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Quantum physics - a text for graduate students r newton

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Đây là bộ sách tiếng anh về chuyên ngành vật lý gồm các lý thuyết căn bản và lý liên quan đến công nghệ nano ,công nghệ vật liệu ,công nghệ vi điện tử,vật lý bán dẫn. Bộ sách này thích hợp cho những ai đam mê theo đuổi ngành vật lý và muốn tìm hiểu thế giới vũ trụ và hoạt độn ra sao.

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Quantum Physics

A Text for Graduate Students

With 27 Figures

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Bloomington, IN 47405 USA newton @i edu Series Editors R Stephen Berry

Department of Chemistry Joseph L Birman Mar! ‘k P Silverman Department of Physics Department of Physics University of Chicago City College of CUNY Trinity College Chicago, IL 60637 New York, NY 10031 Hartford, CT 06106 USA USA USA H Eugene Stanley Mikhail Voloshin Center for Polymer Studies Theoretical Physics Institute Physics Depart ent Tate Laboratory of es Boston University University of Minnes Boston OMA 2215 Minneapolis, MN 55455 USA USA Library of Congress Cataloging-in-Publication Data Newton, Roger G

Quantum physics : xt for graduate students / Roger G Newton p em — ‘Graduate texte in m contemporary Physics) Includes bibliographical ref ISBN 0-387-95473-2 ( paper) 1 Quantum tl 1 IIL Seri QC174 12 : 2 30.12—dc21 2002020945 ISBN 0-387-95 Printed on acid-free paper © 2002 Springer-Verlag ray Ine All rights reserved This w not be translated or copied in whole or in part without the written pet no: ttisher (Springer Verlag New York, Inc., Avenue, New York, NY 10010, USA), except for brief excerpts in connection with re SOL larly analysis

names, trademarks, service marks, and similar terms, even if theo oe are not Mennfed as gd is rot te be taken as an expression of opinion as to whether or

subject to propriet Printed in the United States of America

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Preface I Physics 1 Quantum Basics: Statics 11 1D ical \ 1 l2 Measurements 0.2.2.0 000000000 ae 11 Fields as dynamical variables Đrobabil ties 1.2.1 Correlatio 2.2 Interference xpectation values and variance ¬ Re t5 to 1.3 i ed State 1.4 œ as œ œ 5 š wo š

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Symmetries 143 5.1 The Angular- Momentum Operator ee 143 5.1.1 144 I 2 Anoscillatormodel .0 - 148 1.3 States of sei landspinl/2 150 5.2 The Rotation Group 2 ee 151 1 ier space 155 522 Polari 1 th lensi i 160 5.2.3 The magnetic moment 161 5.2.4 Addi 163 5.2.5 Seo tensors and selectionrules 167 5.3 Time Reversal 172 28 he time-reversal operator . 172 a, in 6.4

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1 Perturbation Theory .0.00.02 0000005 235 8.1.1 Application to ae decay 240

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D.1.2 Associated Legendre functions D 1á D.2 ical D.3 Hermite Polynomial D4 TI Ta D.5 ¬ Polynomials D.6 Problems and Exercises D.1.3 pphewieal harmon Zon: al har: cs E Group Representations E.1 Group: 2 epresentation .2.1 Characte E.2.2 Real vepresentati E.2.3 a Kronecker products Carrier space: 2.4 2.5 Clebsch-Gordan coefficients ONS 26 we E

E.3 Lie Groups E.3 3.1 Coordinate transformations E.3.2 Infinitesimal generators

The Lie algebra E.3.3 B4 R : fT

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Quantum Basics: Statics

1.1 States and Dynamical Variables The aim of vl that of classical vl is to give an it lorstand 1 be set thein bel Therefore the f ks (thoùl lv hị " he đ 1 lofine tl

1 can be determined T! lependent, because a wrong choic in the first will 1 1 from | lized Thus the most basic WI in classical pl £ defined as precisely nature allow 1 1 1 in (ie Hilt Just as the precise form 1 ale of t} th the number of it b fịt degrees of freedom, tl 1 ical variables, etc.—so does $) 1 1+h từ ] ud † kh late f, tem,

