HW #3. 1. Spin Matrices. HW3-soln-improved.nb 1. We use the spin operators represented in the bases where S z is diagonal:
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1 HW3-oln-mproved.nb HW #3. Spn Matrce We ue the pn operator repreented n the bae where S dagonal: S x = 880, <, 8, 0<<; S y = 880, -I<, 8I, 0<<; S = S x êê MatrxForm 0 ÅÅÅ ÅÅÅ 0 y S y êê MatrxForm 0 - Â Â 0 y S êê MatrxForm ÅÅÅ ÅÅÅ y 88, 0<, 80, -<<; (a) Obvouly two matrce commute when they are the ame: =. Alo, t obvou S D ant-ymmetrc n õ S D = -@S, S D. Therefore, t only reaman to verfy S x.s y - S y.s x - I S 880, 0<, 80, 0<< S y.s - S.S y - I S x 880, 0<, 80, 0<< S.S x - S x.s - I S y 880, 0<, 80, 0<< (b) We defne n Ø = Hn cof, n nf, col n x = Sn@D Co@fD; n y = Sn@D Sn@fD; n = Co@D Co@D S n = Smplfy@n x S x + n y S y + n S D 99 Co@D, Sn@D HCo@fD - Â Sn@fDL=, 9 Sn@D HCo@fD + Â Sn@fDL, - Co@D==
2 HW3-oln-mproved.nb n D è!!!!! 99- è!!!!!, I-è!!!!! =, 99 ÅÅ + Co@DM Cc@D ÅÅ, =, 9 ÅÅ Iè!!!!! + Co@DM Cc@D Å HCo@fD + Â Sn@fDL HCo@fD + Â Sn@fDL, === PowerExpand@%D 99-, H- + Co@DL Cc@D H + Co@DL Cc@D =, 99 ÅÅÅ ÅÅ, =, 9 ÅÅÅ Å HCo@fD + Â Sn@fDL HCo@fD + Â Sn@fDL, === Smplfy@%D 99-, =, 99H-Co@fD + Â Sn@fDL TanA E, =, 9CotA E HCo@fD - Â Sn@fDL, === Therefore, one can tae the the normaled egentate to be co y n wth egenvalue + ef and -n e-f y co wth egenvalue -. The tate wth pn along the nø drecton the former, and t probablty to have the potve S when meaured mply gven by» XS = +» S n = + \» = H, 0L co y ƒ n = co ef. ƒ (c) Between Ø n = Hn cof, n nf, coland Ø n ' = Hn' cof', n' nf', co 'L, the probablty» XS n' = +» S n = + \» = Hco ' ' Å, n e-f' L co y ƒ n =» co ' ef co ' + n Å e-f' n ef» = ƒ co Å ' co TrgExpandA CoA E ' + n Å n CoA E ' + co + SnA E co ' n Å SnA E n cohf - f'l + CoA + CoA E CoA E - CoA E SnA E + E CoA E SnA E SnA E Co@f - f DE CoA E CoA E Co@f D Co@f D SnA E SnA E - CoA E SnA E + SnA E SnA E + CoA E CoA E SnA E SnA E Sn@f D Sn@f D Smplfy@%D H + Co@ D Co@ D + Co@f D Co@f D Sn@ D Sn@ D + Sn@ D Sn@ D Sn@f D Sn@f DL Th nothng but I + nø ÿ Ø n 'M = h H + cohl = co, where h the angle between two vector, a expected from the rotatonal nvarance.. Sloppy Hydrogen Atom Accordng to the problem,
3 HW3-oln-mproved.nb 3 Energy = ÅÅ m - e Z ÅÅÅ + Å d d m y d - Z e ÅÅÅ d Solve@D@Energy, dd ã 0, dd 99d Ø Å e m Z == Smplfy@Energy ê. %@@DDD - e4 m Z Th actually agree wth the exact reult. (One hould be cautoned, however, that the agreement wth the exact reult a concdence for th partcular example.) 3. Clacal Uncertanty Prncple (a) The Maxwell' euaton n vacuum are gven by Ø ÿ E Ø = 0 Ø ÿ B Ø = 0 Ø µ E Ø + t B Ø = 0 c Ø µ B Ø - t E Ø = 0 In th problem, there are only x and t dependence, and the only non-vanhng component are E y and B. Then the Maxwell' euaton reduce to x E y + t B = 0 -c x B - t E y = 0 Puttng them together, they reduce to a mple one-dmenonal euaton, c x E y - t E y = 0. Any functon of the combnaton c t - x atfe th euaton, namely Hc x - t L f Hc t - xl = 0. Becaue the form of E y gven n the problem a functon of c t - x only, t olve the Maxwell' euaton automatcally. The form can be etched a To mplfy the problem, defne g = 4 p n ê c. Then the electrc feld : Ey@x_, t_d = E0 * SnA c è!!! è!!! g g Å t - xe E -Hx-c tl^êh L E0 SnA c t è!!! g - x è!!! g E H-c t+xl - Å
