foundathms o f Physics Lelwrs, I bL 7, No. 5, t994

CAN A N O B S E R V A B L E W I T H D I S C R E T E E I G E N V A L U E S BE C O N J U G A T E TO A C O N T I N U U M O B S E R V A B L E ?

R. T. Deck and N. Oztiirk

Department of Phy.dcs and Astronomy University of Toledo, Toledo, Ohio 43606

Received December 30, 1993

We point out that, in combination with the normMization and com- pleteness relations for the eigenkets of the operators, the commuta- tion relation [Oo, Oz] = +ih places severe restrictions on the eigen- values of the two operators Oo and Oz. These restrictions can be used both to determine the character of the pairs of conjugate ob- servables that cml be involved in uncertainty relations and to predict the quantized form of the eigenvalues of a discrete observable which is conjugate to a continuum observable with a finite spectrum.

Key words: quantum mechanics, conjugate observables, mlgular- momentum quantization.

1. I N T R O D U C T I O N

The essentiM content of the uncertainty principle is contained in the notion of incompatible observables the values of which cmmot be known simultmleously in ,'my state of a physicM system. More specifically, quantum theory asserts the existence of pairs of "conju- gate observables" such that, if a and 3 denote the distinct members of a pa r t i cu l~ pair of such observables, the product of the intrinsic uncertainties Ac~ and A/3 in the values of a and/3 must satisfy the inequality

A,~A3 _> hie. (1)

Tile inequality is more quantitatively expressed in tile quaaltum for- malisnl through an equation for the vMue of tile comnmtator of the conjugate operators Oo and O~ corresponding to the observables c~ and ft. In particular, the uncertainty relation of Eq. (1) can be shown to be equivalent to the requirement that the commutator of

419

0894-9875/94/1000-0419507.00/0 iO 1994 Plenum Publishing Corporalion

420 Deck and ()zff, rk

the operators Oo aal(l Off satisfy tile relation [1]

[Oo, = :Lib, (2)

consistent with the commutability of the operators in the limit of zero h.

It is well recognized that the commutation relation of Eq. (2) hms serious implications with respect to the spectrum of eigenvalues of the operators in the commutator. This can be seen in a simple way from the fact that a matrix representation of the commutat ion relation in a finite basis is inconsistent with the fact that the trace of the representation of the commutator on the left hand side of the relation needs to vanish ,as a result of the commutability within the trace of the matrix representatives of O,, and Ot~ in such a basis. Where however the ba.sis constructed from the eigenvalues of the operators is instead infinite, the commutability of the operators in the trace can no longer be proved and it is possible for the operators to be non-coininuting.

On the other band, the argument based on the trace of Eq. (2) fails to distinguish between a denmnerable ml(1 a non-denumerable infilfity of eigenwdues for the operators O~ and Off, and therefore in principle allows one (or more) of the operators to have either all infimte range of discrete eigenvMues or a finite range of continuous eigenvalues. What we claim to show here is that, if both operators have discrete eigenvMues, the commutation relation c,'mnot be satis- fied, and inoreover, if instead one of the operators, say/3, has discrete eigenvalues and the other has continuum eigenvMues, the commuta- tion relation can be "satisfied" only if the eigenvalues of the discrete observable/3j are connected to the (finite) range L of eigenvalues of the contimuun observable c~ according to the qumltization relation

2rrh /~ = - L - , ~ . ~j = 0 , + 1 , + 2 , . . . (3)

The result is remarlmble in that it foreshadows the quantization of tile observable which is conjugate to the angle variable that specifies tile position of an object in sl)herical coordinates, l Since the angle wtriable has contimuun eigenvalues in a range L = 2rr, Eq. (3) cor- rectly restricts the eigenwdues of the conjugate (mlgular momentum) observable to the quantized values 7Zjh. Of course tile form of the uncertainty relation that connects the angular momentum and an- gle variable uncertainties has a controversiM history [2-7] which we return to in Sec. 3. We first provide (in See. 2) a straight-forward analysis of the restrictions on the eigenvalues of the operators in the commutation relation (2) which are imi)osed by the basic elements of the quantum formalism.

