Quantum physics - a text for graduate students r newton

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.

Quantum Physics

A Text for Graduate Students

With 27 Figures

ÂN > Springer

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

87654321 SPIN 10874003

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of stationary action Tl | 1 f tl hacl Ì hicl lud Lt | 1 Julian Schwinger, filtered through and modified I f teacl ing the subject larly for tÌ f 1 Chai m1 Tờ £4] 1 £ 1 hie baal Jed ahold 1 1 lad Tami] 1 TM We +] Fe] and the

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4 1o] 1+1 1 ee : £ 1 table Th nary and partial

Lac «ay 1 £ ling Hill the Dirac delta fimetion, linear ntegral equations, th ired classical and group rep- resentations, 1 1 1 hal lanat + 1 Jotatle 4] 1 1 the 1 © he hort 11 1 nmrtedi Nar & Teellectod

it at the end, and the i ill | I + wl needed TI 1 trẻ the £ |

} inski, John Challifour, and Don Lichtenberg

The configuration representation

na The momentum representation See The number verre enlallon Trenslonnetion Theory 2 Ha ee roblems and Exercises 2 2 ee ee

2.1.3 Tl 37 2.1.4 The Hamiltonian SauaLlons ofmotion 39 1.5 The Schrodinger pictur 40

22 Systems of Particles QẶ ẶẶ 43 2.2.1 Linear and angular momentum 43 2.2.2 The equation ofn motion 44 2.23 TI Schrodi i 46 2.2.4 Example: Gaussian wave packets 48 2.3 Fields 2.3.1 The matter field 51

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

Static Magnetic Fields 217 7.1 The Larmor Precession 217

2 The Al Dohm TE 291 73C] 1 Darticl aL as 299 7.31 Spi bì 1 the fine structure 223

ra The Zeeman effect 2 2 ee ee ee 226 7.4 Peablems and EXxerclses Hs 233

1 Perturbation Theory .0.00.02 0000005 235 8.1.1 Application to ae decay 240 8.1.2 Application m to scai atter Ắaáaáaa 242 8.1.3 § 1 243 8.14 Interaction with electromagnetic fields 244 8.2 ] 253 8.3 Problems and Exercises 2 000.- 0000004 256

9 Multiparticle 257 9.1 Two Pantie Interacting withaenier 257 9.2 Identical particle 264 9.2.1 Fock space 2 2.02 2 ee 269 9.22 Pose-Fi TT; Tra 276 9.3 Problems and Exercises 2 0.-0 0000004 278

10 Relativistic Electron Theory 279 10.1 là sma 279 Relativistic spin 2 .0.02000022000% 281

1012 Nẽẽ ca ee 282 10.2 The Dirac Equallon 292 10.2.1 Electrons in an in electromagnetic field rn 299 10.3 The Dirac Equation for a Central Potential 302 10.3.1 The en atom Q Q Q Q Q Q Q Q Q2 305 10.4 Probl 1E i 308

II Mathematical Appendices 309

A The Dirac Delta Function 311

B Hilbert Space 15 B.1 Linear Vector Space 315 B.2 IfniteDi ional L V rSpaces 318 B.3 Operators 319 B.3.1 Hermitian operators and their spectra 322 B.4 Problems and Exercises 2 0.2 .00-0 0000004 332

C Linear Integral Equations 333 G1 mm edholm vetho ds ee 333 C.2 Tas Lops = : 337 388

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:

2.5 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 E.4 5 Multiple valuedness

Ed The group SO(3) E.5 Problems and Exercises References

Index

Physics

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 (thouøl lv hị "

he ñ 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,

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

ray 7 is a one-dimensional subspace in $) Consequently there exists a vector

We ‘ that spans 7, so that ‘R consists of all vectors of the form aV.

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

t ; in the coeffici After

1 1 : 1 ws :

tl ] È 1 ] 1 fi 4] information that then Spi | “ isel ll 1 means for its description a | Tittle later )

1 icl 1 " 1s là TẾ+I 1 hat 1 £4] 1 Fl 1 fan the d TỊ 1 the al Te whic 1

1 1 1 Lle «nl lated with +] £

1 mm 1 14] 1 Tod weit]

1 | ha d hed

by a state vector Fl lee 1 1 "

1 scl Ne d "w Tung 1 1 Thy 7,

1 tod Lite | 14 1 aeting

1 mỊ fod 1 ase, 1 1

1 f, he f 1 1 tate at

2 le d 1 eid 1

1 F+] 1 hahi] distributions F his foll 1 lassi 1 1 and in particular, the probabilistic rather

1 1 1 1

1 ¢, £ j "

ự 1 lat £4] £ 1.1.1 Measurements

Wi f

tulat I icall 1

®=Mathematicians distinguish between symmetric and self-adjoint operators, the dis-

2 After 1 tl laf 1 h tl A £ ystem originally in

A with 4] lan A(T 1 1 situation T Je] ifi Ì 1 measured, and vice ver a.) As a result, the st bef

because

1 The + A 1 £44

1 1 1 #

bo ads 1 1 taal chanic babil di + 1 4]

