/9/4 FI 33 Quanum Physcs Aleander A. Iskandar Physcs of Magnesm and Phooncs Research Grou Insu Teknolog Bandung Basc Conces n Quanum Physcs Probably and Eecaon Value Hesenberg Uncerany Prncle Wave Funcon n Momenum Sace Aleander A. Iskandar Basc Conces n Quanum Physcs
Frequency /9/4 Revew on Probably Recall he conce of robably densy funcon. Consder he followng eamle A manufacurer of nsulaon randomly selecs wner days and records he daly hgh emeraure 4 3 7 4 37 6 46 8 3 3 3 38 4 43 44 7 3 7 Pu n a class able Probably Densy Funcon Class T Freq. Rel. Freq. f < T 3. < T 3 6.3 3 < T 4 3. 4 < T 4 4. < T 6. =. Hsogram: Hghes Temeraure 3 4 More Aleander A. Iskandar Basc Conces n Quanum Physcs 3 Revew on Probably Consder he followng eamle A manufacurer of nsulaon randomly selecs wner days and records he daly hgh emeraure 4 3 7 4 37 6 46 8 3 3 3 38 4 43 44 7 3 7 The average emeraure can be calculaed usng he robably densy funcon as hus average T md. value frequency number of daa md. value rob. ds. func. T f T P f P Aleander A. Iskandar Basc Conces n Quanum Physcs 4
/9/4 Revew on Probably One oher moran sascal quany s he sandard devaon or T T T T T T T T P T T T T T T f P T P T P T T Aleander A. Iskandar Basc Conces n Quanum Physcs Probably Inerreaon of Wave Func. As saed by Born he modulus of he wave funcon of a arcle s nerreed as he robably densy funcon assocaed wh he arcle P d d Then he wave funcon has o sasfy he followng P d d However one can always normalze he wave funcon by mullyng wh a consan. Hence he condon needed o be sasfed by he wave funcon s ha he nal sae of he wave funcon mus be a square negrable funcon d Aleander A. Iskandar Basc Conces n Quanum Physcs 6 3
/9/4 4 Probably Inerreaon of Wave Func. Eamle : ormalze he wave funcon To normalze he wave funcon we mully wh a number and mose he normalzaon condon Thus he normalzed wave funcon s Aleander A. Iskandar Basc Conces n Quanum Physcs 7 elsewhere 3 4 3 3 4 d d d 3 ~ Eecaon Value from Probably In analogy wh robably conce he eecaon value of he arcle s oson s Or n general eecaon value of any funcon f should be calculaed as And uncerany of he arcle s oson measuremen s Aleander A. Iskandar Basc Conces n Quanum Physcs 8 d d d P d f f d d
/9/4 Eecaon Value from Probably Eamle : For he normalze he wave funcon Calculae and hence. Aleander A. Iskandar Basc Conces n Quanum Physcs 9 elsewhere 3 6 4 3 3 3 4 3 3 d d d Eecaon Value from Probably Eamle : For he normalze he wave funcon Calculae and hence. Aleander A. Iskandar Basc Conces n Quanum Physcs elsewhere 3 7 7 6 3 3 3 6 4 4 d d d 7 4 7
/9/4 6 Conservaon of Probably If we normalze he wave funcon a one me wll say normalzed? I.e. does hold for all me? In shor s robably conserved? Take a me dervave of he robably densy funcon Use he Schrodnger equaon o relace he me dervave and assume ha he oenal funcon V s real. Recall he comle conjugae of he Schrodnger equaon Aleander A. Iskandar Basc Conces n Quanum Physcs d P V m Conservaon of Probably Then Defne he robably flu or robably curren as Hence we ge Aleander A. Iskandar Basc Conces n Quanum Physcs m m P m j j P
/9/4 7 Conservaon of Probably If we negrae over all sace we ge The las se follows from he fac ha for square negrable funcon j vanshes a. Aleander A. Iskandar Basc Conces n Quanum Physcs 3 d j d P d P d Probably Curren The las relaon s smlar o he connuy equaon found n classcal mechancs or elecromagnesm whch saes ha robably s conserved no only globally bu also locally. I means ha f he robably of fndng arcle a a ceran on decreases hs robably does no only urns u a anoher on bu nsead flows o hs oher regon. Hence he name robably curren for Aleander A. Iskandar Basc Conces n Quanum Physcs 4 j P m j
/9/4 8 Eecaon Value of Momenum How do we calculae he eecaon value of he momenum of he arcle assocaed wh he wave funcon? Recall he classcal eresson for momenum Take he eecaon value of hs eresson yelds oe ha he oson does no have a me deendence. The me deendence of comes from he me deendence of. Aleander A. Iskandar Basc Conces n Quanum Physcs d d m mv d m d d d m d d m Eecaon Value of Momenum Use he Schrodnger equaon o oban oe ha Aleander A. Iskandar Basc Conces n Quanum Physcs 6 d d m