Observal)le wllh Discrete Elgenvnlnes 42 ]

2. I M P L I C A T I O N S A N D C O N S I S T E N C Y OF C O M M U T A T I O N R E L A T I O N

(a) Case W h e r e B o t h a a n d / 3 Have C o n t i n u u m E i g e n v a l u e s

The restrictions on thc eigenx-alues of the operators satisfying Eq. (2) derive from the normalization aJl(l completeness relations for the eigenkets aml eigenbras of quantum mechanical operators that resldt from their defining equations

oo1,~' >= o,'1,:,' >, < ~'1oo = ,:,' < ~'1. (4) To develop the arguments used here it is necessary to first consider the conditions under which Eq. (2) c,'m be satisfied for operators ,a.s- sociated with continumn observables. In this c,'Lse the norxnMization and completeness relations for the eigenkets can be written

<~,'1~, '') = ,S(~' - ~" ) , / d a ' l a ' > ( , - , ' I = 1, (5.a)

(/3'1/3") = , s ( Y - / 3 " ) , = (5 .u )

By use of these relations the ordinary (non-operator) equations ob- tained from Eq. (2) by separately multiplying this equation on the h'ft and right by eigenbra.s and eigenkets of c~ and fl have the forms

< ~'1[oo, o,,]1~" >= (,-,' - ~,") < ,-,'1o~,1~" >= iha(,-,' - ~,"), (6.a) </3'1[0o, 0~]1~" >= -(/3' - /3" ) </3'10ol/3" >= ih,S(/~' - /3") , (6.h)

where we arbitrarily choose the positive sign on the right in Eq. (2). The extremely singldar character of Eqs. (6) (for a ' = a" or

#' = /3") requires the quantities (a'lOt~la" } and (/3']O,,[/3") to be proportional to derivatives of 6-fimctions aal(l results in the "equiv- alences"

<o, '10~,1~")--- ihb~,,s(- ' - ~ " )= - i ha - - ~ < -'1o," >, (7.a)

< 'IooI/3"> = - < > , (7.b)

These last equations caal be reworked into more general relations (aa~d their adjoints) by expansion of an arbitrary ket [~) (relating to the degrees of freedom of the observables a or /3) in terms of eigenkets of either a or/3 in the forms

f do,"lo/')(~,"l,I,>, I'.I,)= f, 't/3"l/3")<Y'l 'I '). (8) I,I,> J . /

422 Deck and Ozlfirk

The expansions allow Eqs. (7) to be converted to the relations

(o'lOal~) = -ihb-~, (dl~), (a' lO.l~) = ih~-~7(fl'l~),

(9.a)

(9.b)

which (in combination with their adjoints) establish the well-known prescriptions that define the result of an observable operator acting on a (continuum) eigenket or eigenbra of a conjugate observable:

(c,'lO,~ = - i h o--OgT(.'l,

(felOn, = U,O@, (/~'I,

Oalc, '') = iho@lC,"), (10.a)

O,,lfl") = -ih o@ifl"), (10.b)

(with the lack of complete symmetry between a mad fl a consequence only of the particular choice of sign in the commutation relation of Eq. (2).) The prescriptions (10) combined with Eqs. (4) lead imme- diately to a differential equation for the bra-ket product < a'lfl' > in the form

(~'1o~1/~') = - i h ° ( c , ' t 3 ') = fl'(dl3'), (11)

which has the solution

(o/I,~') = (~'1o,') = c J '~'/h, (12)

where C is an arbitrary complex constant (independent of a ' and ~'). The restflt guarantees that the absolute square of (a'lfl') is indepen- dent of both a ' and fl', in agreement with the uncertainty relation of Eq. (1) which requires that, in an eigenstate defined by a zero uncertainty in the eigenvalue of one member of a pair of conjugate observables, all eigenvalues of the other member of the pair be equally likely, consistent with an infinite uncertainty in that observable.

