Matrix Mechanics Exercises Using Polarized Light
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1 Matrx Mechancs Exercses Usng Polarzed Lght Frank Roux Egenstates and operators are provded for a seres of matrx mechancs exercses nvolvng polarzed lght. Egenstate for a -polarzed lght: Θ( θ) ( ) smplfy cos sn cos sn cos sn Operator for a -orented polarzer: Θ op ( θ) ( ) ˆ cos cos sn sn Egenstates for vertcally, horzontally and dagonally polarzed lght: V Θ( ) H Θ D Θ float V ( ) H ( ) D ( ) V V H H D D Operators for vertcally, horzontally and dagonally orented polarzers: V op Θ op ( ) H op Θ op D op Θ op Vˆ V V Hˆ H H Dˆ D D float Demonstrate that a polarzed photon s an egenfuncton of a orented polarzer, wth egenvalue. smplfy ˆ
2 Θ op ( θ) Θ( θ) or smplfy Demonstrate that a polarzed photon s a lnear superposton of the vertcal and horzontal polarzaton states. Demonstrate that a vertcally polarzed photon s a lnear superposton of the / and / polarzaton states. cos sn cos sn Calculate the probablty ampltude and probablty that a / (6 degree) polarzed photon wll pass a vertcal polarzer. V Θ.5 V Θ.5 or Θ V op Θ.5 V.5 V.5 ˆ V V V V.5 Calculate the probablty ampltude and probablty that vertcally polarzed photon wll pass a / (6 degree) polarzer. Θ V.5 Θ V.5 or V Θ op V.5 Calculate the probablty ampltude and probablty that a / (6 degree) polarzed photon wll pass a dagonal polarzer. D Θ.966 D Θ.9 or Θ D op Θ.9
3 Calculate the probablty that a / (6 degree) polarzed photon wll pass the followng sequence of polarzers: vertcal, dagonal, horzontal. H H op D op V op Θ.6 H Hˆ DV ˆ ˆ.6 Calculate the probablty that a / (6 degree) polarzed photon wll pass the followng sequence of polarzers: vertcal, horzontal, dagonal. Explan the result. D D op H op V op Θ Wth ths sequence the frst two polarzers are crossed (have a 9 degree relatve angle). hus the vertcally polarzed photon emergng from the frst polarzer s stopped by the second polarzer. Confrm the results shown n the dagram shown below. In other words, show that.5% of the ncdent unpolarzed lght wll pass an arrangement of vertcal, dagonal and horzontal polarzers. H H op D op V op Θ( θ) dθ.5 H HDV ˆ ˆ ˆ d hs calculaton can also be performed assumng that unpolarzed lght s a 5 5 mxture of vertcally and horzontally polarzed lght. H H op D op V op V H H op D op V op H.5 Now a /6 polarzer s placed between the vertcal and dagonal polarzers, and a / polarzer s placed between the dagonal and horzontal polarzer. Calculate the fracton of lght that emerges from the fnal horzontal polarzer and explan the result.
4 H H op Θ op D op Θ op V op Θ( θ) θ d.5 6 hs calculaton can also be performed assumng that unpolarzed lght s a 5 5 mxture of vertcally and horzontally polarzed lght. H H op Θ op D op Θ op 6 V op V H H op Θ op D op Θ op V op H.5 6 he ntal and fnal polarzers are crossed and wll not transmt lght. A sngle p/ polarzer sandwched n between allows lght through for the reasons presented earler. he addton of two more polarzers ncreases the fracton of transmtted lght because the relatve angles between successve polarzers has been reduced. Sgnfcantly more lght gets through each set of polarzers because the angle between them s smaller. Calculate the probablty that unpolarzed lght wll pass the followng sequence of polarzers: vertcal, / (6 degree), dagonal. D D op Θ op V op Θ( θ) θ d.7 or D Θ op V op Θ( θ) θ d.7 Calculate the probablty that unpolarzed lght wll pass the followng sequence of polarzers: vertcal, dagonal, / (6 degree). Explan the dfference n the results. Θ Θ op D op V op Θ( θ) dθ. or Θ D op V op Θ( θ) dθ. he operators representng the measurement of dagonal and 6 degree polarzaton do not commute. Calculate the polarzaton of the ncdent photon such that the probablty t wll pass three polarzers (vertcal, horzontal, dagonal) s.. θ 75deg Gven H H op D op V op Θ( θ) = Fnd( θ) deg he next few exercses nvolve crcularly polarzed lght. he base states for crcularly polarzed lght are: L R
5 Show that they form an orthonormal bass set: L L R R L R R L Show that they are lnear superpostons of the vertcal and horzontal polarzaton states: ( V H) ( V H) Show that vertcally and horzontally polarzed lght can be wrtten as superpostons of crcularly polarzed lght: ( L R) ( R L) he angular momentum operator n atomc unts s: Pang Calculate the expectaton value for angular momentum for a vertcal, horzontal and dagonal polarzed photon. V PangV H PangH D PangD Calculate the expectaton value for angular momentum for a polarzed photon. Θ( θ) PangΘ( θ) Calculate the expectaton value for angular momentum for left and rght crcularly polarzed photons. L PangL R PangR he remanng exercses deal wth the effects of half and quarter wave plates. Half wave plate and rotated half wave plate: W W ( θ) Quarter wave plate and / rotated quarter wave plate (whch has the same effect as a 5 5 beam spltter): W BS
6 Show that, When a half wave plate s placed between algned polarzers all the lght gets through. V W ( ) H V W ( ) V When a half wave plate s placed between crossed polarzers no lght gets through. V W ( ) H When the half wave plate s rotated by an angle of / all the lght gets through. V W H When the half wave plate s rotated by an addtonal angle of / no lght gets through. V W H When a half wave plate rotated by / s placed between two vertcal or horzontal polarzers no lght gets through. V W V H W H here s no effect when a quarter wave plate s nserted between ether algned or crossed polarzers. V W V H W V If the quarter wave plate s rotated by / 5% of the lght gets through. V BSV.5 H BSV.5
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