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The TW] 1 Tine +] lst T, £ Ts L tự, ] Thị tet of the evetem ¥ hold 1] 1 : 1 1 1 1] Lert] Tine af4] NI d We } † fs taps, bh iti 1 14] h lữ {ữ I 1 the state ©’ w+ nerally differs from tữ Dp 1 } 1 NI £ +} 4, Tod + d † ]

ds, i Jer to d Ì tat “as precisely nature allows” it | I ified j in i i

hat ial rel 1 Ll otk Am ibl “completely specie 1 tom Ina f : the vector Ww “has hair,” while the ray spanned by W “is shaved.” It is the da 3 let lett 1 omtanal, L their ° er te forms V Từ 1 an 4] 1: hb al Tế only slit #1 is ] ] ur only 2 it is in the state Wo Botl Ì † 1 Ï d 1h NI d dlr ] + 11 owever if both slits are open, tl ] Wy + coWe, with d pat the t toa Nữ dữ ữ taneled A 11 1 Jee pi} 1 £ 1 Pp £ } 1 1

] ] | There is no ana- 1 1 for classical 1 but for wave ystems such as light, i is inf ined i he fall deacrintt £1iebt T £ the e] wo field as nec

lo for the deacrints r hi ll bed bel mm ] 1 4 both “particles” and not be pushed too far.) Tand TT wl TT] T and and II trai Lined j he Ul F+] Lined ay : 1 1 : (see Am Appendix B) § = §'@.S", and its tờ] — I 1 II ] £ — glo which +] 1 Hl Fl the form than = #' & le the Wi} £ 1 1

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formed out of li binati ÿƑ "- Ì @ ÿH of ind 1 icl the fin t one in state vw _and the second i in Tin WH Jeti 1

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2 After 1 tl laf 1 h tl A £ ystem originally in A } p 1 CA at e " Th Ned +4 Ct P postulate) Te +} + tat A with th A the + 1 ifi f tl together with the third 1i the same result, el 1 if if tl laf lq 1 + follows f ] ] 1 "hãm } 1 ĐỊ in identi- 1 the value A } 1 T

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bị ble A li f tl (the spectrum fo TH ti ] li ] ] lnel), and (2), (3) A Te of fA wit] 1 AA 1 fA } Taft dc hich A has th ted val L 4 ; > É to make it so a: 1 f H r Tat (see Appendix B), a stat „ : Pa £4] ] 1 foment lead 1 loa A and B unless t] 1: A and RB Tr ond “4 ¢ 1 led F 1 Iolo cof of , ĐỊ 11, +1 1 f kL f£ hich beled buy +t 1 1 fone, +] ] bled 4] } of the system, which fail t th th , are then not sharply defined

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AY), (1.3) which implies }>, |c4|? = 1 and |c4|? is the probability P(A|) of obtain- i t A fAi I WF so that appropri- 1 h ately 3>, P(A|¥) = 1 Tl tate of " | | i f i d the coeffi- ents ca are the * ‘probability amplitude (the squares oe whose > magni- tudes are the Ị upon m of A: P(A|Y f A the sum in (1.2) I Fourier series by a Fourier an © | = Ít 2)? If th he 11 t analogous to replacing T= " dAc(A)fA, 49 | i,j es, 4 Ay A Here, however, U4 ne 5 since it does

ave a finite nom, it is not normalizable [TI analogou to exp(ikx), witl 1 Fourier i 1 1] These that (arte) ~ =d(A— 2 củ 5) We call cll them “d-normalized” and refer to a 1 set of vectors: that atsy imply that

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respectively, is given be I 2 I II P(A, BIW) = [Uh © UB DP? = |S eam (Uy, VL) UE, Y ny nm Tharaf, 1 hahilty of Gndi Tena] Ul iprespective of n