It follows that the equations derived from the commutation relation in Eq. (2) are consistent with the underlying uncertainty re- lation (1). But it remains to check the consistency of Eqs. (7) with the normalization and orthogonality equations expressed in Eqs. (5). Because Eq. (12) is a direct consequence of Eqs. (7) and the com- pleteness relations, in Eqs. (5), provided Eqs. (7) are consistent with the completeness relations, Eq. (12) should be usable for construc- tion of representations of the quantities (dlO~l~") and (~'lOol~")

OI)ser~able with Discrete Eigenvulues 423

which have the properties required by Eqs. (7) (and (6)). Moreover, provided Eqs. (7) are in addition consistent with the assumed or- thogonality relations for the eigenkets Io') and 18') (so that the left and right hand sides of Eqs. (7) are correctly equivalent), Eq. (12) should be usable for construction of representations of the quantities (c~'la"), (fl'lfl") with the properties defined by Eqs. (5) To check the consistency of Eqs. (7) with the orthogonality and completeness relations for the component bras and kets it is therefore sufficient to check the validity of the representations constructed from Eq. (12) for the quantities <WlW'>, <YlY'), (WlO~lW'), and <fl'lO~ly'>,

By use of Eq. (12) along with the assumed completeness re- lations, representations for the listed quantities can be constructed in the forms

/ ! • t tt t ( - ' 1 - " ) = d /~ ' (~ ' lY)( /~ ' l -") = ICl ~ d/3 ~'(° -~ ) a / h ,

(13.a)

(13.17)

(~,'lOalo~") -- / aY(o/IOalY)(YI~,") = ICl 2 / dYYc '(' '- '~'')a'/h,

(14.a)

(/~'1o~,16'") do,'(YlOol,~')(~'lY') ICl 2 , , -~(a'-~")~,'/h

(14.b)

where the integrations extend over the entire rmlge of allowed eigen- values. Equations (13) and (14) will be consistent with the assumed orthogonality relations had Eqs. (7) respectively provided it is pos- sible to satisfy the relations

IV? / d Y ~ ±k(" ' -" ' ' )a = ~(~' - ~"), (15)

and

: o " ) , (16)

and their counterparts with the roles of a and fl reversed. 2 By eval- uation of the integral on the left hand side of Eq. (15) between the limits defined by the minimum and maximum eigenvalues of fl (and

424 Deck and (-)zlfirk

use of the properties of the &function), Eq. (15) in the cases c~ I = a" and c d ¢ a" , respectively, reduces to the two relations

ICI2( fJ , , , , , , . - /3 , , ,~ , , . ) = oo, o,' = o,", (15 ) '

a n d

i h l C I 2 [ :~ ,,~,, ,, . . . . 1 . . . . - c~" ) t e ~" - ,,-,,.,,i,,.j a . ' c J ' ,

( 1 5 ) " both of which demand an infinite rmage of eigenvalues of/3. In par- ticular, Eq. (15)" can be satisfied for a t ~ ~" only if tim exponential exp[+i(a ' - c~")/3'/h] ha.s a common value at both the upper and lower limit of /T, which requires the exponential to be a periodic flmction of fl' with a period equal to (/3m~x. -/3,, , i , .)/n, where n is a non-zero integer. But since the exponential flmetion with imaginary axgument is periodic with respect to its argument with period 2rr, the latter requirement restricts the values of Ic~' - a" I by the condition

2rrh I , , ' - , ' , " 1 = (/~ .... _/3, , , , , , . ) n, ~ = 1 , 2 , 3 . . . . , ( 1 7 )

which can be satisfied for continuum (non-discrete) eigenvahms of c~ only if the spacing between the allowed eigenvalues of a ' defined by 27rh/(~ ...... -/3~,i,.) is zero, or equivalently if (/3m~x, -- /3mi,.) is infinite.

