roduce an opera tor A ( x , ~ /~x , t) with the p rope r ty tha t , once appl ied to a solut ion of the SehrSdinger equat ion , i t gives another solution. The expl ici t form of this opera tor A is obta ined by means of a simple differential equat ion wi th one and only one solution. W e then find an explici t solution of the t ime-dependen t SchrSdinger equat ion , as a simple general izat ion of the vacuum wave function. App ly ing i tc ra t ive ly the A opera to r we obta in an explici t form for a set of solutions. Noth ing can be said, however , abou t the general case.