] ] is ] b m | h, lanm|2 If the tw I h lữ + Pl ot hi 1 only if is a product

baem, in which instance WV is of the form I (1.6) Tr +} 1 babil cod 1 @liaind is independen 1 of the state of system IT 7, ] F tho £ th wl @ wll oe 4] 1 nm n m 1 he se (1.7) IS can be assumed to be nor- def where Om Pm = Yo, Gum Uh; the states bi,

lized | il ll 1 Eq.(1.7) clearly 1 1 de houch t} 1 ' i Fad TT vị 1 T I } £, J} m be T yl m É le | tat A 14] Ine 4G II h A and the state: ss may be eigenstate: €B with ¢t lue B Wh tin tỊ i TỊ 1 1 to † cA B However tof A TT all Por yy]

lt of fB I would have been, had it been performed † CPR with :

r the latter, and theref listurbing i Fsuch a sì : 14 E EPR EPRI L T } } † Tì 1 Ra} 5 the Geéannren €x- i} lai 1 periment goes as follows A icle of spi i in-1

ich fly off j trẻ đã TẾ 1 1 are far apart, tl f fr t A the

ha £ 1 £ ] ] lay] £ B But

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lated | her Therefore if tl tical Bf 1 be down On the other hand “€ the horizontal epi fA had | lạng found be Leff a1 et 1 “al hine it that he | ] f B is right (Since, by EPR’s definition 1 1 ctions are “elements of reality” but quantum be simal «oy and | 1 1 en 1 1 B imply that both spin proje Lanine d 1] 1 ¢ + 1 le 4] lite 1 “com- " F xesliEs 1 1 see sy 1 3m 1 dat £ an] AT) whi] 1 le 4] £ ena} 1 1 11 mà 1 lol 1 Fl 1 hanlas d yo Pp «naducihle dieturl 4] + of ta is sometimes claimed N ] if | (1.7) + C4 : le ot 1 of the other, although tl I ion (at tÌ tt 1 1 £44 1" 1 1 diet hat: + M1 + f | 1 hal fl £ tre THỊ £ lation 1.2.2 Interference The abet Leeteal } £4] £ in = (+ 2) /⁄2, is that ïf tÌ is in tl Ữ, it can be £ 1 1 tna hath of +} Nữ đ of the state P2 As- that + and w botl lized and ll 1 L th Lahilite of findine +] 1 ha cl «tice af ait] ia 1 whic 14] ual however for the jj Ter ind ‘| state W/ = (cP, + Wy)/V2 with any real y £0 ° 1 tin lessenbed bự + lived

© 1 ki fi Ì babili Ì upon measurement it will be found i ith tl b= (Uy + W)/V/2 This probability is

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t ] t ily equal he nho P(|® ing it i ¡th fị and Pel) of finding it in agreement with Mộ even if the two states Va and Wo are mutually *(iÿ„.®)l L ME J) ct hat tech 1 ro} 1 L 1 1 Th 1d wl £ Since 1 fl 1 1 1 if Ð(W,„.@Ẳ®) - P(ữ.,@®) — , numbers 8) and ( ®) = th P(#,#) ip (8i 4 1 1 1; 1 Tag P, 1 and (V2, ®) Ww 1 Ha £ heel Ladd 1 bại £ £ | 1 1 that for £ I t led; tk i lent As already noted earlier lie ld 1 1 pectral decomposi ition Ì Je of tl ] f † tral component Ì ffi ify tl wave uniquel } 1 ] fe f particle } fi } 1+1 1 £ 18) 1 1 Trosb°l Fe] 1 lability +}

1 he off, Co kind of lat 1 £ | 1 I 1 ] 1 hat Labi i Tự) TT uch as wave system 1a] ee sy leah] 1 1 Thal 1£ 1 f+} n between het: 1 1 14] £ 1 Đ K † Œ ae tran] d by th aoe + Toealzz in} + + 1 Qo} lncal 1 f J I ] les, usually regarded 1 1] Lle £ physically unacceptable; for however leed | There is, therefore, no reason