/9/4 9 Eecaon Value of Momenum Thus Snce he wave funcon s square negrable funcon means ha vanshes a hence he frs erm does no conrbue o he evaluaon. Thus we have whch sugges o assocae he momenum o he oeraor Aleander A. Iskandar Basc Conces n Quanum Physcs 7 d d d d Eecaon Value of Momenum Eamle : For he normalze he wave funcon Calculae and hence. Aleander A. Iskandar Basc Conces n Quanum Physcs 8 elsewhere 3 3 3 3 3 d d d
/9/4 Eecaon Value of Momenum Eamle : For he normalze he wave funcon Calculae and hence. Aleander A. Iskandar Basc Conces n Quanum Physcs 9 elsewhere 3 3 3 d d d Eecaon Value of Momenum Eamle : For he normalze he wave funcon Calculae and hence. Aleander A. Iskandar Basc Conces n Quanum Physcs elsewhere 3 7 Recall he revous resuls of hen Whch s conssen wh..6 7
/9/4 Eecaon Value of Momenum The eecaon value of oson s a real value as can be easly seen from he defnon However because of he form of momenum oeraor ha nvolves he magnary number as well as a dfferenaon wll he eecaon value of he momenum be a real value? In fac can be roved ha he momenum eecaon value s always a real number. Aleander A. Iskandar Basc Conces n Quanum Physcs d d P d Eecaon Value of Momenum In he las se he square negrably roery of he wave funcon has been used where saes he he wave funcon s a localzed funcon so ha as. Aleander A. Iskandar Basc Conces n Quanum Physcs d d d d d
/9/4 Hesenberg Uncerany Prncle The resul of he las eamle on roduc of he eecaon values of and was found ha Ths relaon s called he Hesenberg Uncerany rncle. I saes one of he fundamenal conce of quanum world ha s measuremen of oson and lnear momenun canno be done smulaneously wh he hghes accuracy. If measuremen of oson s done very accuraely = hen he value of he lnear momenum s no known snce accordng o Hesenberg uncerany rncle yelds and vce versa. Poson and momenum are sad o comlemenary varables. Aleander A. Iskandar Basc Conces n Quanum Physcs 3 Hesenberg Uncerany Prncle One observaon can be used o see hs rncle. From wave ocs he sread of he dffracon aern as a Consderng he lgh as hoons a he sl he oson of he hoon s whn he measuremen s uncerany of a bu he momenum s uncerany has sread. Thus a where k. a h Aleander A. Iskandar Basc Conces n Quanum Physcs 4
/9/4 Hesenberg Uncerany Prncle oe ha he Hesenberg uncerany rncle beween he oson and momenum holds for all drecon y and z. One oher Hesenberg uncerany relaon s he uncerany beween energy and me eced sae E E E eced sae ground sae E ground sae Aleander A. Iskandar Basc Conces n Quanum Physcs Wave Funcon n Momenum Sace Recall ha he secral dsrbuon funcon of he wave acke s none oher han he Inverse Fourer ransform of he wave funcon a = Calculae he followng e d d d e e d d d d Aleander A. Iskandar Basc Conces n Quanum Physcs 6 3
/9/4 Wave Funcon n Momenum Sace d d Fourer ransform of a normalzed wave funcon s normalzed. e consder he followng d d d e d d e d d d Whch s a saemen ha he momenum eecaon value can also be calculaed from usng he momenum oeraor self. Aleander A. Iskandar Basc Conces n Quanum Physcs 7 Wave Funcon n Momenum Sace Ths las saemen s smlar o Thus f s he wave funcon n saal doman should be nerreed as he wave funcon n momenum sace wh he robably densy funcon of fndng a arcle wh momenum s gven by. In hs momenum sace he oson oeraor s gven by Hence he eecaon value of oson s d Aleander A. Iskandar Basc Conces n Quanum Physcs 8 d d 4
/9/4 Wave Funcon n Momenum Sace Eamle.4 Consder a arcle whose normalzed wave funcon s e a. For wha value of does P = eak? b. Calculae and. c. Wha s he robably ha he arcle s found beween = and = /? d. Calculae and use hs o calculae and. elsewhere / Aleander A. Iskandar Basc Conces n Quanum Physcs 9 Summary Physcal quany an observable s reresened by an oeraor. Measuremen of observable s evaluaed as calculang eecaon value There are wo ways o calculae he eecaon value n saal sace or n momenum sace. hys. quany O ˆ d Aleander A. Iskandar Basc Conces n Quanum Physcs 3
/9/4 Summary Saal sace Momenum sace Wave funcon Poson Momenum E e e Aleander A. Iskandar Basc Conces n Quanum Physcs 3 d d 6