Given an infinite r,'mge of eigenvalues of/3, it is necessary to investigate the convergence of the integral in Eq. (15) at the infinite limit(s), and in addition it is necessary to verify Eq. (16). But the latter equation call be shown to be consistent only if the upper and lower limits of the /3' integration are equal and opposite in sign; which metals that the integration must range between plus and nfinus infi,fity. Given these infinite limits, the integrals in Eqs. (15) and (16) have vMues represented by the equations

i+oo~ [sin ( ~ ) L] ICI2 d/Y¢+ ~(°'-°")~' = 21C12 )in~- (~"-~'")h (18.a)

a l l (1

IC l~ / /2 d/:7'/3'~+i(°'-°")~'/h

{}bser,-M~le wllh Discrele Ell~envMues 425

By use of the usual definition of convergence the right hand sides of the above equations do not converge to the required value of zero for a ~ ~ cFq However, this value can be assigned to the integrals on the left sides of the equations by the well-known modification of the definition of the value of all integral at a limit of integration which tends to plus or minus infinity. In particular, the definition of the integral at a limit which tends to plus or minus infinity needs t o be nmdified so that the value of the integral at the limit is taken to equal the mean of its values in the vicinity of the limit. With this definition, provided the upper and lower limits of integration in Eqs. (15) and (16) are infiIfite, the left and right hand sides of these equations are made consistent for cF ~¢ c~".

In order for Eqs. (15) and (16) to be also satisfied for cF = cF', and the left-hand sides of these equations provide representations of a ~5- flmction and a derivative of a (-flmction respectively, the integration of the left haald side of Eq. (15) over cF between may limits which enclose the point ce" must result in the value unity. This requirement c,'m be satisfied provided

[C[ ~ = 1/2~h, (19)

which value for ]C] 2 allows Eqs. (15) and (16) to I,e re-expressed ms

1 / / o o ~ ~ +}(,~,_~,,,)~, d/~ ~ ,. = ~(a" - a ' ) , (20) 2rrh

i l / / ~ _ ~,. ) ~, 0 0 + ~ d~3'/3'~ ± ~ ° ' = ,~(~' - ,~,,). (21)

The above derivation of the well-known representations (20) and (21) for the ~-fimction aald the derivative of the 6-fimction is included here only to review the argmnents used below in the case where one of the pair of conjugate observables is assmned to have dis- crete eigenvalues. On the other hand it is significant that the above analysis makes it evident that Eqs. (7) can represent "solutions" of the relations (6) consistent with the axioms of the quan tum formal- ism only if both the continuum observables c~ and/3 have ranges of allowed eigenvalues that cover the entire real line from - o o to +co. Therefore in any case of continuum observables which have more re- stricted ranges of allowed eigenvalues, since it is not evident that any consistent solutions of Eqs. (6) exist, it can be a.ssumed that the underlying commutat ion relation (2) is invalid. As a consequence, pairs of continumn observables which have more restricted ranges of continuum eigenvalues nmst either be not conjugate in the sense of

426 Deck and Ozlfirk

Eq. (1) or their "conjugateness" must be expressed by a conmmta- tion relation more complex than Eq. (2). Assuming (on the basis of simplicity) that it is the first alternative which is correct, and under the assumption that the quantum formalism is sufficient to allow for a complete expression of the basic uncertainty relation (1), it is pos- sible to conclude that no conjugate continuum observables exist with a less thasl infinite range of positive and negative eigenvalues. The conclusion represents ml interesting but well-known result. What we claim to be new in the present paper is the result of a similar anal- ysis in the case where one (or more) of the operators in Eq. (1) h,'~s discrete eigenvalues.

( b ) C a s e W h e r e (~ H a s C o n t i n u u m E i g e n v a l u e s a n d /3 H a s D i s c r e t e E i g e n v a l u e s

In the case where one member of the pairs of conjugate ob- servables, say c~, has continuum eigenvalues and the other inember of the pair,/3, has discrete eigenvalues the eigenkets of c~ and fl need to satisfy normMization ,'rod completeness relations of the forms

(,¢1 ~'') = 6(a' - ,~"), / d ~ ' l , ¢ ) ( ~ ' 1 = 1, (22.a)

</3jl/3k) = 6jk, ~ I/3j)(/3jl = 1. (22.1)) t~s

By separate multiplication of Eq. (2) on the left and right respectively by eigenbras and eigenkets of a and /3, we obtain in this case the equations