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Pp by definition, given by def A) HỆ AP(AIN), A faahich | HÌ 1 11 ff] 1/19 with the result (UAB) (JAI) (A) = +2 a || © |? | (1.9) The mean value, of course, depends on the state V, but this dependence i il indi 1 in tl A\ The variance, di def = V((A— (A))?) = v(A?) — (A)? (1.10) To see this, a (A*) = (4, ay = = A? = (A), and hence AA = A Schwarz’ s taauHợ that if AA = Os % must be an eigenstate of A: assuming that AW

eigenstate of A, and that || © ||= 1, we have o that © is not an

(A)? =|(E, AW)? <| |P|LA# |Š= (A#,A) = (W, A®8) = (A?),

L

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then hav Schwarz’s inequality |(#,[A, BỊ#)| |(#, -(,BA'Đ) < JW.AB h +\(U,B/AN)| < 2) AG || BV |= 20448; SO 1 AAAB> 3 (A,BỊ)| (119) tant (that ltipl ftl as it is in the special for each component hall j fi 1.3 (de, Pi] = thea k,i=1,2,3 (112) + } Tk J hal hick hich equals 1 when & = ỉ and ] 1 + Ï do otherwise) and therefore, by (1.11),

1 AgApe> 5h R— 1,3,8, (113) 1 hitchand eide of which la 1

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it

only to oe in a “coarse grain of the Hilbert space 5 that is, to be in in one eof 1h +†E », In that potas 11 lan] 1 Liliter fon 4] one would calculate this probability by

P(8) = Spa POW) = Spal (&, Ua)? = (| 7 paln)(n) Now Py “ |n)(n| may be regarded perator in tÌ that def : Pn ae _ #}W I ector on Uy, Py Ty, ] + 1 1 p2 p ne where the numbers 7, ti h that >, pn = 1, and PyP Ds cs is described by p, th hahilite of P(®,p) = (®, p®) = (|p|) = (ø) (115)

Pm = 0 for all n but one, the density woe 1 isolated, “Shaved state of the System Fo or examp : trhe + lated ] l1 từ ith P bability 2 5 ) its density operator is p- = Le + ay and the probability of finding it to agree with the state ® is FG le) = (@, 81)? + |(®, #2)|?], without the interference term present if it w n a superposition of the two states ¡ and ¿ Equation (L15 ) that if tem is in the stat described by p, the probability of obtaini ] lt A of the variable A is given by i 2 _ (A,pWa) _ (AlelA)

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1 £4] đ đợi by Pub (rv, 6) oO come A AV#A,®) the other hand, tl f tl f I t f 4h + 1 “Lad 4 he d of the outcome > > Ừ ‘a > & “he ở > | >ừ/ & đ a ơ ơ a 1 ] | fy | ®)I3 ] £ After the measurement, tl ] fi ] † 10 WN TỊ fay bul lM a Mixed slate ate after the measurement of A, with the “outcome ignored, si 1s tuoi by ØA =`PApPa (1.18) A Ifa system, consisting of two subsystems with Hilbert spaces $! and 9", is in the eure s state F = Ÿ2 „2xx @ WI, then the State of system I, with s m II ignored, is aed by the de ensity o P= SP drm, Pram = =a (1.19) n k where =|n)) Obl, meaning that Prin or = oe ww 6)) Thi I 1 in tl in a mixed state, po = = tp, (1.20) 1 Even 1]

8This is often expressed by saying “the outcome red.” Such a formulation, however, gives the misleading impression that what the — knows or has in mind influences the state of the physical system No such inference should be drawn

®Suppose a standard two-slit experiment is performed with electrons made visible on a screen and photographed Describe the state in which “the come is ignored”; how would you record it?