4 HW3-oln-mproved.nb 4 Plot@Ey@x, td ê. 8t Ø 0, g Ø 400 p, E0 Ø, Ø 0<, 8x, -30, 30<, PlotRange Ø 8-, <D Ü Graphc Ü It ocllate ut le the plane wave, but localed. The "uncertanty" defned ung the formula analogou to the uantum mechancal wave functon. Frt the "norm," norm = Integrate@Ey@x, 0D^, 8x, -, <, Aumpton Ø 8g > 0, > 0<D General::pell : Poble pellng error: new ymbol name "norm" mlar to extng ymbol "Norm". More -g H- + g L E0 è!!! p We could et the overall normalaton E 0 = throughout nce t wll drop out after tang the norm correctly nto account. But we can alo leave t n a a chec that everythng worng correctly. Next the expectaton value Integrate@x * Ey@x, 0D^, 8x, -, <, Aumpton Ø 8g > 0, > 0<D 0 OK, th vanhe. Fnally the varance, temp = Integrate@x^ * Ey@x, 0D^, 8x, -, <, Aumpton Ø 8g > 0, > 0<D 4 -g E0 è!!! p H- + g + gl 3 varance = temp ê norm General::pell : Poble pellng error: new ymbol name "varance " mlar to extng ymbol "Varance ". More H- + g + gl ÅÅ H- + g L Note that the E0 dependence dd n fact drop out. FullSmplfy@varanceD J + g - + N g
5 HW3-oln-mproved.nb 5 One can wrte t a HD xl = I + Å g g - M, where g = 4 p n ê c. It epecally mple when g p, when HD xl =. (b) The Fourer tranform to the freuency doman a a functon of the varable "f" calculated below. We mght a well pc a pecfc poton le "x=0" to evaluate the Fourer tranform. fttemp = Integrate@Ey@0, td * E I p f t, 8t, -, <, Aumpton Ø 8g > 0, > 0, c > 0<D Integrate ::gener : Unable to chec convergence. More Ic è!!!! g + f p M E0 IfAf œ Real,  - Å c J- + 4 f p è!!!! g Å c N "##### ÅÅÅ p ÅÅ ÅÅÅ, IntegrateA c  f p t- c t Å SnA c t è!!! g E, 8t, -, <, Aumpton Ø c > 0 && g > 0 && > 0 && f RealEE ft@f_d = fttemp@@dd * fttemp@@, DD è!!!! g + f p M  - Ic Å c J- + 4 f p è!!!! c g Å N E0 "##### ÅÅÅ p ÅÅ Å c Agan tartng wth the norm (notng that ft@ f D * ft@ f D = -ft@ f D^ ): norm = Integrate@-ft@fD^, 8f, 0, <, Aumpton Ø 8g > 0, > 0, c > 0<D -g H- + g L E0 è!!! p ÅÅÅ Å 4 c Integrate@-ft@fD^, 8f, -, <, Aumpton Ø 8g > 0, > 0, c > 0<D -g H- + g L E0 è!!! p ÅÅÅ Å c Next, the average freuency, exptf = Integrate@f * -ft@fd^, 8f, 0, <, Aumpton Ø 8g > 0, > 0, c > 0<D ê norm c g è!!! g Erf@ è!!! g D ÅÅ H- + g L p exptf ê. Ø c è!!!! g ë H p nl g n Erf@ è!!! g D - + g LmtAErfA è!!! g E, g Ø E One can wrte t a n Erf@gê D, where g = 4 p n ê c. It epecally mple when g p, when t reduce to nothng but n. -E -g Fnally the varance n the freuency (remember to normale!)
6 HW3-oln-mproved.nb 6 exptf = Integrate@f * -ft@fd^, 8f, 0, <, Aumpton Ø 8g > 0, > 0, c > 0<D ê norm c H- + g H + gll ÅÅÅ 8 H- + g L p FullSmplfy@%D c H- + g H + gll ÅÅÅ 8 H- + g L p % ê. Ø c è!!!! g ë H p nl êê Smplfy H- + g H + gll n ÅÅÅ H- + g L g Lmt@%, g Ø D n Then the uare of the dperon n the freuency : dperon = exptf - exptf^ êê Smplfy - c I-H- + g L H- + g H + gll + g g Erf@ è!!! g D M ÅÅ ÅÅ ÅÅÅ ÅÅ 8 H- + g L p dperon ê. Ø c è!!!! g ë H p nl êê Smplfy - n I-H- + g L H- + g H + gll + g g Erf@ è!!! g D M ÅÅ ÅÅ ÅÅÅ ÅÅ H- + g L g Lmt@dperon, g Ø D % ê. Ø c è!!!! g ë H p nl c 8 p n ÅÅ g Namely, HD f L = n IH-e -g L HH-e -g L+ gl- g Erf@g ê D M ÅÅ whch mplfe to g H-e -g L HD f L = n ÅÅ g = ÅÅÅ c when g p. Therefore, HD xl HD f L = Å c. 8 p 6 p Once nterpreted a a photon, HD f L = c HD pl ë h, and hence HD xl HD pl = 4, a expected.
HW #3. 1. Spin Matrices. HW3.nb 1. We use the spin operators represented in the bases where S z is diagonal:
HW3.nb HW #3. Spn Matres We use the spn operators represented n the bases where S s dagonal: S = 88,
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