( ~ ' 1 [ o o , o~11~") -- (~' - ~ " ) ( ~ ' 1 o ~ 1 ~ " ) = ih~(a ' - a" ) (23 .a) </3jl[O ~, O~]ll:~k ) = (/}j -/:tk)(/3jlO~l/3k) = --ih6jk (23.b)

(where the positive sign has been chosen on the right in (2)). Because Eq. (23.a) is identical to Eq. (6.a), it can be solved for the qua~tity (~'1o~1~") in the san~e form expressed in Eq. (7.a),

(cr'[O~lcr") = - i h o @ 6 ( a ' - a " ) = - i h ~ , < a ' l a " > . (24.a)

By anMogy we then assert that the similarly singtflar equation (23.b) can be solved for the quantity </3jlool/3k) in the analogous form expressed by the relation

• 0 . O (/3jlO~,l/3k) = ~ h -K s -@~ = i h - ~ j (/3jl/3k). (24.b;

upj

()h~er~Mile ~ilh Discrt.le Eigen~al.es 427

In fact, as a consequence of the connection between discrete and continuum eigenkets [8} Eq. (24.1)) can be shown to go correctly into Eq. (23.b) m the hnnt m which the discrete eigenvMues/31 become tile continuous eigenvalues /T. Nevertheless it is necessary to deter- mine if Eqs. (24) are consistent with the remaiifing equations of the formalism in this case where the conjugate observables do not both have continuum eigenvMues.

In the manner described in the derivation of relations (10) and (11), Eqs. (24) lead directly to the presccriptions

: - ) (~,'1o3 = - i h a~, (,~'1, (25.a)

(3jlO,, = i ho@(3 j l , (25.b)

and to the resulting equations

(c~']O~]3j) = - i h o@(c~']3j) = 35(~']3i), (26.a)

= t, 0 (3j lOola ' ) +i,=-:- . ( f l i t , ' ) -- c,'(3slc,'), (26.b)

which have tile general solution

(a'lflj) = (flj[•') = C 'ci~'&/h, (27)

with C' an arbitrary complex constant (indepemlent of c~ I and/3j). For Eqs. (24) to be consistent with tile assumed orthogonal-

ity and completeness relations for the eigenkets la) m~d [3i) the representations for @2 la."), (ill Ilk), (o: 103 ]c~"), aaM (ill ]Oa ]/'3k ) con- structed from Eq. (27) must be consistent with the Eqs. (22) and (24). The requirement leads to the relations

<~,'l~"> = ~ ( " ' l ~ i > < ~ s l e ' " ) : IC' I ~ ~ F _ _ , ~ ( ° ' - ° " ) " = a(~, ' - ,~"), /si &

(28.a)

f ~ °rmLx = dc~ e - r ( & - ~ > ' = ~5~j~,, (28.b) (13ll3~) I C ' l z ' - ' ' m l n

( ~ ' l O e l ~ " ) = IC ' l ~ ~ ~ ° ' - ° " ) ~ ' = - i t , a @ 6 ( ~ ' - ~,") , (29 .a )

j f . . . . . dz 03 i ~& #~,, = da a e = - - . (29.b) ( / 3 i 1 0 o l 3 k ) iC,i ~ , , -~o~,-3~>,' • o

m m

428 lit'ok and ()zlfirk

where the integrations aald smnmations extend over the complete realges of allowed eigenvalues of c~ as~d/3 respectively. By evaluation of the integral in Eq. (28.h) between the limits defined by the mini- mtun mad maximmn eigenvalues of c~, this equation, in the separate cases/3~ = fl~ m~(1/3j # fl~,, yields the respective equations

IC' l~ ( .m~ - . . , ~ . ) = 1, fli = ilk, (28.b)'

m~d

flj ¢/3~, (2s.I,)"

1 f f ~ m a ~ -- ~t~ -- do:to t. ( ' ~ - k ) a '