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tk f | | 1 o that the final state of † f+} 1 1 ` schematically given by

W—ÝÝ flor,0s)¥a(or) @ Voou(aa), ayo ] ] tat lect Wreu(@o) is the stat f th + d 1 : va 1 c t BỊ 1 1 Hirection the angular momentum, etc If tl i is 1, the probability of teint 1 @ ha 1 + og by Ý- Hanes(a(ei).0a6) ÝIŒ.9a6) 9 822.G))Ƒ = } x (Pnau(02), „„@))| =_ (Pa(9),paai(8)), where

pa ff Hains tek 9) lah ara at hi at, as remarked earlier, the state of the elect ill] ixt thar £ di forant + i mixture momentum projections

For another example, take the vectors |E,) = Vg, to be normalized tates of the Hamiltonian H(t] + let IP \IE.| bet} my Teed Labi] Find 1 ; Lm m } 1 } ] } £TT †

ø= >`p(E,)P„ = È `p(H)P„ = p(H) n n because HP„ = En Pa and Son Pr = 1 [see (1.23) below] for a system in tl 1: 1 1 1: ï Lp 1 Teeth at ” with con stituent a (yy ex b+ the Bol | 1 in the simple form

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(Un, Um) = 4 re the eigenvalues of p Since Pn > 0 for all n, the den ity operator is 1 positive 2-de fin ate tl to add up unity, 7, Pn = ace, i.e f its ei qu nity: =1 (1.22) Furthermore we have p? = >>, Paln)(n, so > that (op - p?) =, (Pn — ø2)|@|)|* > 0 becau e for all n, 0 < and therefore P — Đn >Ũ On Pn = 5 unless py = ly if a we have (p — "2 =0 TI ] hich tl I 1 1 1Ì + 7 1 by 1, tl ye ja), and the density thi p= | I The £ boệng 3d Lont 2 mixed, isolated Seat 2 The trace of p? 0 < trp? = DnB e Š 2P e one of 1 1x; holde ‡£ 1 Gf = Pa 1 fi 1 ma’ be taken to be a relative measure of the “ogre of mixing” or “coherence” of its state How h fl 1 £ are + siates, the mỉnimum oŸ trø2 is 1/n.1Š 1.4 Representations ¬ mì > Ua, ie, tl 1 } r I led | r+} 1 1 ce

f coeffici i (1.3 ƑA,) = (A N thovetone uniquely Ƒ; tÌ f | {(A]}}, where A runs over the pectrum of A, is the A tate f |) If the ere is «de ‘legen tac, Le., if than one Trai all of tt

the term “non-mixed” rather than “pure” because the state is not as well defined a: as ® nature permits; it is isolated and has “no hair.” States represented by a density operator never have “hair.”

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calculations mỊ hi Tittle etickier if the Tamniti tor A ha a continuous Gan A 1 B24 for dataile) TH spectrum, i hick Ì eri 1 ned đ man] ll the £ 1 esigned pectra E q oo is ¬— by @ 28) in the ee and with the “normalization” (B.24) we obtain the gen eralized Fourier integral

= | 4A|4)(AD),

The “matrix” XI he | 1X (A’, A”) of an integral operator, since bà (B.23)

(AIXIA) =_ (A|IIXI|A) = JaArdAr (AJ A’ A" |X| AM (AA) [as dA” (A|A”)X(A, AN) AMA)

however, in some instances they become equivalent to diferent cpertore ‡ f tÌ ‡ A with the dis- crete eigen ah ue Ay T1 its A i {(A] Ay) } i itl A 1 + ML 1 1 1 1 1 TÈ +} + 1 1 1 Tn the A +} tor act 1 Tự 1 Tita] 1 AI yy | in the sequence, {A|A]) = A(4]), hic} 1 1 1 A in rỊ 1 with +] the diagonal, b fore to +} bị 4JAIA” £ Endi ll the Al’ Al AN ef A”6 4: au One therefore Ệ 1 ble al +] bị bị basis (We chall d 1 lotai) a little | leo A fl

