k m The reason that his is very useful can be seen by examining the Taylor series expansion of some potential V(x) about a minimum point:
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1 roic Oscilltor Pottil W r ow goig to stuy solutios to t TIS for vry usful ottil tt of t roic oscilltor. I clssicl cics tis is quivlt to t block srig robl or tt of t ulu (for sll oscilltios bot of wic r govr by ook s lw: F k t ( t si ( t B cos( t F ( ( k k T rso tt is is vry usful c b s by iig t Tylor sris sio of so ottil ( bout iiu oit: ( ( ( ( ( ( L Now t scl of t ottil is rbitrry so w c subtrct t costt tr ( witout ffctig ytig w ow v: ( ( ( ( ( L but w v clr tt w r kig tis sio rou iiu oit i ( t coitio for wic is tt t first rivtiv of t ottil t tt oit is zro so: ( ( ( L Now if w sty sufficitly clos to t iiu oit t (- is vry sll w c igor igr owrs of it t:
2 ( ( ( iol s log s you sty r t iiu oit i y ottil it c b crctriz s sl roic otio bout t iiu wit srig costt k ( OK giv tt if w c solv t qutu cicl roic oscilltor robl t w v isigt ito rly ll qutu oscilltory otio so log s t litu of oscilltio is sll. so lt s tk t ottil: ( solv t TIS: Now tr r two wys to roc tis robl lytic to lgbric to. I will go troug t lgbric to ot covr t lytic to sic it is tticlly cubrso I o t tik tt it rlly s lot to t iscussio of qutu cics. It os owvr rrst so ic tticl tools o soul go troug it t so ltr ti just to s it o. OK lt s rwrit t TIS bov i littl or suggstiv for by usig t iltoi: [ ( ] look for wys to fctor t lft si of t qutio. O coul orlly writ tt u v ( iu v( iu v iu iu iuv viu vv u iuv ivu v t ctr trs woul ccl. But w v lry s tt ortors o t work tt wy (o t lwys cout i fct if w fctor t iltoi s:
3 ( [ ] ( ( ( ( [ ] [ ] ( ( ( ( [ ] [ ] ( ( ( ( ( [ ] ( ( ( ( [ ] i i i i Now lt s k two w ortors fro t fctors bov: ( ( ( ( s wt t rouct of ts givs: ( ( ( ( ( [ ] ( ( ( [ ] ( ( i i Just out of curiosity wt bout t rouct t otr wy rou?
4 ( ( ( ( ( [ ] ( ( ( [ ] ( ( i i So w c s tt t couttor of t two ortors is: [ ] T iltoi c ow b writt i trs of t ortors - : w c ow rwrit t TIS i trs of ts ortors: ± ± Now so fr w v oly rwritt t TIS i trs of so ortors tt r fctors of t iltoi. ow s tis l us fi t solutios tt w sk? Lt s try sotig (tt I kow will work but I ot sur wy o woul tik of tis witout tt kowlg: Giv tt so is solutio to t TIS wit igvlu lt s s if is solutio lso:
5 ( ( ( ( ( ( w s tt is lso solutio to t TIS wit igvlu ћ. W c follow t s rocur for - w fi it too is solutio wit igvlu ћ. T ortors - r t rfrr to s t risig lowrig ortors i tt ty k so solutio of t roic oscilltor TIS wit igvlu ito otr solutio wit igvlu igr or lowr by ћ. Now t tis oit rbr wy bck t t bgiig of clss wr w iscuss blckboy ritio Plck s ostult tt oscilltig syst c oly v rgis tt r itgrl vlus of ћ. It ss w v riscovr tis coitio but wit Wt s if w ly t lowrig ortor twic? If w strt wit igstts wit igvlu t first lictio givs us igstts wit igvlu ћ t t sco lictio (o tis w stt will giv us otr igstt wit igvlu - ћ. c lictio lowrs t rgy by ћ. But t so oit wo t t rgy go blow t iiu vlu of t ottil (wic r w fi to b zro sic w ro t costt tr i t sio of t ottil so <. Sic w kow tt igstt is ot orlizbl tr ust b so oit wr tis lowrig cis fils -. W c t us tis to fi t lowst igstt :
6 costt l Lt s gt t orliztio costt: so tt Wt is tis stt s igvlu? Plug it ito t TIS us t fct tt - :
7 Wll tt s ot quit wt w ct. It ss tt Plck s yotsis works (s w of cours urstoo for t iffrc btw rgy stts of oscilltig rticl but ot for t rgis of t stts tslvs sic ow c rgy igvlu bov tis grou stt s rgy sic t risig ortor s ћ c ti it is li. ow o w gt t corrsoig rgy igstts? W just ly t risig ortor to our grou stt s y tis s : ( For istc t first cit stt is giv by: ( ow o you gt? Tt s rigt you c just ly t risig ortor oc gi. Lt s k ts look littl clr t I will itrouc t st of fuctios tt ts igstts r scrib by. Mk t substitutio: I us t fct tt wic you c rov to yourslf if you wt. T grl for of t solutios is:
8 ( (! wr t ( r t rit olyoils t first fw of wic r: ( ( ( ( ( ( Notic tt ts r ltrtig v o fuctios.
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