Time Dependence Up to this point the systems we have dealt with have been time independent The variable t s t d n y f r s d s e sm e ve d s t t e o scr e y d f s s t s h r g f e s y ve n r o sc e m e d e l ID: 81829
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V. Time Dependence Up to this point, the systems we have dealt with have been time independent. The variable t h a s n o t a p p e a r e d i n a n y o f o u r e q u a t i o n s a n d t h u s t h e f o r m a l i sm w e h a ve d e v e l o p e d i s n o t y e t a b l e t o d e scr i b e a n y k i n d o f m o t i o n . T h i s i s a t o d d s w i t h o u r u n d e r s t a n d i n g o f r e a l i t y , w . ¶ t N o t e t h a t i n q u a n t u m m e ch a n i cs t i m e , t , p l a y s a f u n d a m e n t a l l y d i f f e r e n t r o l e t h a n t h e p o si t i o n , q ˆ , o r m o m e n t u m , p ˆ . T h e l a t t e r t w o a r e r e p r e se n t e d b y o p e r a t o r s t h a t a c t o n t h e st a t e s, w h i l e t y ( q ˆ ) i s n o t . I n m o st ca s e s, t h e d e p e n d e n c e o n t i s u n d e r st o o d a n d w e ca n w r i t e t h e s h o r t - h a n d v e r si o n o f S c h r ö d i n g e r ’ s e q u a t i o n : ˆ i y = H y . T i m e d e p e n d e n t q u a n t u m s y s t e m s a r e t h e r a m e w o r k w e ’ v e d e ve l o p e d . A . E n e r g y E i g e n s t a t e s A r e S t a t i o n a r y S t a t e s F i r st , w e w a n t t o a d d r e ss t h e ve r y i m p o r t a n t q u e s t i o n o f h o w e i g e n st a t e s o f t h e H a m i l t o n i a n ( i . e . e n e r g y e i g e n st a t e s) e vo l v e w i t h t i