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1 ] + £4] d ff, J M1 + J] 1 TỰ, +1 £ | 1 The x 4] 1 ] SC Euclid H the vector g, a d 1 hI 1 1 +1 + & Top + tie] cad £ 1 | lai đ 1 ĐỊ he 1 1” | Tự 1 ] 1 } d hịc] 1 he atv.d ] 6 Only c 1 had Ciba nh 1 Ea] 6 tr +Ị 1 distinct and must not be confused S% the d ta santables o£a sinøl 1 1 1 1 mm" 19) Tang lhe d lhe @ W, going to construct their spectra

Tet the 6 tai F +] ta of đ say +1 1 1 a | that al Ai

1 1 #2 | toa] 11 han fnd & TT xoa] ¢, to first order in ¢, that

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/ đŠqlä (ấi =1 (1.27)

4] ~ s Ls: : soos 1

Sinee (iđ|) =

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the infinitesimal neighborhood d' f tl fi is given by |b(Q)|?< ] Trị 1 iat 1 ha bolst + density of a particle of M ini fi i I a def th, * ID = =z|”"(@Vø() — (V9*(8)19()], (1.31) Lied 1 1 4 1 hat ->/N 1 re F+] 1 my 14] lad ea +] fl 1 Tha Jon ect , 8 £4] 1 ald.) = (Geld -¥ patla (8007 (9 (132) ® | f Liti f [The configura- 1 1 + 1 Leal 1 +] 1 1 +} £4] two particles I ] ] ] ] 1 1 Pa 1 1 Without ] + a 1 Lvaical 6 Jer led 3 1 Ta] h d £ R 1 £ Ii 1 £ ble A with ¢} + + tat + f A ith th 1 Á le it + tum physics is described in terms of “coordinate space” wave functions, it di ff 1 istic of tl Tn reality it Ì he inevi f the probabilistic n: nature of abilistic — zs confronted with an quantum "¬ Whenever a Poll

lt © probability distribution instantaneously L jl L “collapses” 10 1 la hall

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tions of 3n variables g, & = 1, ,n, | while the momentum of the k* icle i 1 by the diff ~V a, „ and zt 7.) \2d2q, - da, is th " lÚ(1, đn)|?đ S4 in tl 8 | ™ Bq, - hahilite denatty dq, near the pọnt đi of F+t g, of their joint ôdae tx ol ơ \|2 1 other, WY 1Ị f£ +] : : | fi a g,,), then particle #1 Fl Tf and tanlated 7 1112 đong : 1s given by PG.) = | aaa Aan UCase YO" dan (1.33) he Tt £ 1 te +] " £ TT] £ 1 1 ¢, 1 1a] £ i £ 1 f the form WG-29) = So aey,b "¬ Per GF) Ve, (He); (1.34) £4] £ £Œ † 1 1x T11 14 1, 1# 2 The momentum representation

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a=] pp)" AV )OW 13 1 i Wad article to have its mornentum in the neighborhood ip of p Dp Ag - The slilbert 1 Bla £ : ˆ 1 d tl + f tho pth tre] +

lai d¿ = et The ohysical in- terpretation 1 of OG 1 Ủ, that of £44 bability-density amplitud: 1 Lharhand Boe Be of the point B1, ,Bn is JĨ(đ, , 8, ‘in Bp

There is, however i inui ft] £ 1 thai fa] en ta]

£ 1 1: 1

and assume the “box” has len by repeating the “box” 2 pia setting |g + ye

ant As a result we obtain by (1.29) b) = ".¬ (ol n8 ePP/* (nla), = i where ¢ is a therefore consist of the points p, = 2h “box » | Thus ‘he esult, of cour: f tl linat 1 1 af the } 1 1 F] 1 =1, 2, 3, the f tl th ] nụ = 2m Đan Lao nị = 0, +1, +2 mỊ lin (B92) than } o be eplac " i f i he fact that, wea whereas Fourier ion the

1 fae Ê 8 1 (in that case, too ftl le tl 1 of deđniti is necessarily periodic.)

1.4.38 The number representation Tet 119 A22) ca.) a 4] tại Tf d 1 £ the fald 1 + i i hall 1 1

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