Ệ

the measurement | might have been quite nonuniform.) TỊ 1 1

leah] 1] 1 TH pace ] f +} Ty, +

A are denoted by A, tl ] 1] | HW defined 4] functions of A (In other words, tl J ¢ LA

fi † heir definiti 31 ãn lied] 1 1 bed utside $ 9+ † £ ]

bị ble A li f tl (the spectrum

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

1.1.2 Fields as dynamical variables

W ] 1 he 4] | Ea the gs and Ø8, are ob- 1

1 ta Leld whicl

we shall denote by U(r) tor-functi f 4 This operator i

TỊ Ty, J 1 1 1 + hI Hài 1 1 TỊ, T†

he 4] ] p= LT] a 81 Since the

kat

T(#|#) = |(®, #)/ || # |Ƒ II IP a.) asurement of the observable A t (4) / || a 2 | Ỷ IP, where Wa is the

g to the eigenvalue A, refore often

Jf P(s, W) = I I 1

i A A 1 1 + nh 1 | Id It

1 he P(& Wh Pir &) +} hal 1

£ +1 1 1 ahi} hat @ has those mỊ of WV The of 1 hie} 1

1 Te 1 lot

£4] fal] TT bial

ha haeie of +] Te, any Hermitian operator A:

Y= Veal, (1.2)

A } +] +} fA hick £ 1

ta Tew J ol] +} Ti

1 f¬I1 q 1 1 ort} a4 £41

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-

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

c(A4) = [aa (ha, ©),

1 the mÌ 13 £ Inf A\2 io that of habalite: denest: la( ANIZTA fe 4] Lay ced 1 1

observable A to lie in the int ] đA around A If an operator has a

Ì i if we wi h to

£ 1 hI 1 £ hall indi +1 + + 1# E 1.2.1 Correlations

with the Hilbert spaces pie and aa so 0 that § = 9! noah, and a pure

11 1 Tr 5 lị 1 £ me @Uy (We |" ily

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-

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

Š[Bohm, D.] p 614.

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 ctions are “elements of reality” but quantum be simal «oy and | 1 1 en 1

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

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

P(W8) = (V8)? = 5/8) +, 8)? 5) BP + 2002, 8) + 8(0y,#)*(W2,8),, (18)

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 Ñ 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 E=nđ H 1 local Trad by 4]

111 ] d th + 1 though we call them “par- ticles,” are not little billiard balls.®

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*) = (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?),

1 define A’ “2 A — (A) [this should really be (written as A) & A (A)1, where 1 is the unit hat for all j, 1Y = WP), but we shall forego this bit of pedantry] and 1B a BL (B), so that (AA)?

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 hine but Heisent 1

TP the two ol bles A and B 111

1 1 ] There are, then, states in which } harpl 1 we oe ay tus permit h the fact tk Hermiti

1 f Commuting ob-

1 + hvsicall Lo

1 Cod ] ly Ty 1 ca] 1 1 1 1 Ine of onl

Am 1 1 thị f 1.3 Mixed States

mỊ ¢ ] £ ] Cod isely as nature allows In ord h less-tl | tet £ ] ] 1 +1

r 1 divided 1 1

1 1 R heel Fan o† Ps Wi Cod to |

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)

P(A| “10 Ta) (4A) = tr(P (Pap), (1.16) 1.16 here P 1 1 PA | lue A and

tr denotes the trace (see Appendix B) g r ble A t

1 £4] ñ đợi by Pub (rv, 6) oO come A AV#A,®) the other hand, tl f tl f I t

† 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

®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?

a terms of ensembles, the measurement of A with the result A yields

of states in each of which A has the same value A (this 41 mhei is “he seeribed

bự a an sigs of A with the eigenvalue A) The measurement of ‘A “with the result

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

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

pae HAT cứ 20

(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

h fl 1 £ are + siates, the mỉnimum oŸ trø2 is 1/n.1Š 1.4 Representations

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.”

18 Prove this as an exercise.

uniquely A 1 1 1 TẦNG haf le [ l A A | 1 forall A D4 A

BỊ

Sˆ|4j(AI=1, (1.23)

A

1 4 Je fon 4] 7, : F (19391 ] both sid ‘bi | in which case it reads

or more generally,

Ie KW ƠŒIXIB Ơ|1X1|B N(G|AHAIXIAOD(4|B), AA’ A|X|A’ d the A tati f |B 1 |C) In other words, in the ñ 11 hot 1

| + hb AIX|A Tdinh simply becomes matrix multiplication,

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

ñ 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

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 For continuous spectra, tl hi fi lly tl though

1 let The A

|4) is the Di 4— Ái), a “funcbion” that differs effec-

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

L ell Le)TT—¿(/E pH GiL1 — ?/5)BI] |đi (de Piqi + €||a2 4+ €)|[ ale/h)pi] |g),

hi hich implies that [1 £ i th the li that F1 eŸ 1 || L [1 — 2/ñ)Ði| |) = la + $) (1.25)

Mh £ + 1 1 hi + ] 14 TỊ 1 1 That 1 eal) 4] ta 1

tự 1 £4} + + L £ 1] 1 Th +]

EI +† Time £ to 4 1 +] 1 shie† 1.25) could break down +1 Ty, is for [1 1 to +1 vanish But this would : z

1 ey hịch ï le H +] fa | 1 Jed al and below

Si | the simulta-

| Fe +] £ boat

1 1 Jad main a Boyer | ]

def o@ = 4)

† th

representation, tl tt tor is p BY@) = -thV GO), (1.28) formula

= | Pv@' nv WO,

assuming that f d°q\b(q)|? =

Hi ft 1c ae : 11 This leads to the result

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

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| 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

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 «dae tx ol

† ¬ \|2 1 other, WY 1Ị f£ +] : :

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 ] lati 1.1 which apart from ¢ a sign, is symmet-

11 a 1 that 1 4] het] 1 must generally | fi t and a one-particle system’s Hilbert s I 1 by tl f i bl } fs | , th th P d

14 As an exercise, prove that UV! @ Uv" is mapped into the produet ¿HH o; the corresponding wave functions.

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 JÓ(ñ, , 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 deñniti

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 IW().0!@0] =3°œ-=?), IW(),0()] =0, (1.36) which will be j ju stified along with (1.12), in Cl in f