HW #3. 1. Spin Matrices. HW3.nb 1. We use the spin operators represented in the bases where S z is diagonal:
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1 HW3.nb HW #3. Spn Matres We use the spn operators represented n the bases where S s dagonal: S = 88, <, 8, <<; S = 88, -I<, 8I, <<; S = S êê MatrForm ÅÅÅ ÅÅÅ S êê MatrForm - Â Â S êê MatrForm ÅÅÅ - ÅÅÅ 88, <, 8, -<<; (a) Obvousl two matres ommute when the are the same: =. Also, t s obvous S D s ant-smmetr n õ S D = -@S, S D. Therefore, t onl reamans to verf S.S - S.S - I S 88, <, 8, << S.S - S.S - I S 88, <, 8, << S.S - S.S - I S 88, <, 8, << (b) We defne n Ø = Hsn osf, sn snf, osl n = Sn@D Cos@fD; n = Sn@D Sn@fD; n = Cos@D Cos@D S n = Smplf@n S + n S + n S D 99 Cos@D, Sn@D HCos@fD - Â Sn@fDL=, 9 Sn@D HCos@fD + Â Sn@fDL, - Cos@D==
2 HW3.nb n D è!!!!! 99- è!!!!!, I-è!!!!! =, 99 ÅÅ + Cos@DM Cs@D ÅÅ, =, 9 ÅÅ Iè!!!!! + Cos@DM Cs@D HCos@fD + Â Sn@fDL HCos@fD + Â Sn@fDL, === PowerEpand@%D 99-, H- + Cos@DL Cs@D H + Cos@DL Cs@D =, 99 ÅÅÅ ÅÅ, =, 9 ÅÅÅ HCos@fD + Â Sn@fDL HCos@fD + Â Sn@fDL, === Smplf@%D 99-, =, 99H-Cos@fD + Â Sn@fDL TanA E, =, 9CotA E HCos@fD - Â Sn@fDL, === Therefore, one an tae the the normaled egenstates to be os sn wth egenvalue + ef and sn e-f os wth egenvalue -. The state wth spn along the nø dreton s the former, and ts probablt to have the postve S when measured s smpl gven b» XS = +» S n = + \» = H, L os ƒ sn = os ef. ƒ () Between Ø n = Hsn osf, sn snf, osland Ø n ' = Hsn' osf', sn' snf', os 'L, the probablt s» XS n' = +» S n = + \» = Hos ' ', sn e-f' L os ƒ sn =» os ' ef os ' + sn e-f' sn ef» = ƒ os ' os TrgEpandA CosA E ' + sn sn CosA E ' + os + SnA E os ' sn SnA E sn oshf - f'l + CosA + CosA E CosA E - CosA E SnA E + E CosA E SnA E SnA E Cos@f - f DE CosA E CosA E Cos@f D Cos@f D SnA E SnA E - CosA E SnA E + SnA E SnA E + CosA E CosA E SnA E SnA E Sn@f D Sn@f D Smplf@%D H + Cos@ D Cos@ D + Cos@f D Cos@f D Sn@ D Sn@ D + Sn@ D Sn@ D Sn@f D Sn@f DL Ths s nothng but I + nø ÿ Ø n 'M = h H + oshl = os, where h s the angle between two vetors, as epeted from the rotatonal nvarane.. Slopp Hdrogen Atom Aordng to the problem,
3 HW3.nb 3 Energ = ÅÅ m - e Z ÅÅÅ + Å d d m d - Z e ÅÅÅ d Solve@D@Energ, dd ã, dd 99d Ø Å e m Z == Smplf@Energ ê. %@@DDD - e4 m Z Ths atuall agrees wth the eat result. (One should be autoned, however, that the agreement wth the eat result s a ondene for ths partular eample.) 3. Classal Unertant Prnple (a) The Mawell's euatons n vauum are gven b Ø ÿ E Ø = Ø ÿ B Ø = Ø µ E Ø + t B Ø = Ø µ B Ø - t E Ø = In ths problem, there are onl and t dependene, and the onl non-vanshng omponents are E and B. Then the Mawell's euatons redue to E + t B = - B - t E = Puttng them together, the redue to a smple one-dmensonal euaton, E - t E =. An funton of the ombnaton t - satsfes ths euaton, namel H - t L f H t - L =. Beause the form of E gven n the problem s a funton of t - onl, t solves the Mawell's euatons automatall. The form an be sethed as
4 HW3.nb 4 PlotASnA p n Jt - - tl êês NE E-H ê. 8t Ø, n Ø, Ø, s Ø <, 8, -3, 3<, PlotRange Ø 8-, <E Ü Graphs Ü It osllates ust le the plane waves, but s loaled. The "unertant" s defned usng the formula analogous to the uantum mehanal wave funton. Frst the "norm," IntegrateAJSnA p n Jt - - tl êês NE E-H N ê. 8t Ø <, 8, -, <E IfAImA n p E == && Arg@sD > - 4 && Re@s D > && Arg@sD < p 4, J - 4 p n ÅÅÅ s - N è!!! p è!!!!!! s, - Å s SnA p n ÅÅÅ E E - norm = SmplfAPowerEpandA p n s ÅÅÅ è!!! p è!!!!!! s EE J p n s ÅÅÅ N è!!! p s We set the overall normalaton E = throughout as t drops out after tang the norm orretl nto aount. Net the epetaton value IntegrateAJSnA p n Jt - - tl êês NE E-H N ê. 8t Ø <, 8, -, <E IfARe@s D >,, - - s OK, ths vanshes. Fnall the varane, SnA ÅÅÅ p n E E IntegrateAJSnA p n Jt - - tl êês NE E-H N ê. 8t Ø <, 8, -, <E IfAImA n p E == && Arg@sD > - 4 && Re@s D > && Arg@sD < p 4, - 4 p n s ÅÅ è!!! p Hs L 3ê J J- + 4 p n s ÅÅ N + 8 p n s N ÅÅ ÅÅ ÅÅÅ Å 4, - - Å s SnA ÅÅÅ p n E E
5 HW3.nb 5-4 p n s ÅÅÅ è!!!! p Hs L 3ê p n s ÅÅ + 8 p n s SmplfAPowerEpandA ÅÅ ÅÅ ÅÅ ÅÅ ì normee 4 s J J- + 4 p n s ÅÅÅ N + 8 p n s N ÅÅ J- + 4 p n s ÅÅ N One an wrte t as HD L = s H - e -g L + ÅÅ g e-g -e, where g = 4 p n s ê. It s espeall smple when g p, when -g HD L = s. (b) The Fourer transform to the freuen doman s gven b IntegrateASnA p n Jt - NE E-H - tl êês 8t, -, <, Assumptons Ø 9ReA s E > =E IfAIm@f - nd == && Im@f + nd ==, 4 p If +n M s -  - ÅÅ J p Hf-nL s - p Hf+nL s Å N "##### ÅÅÅ p ÅÅ ÅÅ, "####### s E I p f t ê. 8 Ø <,  f p t- t s Sn@ p t nd te  - 4 p If +n M s p Hf-nL s - p Hf+nL s "##### p ÅÅÅ SmplfAPowerEpandA- ÅÅ ÅÅ ÅÅ EE "####### s 4 p If +n M s -  - J p Hf-nL s - p Hf+nL s Å N "##### ÅÅÅ p s ÅÅ ÅÅ ÅÅÅ Agan startng wth the norm(droppng the overall f -ndependent fators), IntegrateA 4 p If +n M s - ÅÅ p Hf-nL s IfAReA s E >, - 4 p n s J- + 4 p n s ÅÅÅ N ÅÅÅ è!!! p "#######, s - p Hf+nL s, 8f,, <E p If +n M s Å J p Hf-nL s Å - 4 p n s ÅÅÅ p n s ÅÅÅ norm = SmplfAPowerEpandA ÅÅÅ è!!! p "####### EE s J p n s ÅÅÅ N ÅÅ è!!! p s Net, the average freuen, - p Hf+nL s N fe
6 HW3.nb 6 IntegrateA 4 p If +n M s - ÅÅ p Hf-nL s IfAReA s E >, n ErfA p "########### n s E è!!! p "###########, n s - p Hf+nL s f, 8f,, <E - 8 p If +n M s Å J p Hf-nL s Å SmplfAPowerEpandA n ErfA p "########### n s E Å è!!! p "########## E ì norme n s n Erf@ p n s D ÅÅÅ p n s ÅÅÅ - p Hf+nL s N f fe One an wrte t as n Erf@gê D -E, where g = 4 p n s ê. It s espeall smple when g p, when t redues to nothng but n. -g Fnall the dsperson n the freuen s IntegrateA 4 p If +n M s - ÅÅ p Hf-nL s - p Hf+nL s f, 8f,, <E IfAReA s E >, - 4 p n s J- + 4 p n s p n s ÅÅ p n s N ÅÅ ÅÅ ÅÅÅ 6 p 5ê s "####### s - 8 p If +n M s SmplfA J p Hf-nL s - p Hf+nL s, N f fe - 4 p n s ÅÅ p n s p n s ÅÅ p n s n Erf@ p n s D PowerEpandA ÅÅ ÅÅ Å 6 p 5ê s "####### ì norm - ÅÅÅ EE s p n s ÅÅ J- + 4 p n s ÅÅÅ N J J- + 4 p n s N p n s p n s N p n s ÅÅ 8 J- + 4 p n s ÅÅÅ N p s p n s ErfA ÅÅÅ p n s E ì Namel, HD f L = n HH-e -g L HH-e -g L+ gl- g Erf@g ê DL ÅÅ ÅÅÅ whh smplfes to HD f L = n g H-e -g L HD L HD f L =. 6 p Å g = One nterpreted as a photon, HD f L = HD pl ë h, and hene HD L HD pl =, as epeted. 4 ÅÅ when g p. Therefore, 8 p d
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HW3-oln-mproved.nb HW #3. Spn Matrce We ue the pn operator repreented n the bae where S dagonal: S x = 880,
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