L L ~n~n

= 27ril e -12'r (;lJ--llk)~min ( 1 - C-27Ti(llj-111c) --nj ---- 72 k - : ~'ti ,lk,

which, for symmetric limits of integration, rrmi. = --C~m~x = - L / 2 , becomes the valid equation

1 f + L / ~ , -,:.(,,j_,,,)~,, sin7r(nj -- nk) - - / d a e L = = ~ j . ( 3 3 ) L 3-L12 7r(nj - - i, l k ) n~,

(32)

the first of which requires

IC't ~ = 1/n , L - (c,,,,~,x - c~mi,). (30)

But E(I s. (30) an(l (28.b)" c~tn be satisfied for a non-zero walue of C' only if tile range of eigenvalues of c~ is finite so that L 5~ oo. In

,-) rt l)artictll~tr Eq. (~8.1,) requires the exponential exp[-i(/3j -/3k)c~/h], for /3j -~ [~k, to be a periodic fimction of c~ with a period equal to L/T~, where I~ is a non-zero integer; ,'rod the latter condition restricts the values <)f I/4j -/3j, I to the values

2rrh I / J , -13k l=- - -~ '~ , , , = 1 , 2 , 3 , . . . , (31)

which can be guar~mteed in general only if/3 i is restricted by the equation

2rrh /3j = - z - r , ~, ,~ = 0 , + 1 , + 2 , . . . . (3)

But, since the values of flj are by assumption discrete, it follows im,nediately from Eq. (3) that L cannot be infinite. Moreover, by use of Eq. (3) for finite L, Eq. (28.1)) is reduced to the relation

( )b~ ,e~ al'~l¢ * i l h l J i s c r r l t ' E i l l en snh , e s 4 2 9

Therefore, with L finite, Eqs. (30) ,'rod (3) casl be said to insure the validity of Eq. (28.b), and it is left only to determine whether the remaining equations in the set of equations (28) and (29) can be satisfied for a finite r,'mge of continuum eigenvalues.

Note first that by use of Eqs. (30) and (3), Eq. (28.a) can be rewritten in the form

1 ~. ~ c~Z-(,','-c,"),,, = 5(cr' - o/ ' ) , (34) r l 1

which reduces in the cases c~ l = a" a~lct a I ¢ a " to the respective relations

1 ~ Z l = e ~ (a ' - a " ) (34.a)

II 1

and 1

Z c~.'("'-"')'b = O, (a' ¢ o"). (34.},) H2

These two relations (h'mmld a range of eigenvahws of the discrete observable /3 which is infinite. Consider in particular Eq. (34.b) which can be satisfied only if the sumlned terms on its left hand side exactly cancel. Since each of the summed terms in the equation is a complex number with modulus unity, the real and imaginary parts of each of the terms defines a point on a unit circle centered at the origin in the complex plane, and the real ,'rod the imaginary parts of the summed terms can therefore separately cancel identicMly only if the defined points on the unit circle are unifi)rmly distr ibuted over the circle. For all values of c J - c d ' and integer n l this is possible only if the number of terms in the summation is non-linite, or equivMently the munber of eigenvalues of/3 is infinite. Given this, the validity of the left hmld side of Eq. (34) ms a representation of a 5-function requires the additionM equation

i z:: £: a"+A 1 ~2 . . . . 1 sin (--U- J/ dc~' ~ e-r-(~,-c, ),,, = _ - I . (35)

In a more precise form this equation is later shown to be a valid identity.

It remains to examine the validity of Eqs. (29) the second of which is the analogue, in the present case of one discrete observable, of the contimnun case equation

( Y l O o l Z " ) = ICI = U, 6(/3' - / 3 " ) . r a i n

(36)

4 3 0 Deck and ( ) z t a rk

Since in tim limit in which tim discrete observable becomes contin- uous Eqs. (29.a) and (29.b) nmst reduce to the two equations (16) and (36) respectively, it can be inferred from the above analysis of Eq. (16) that Eqs. (29) will be ~alid only if the upper mad lower lira its on the summation and integration are symmetric about zero, so that

a,,,i, = -c+ ..... = - L / 2 , and nj = 0 ,+1 , : t : 2 , . . . , +oo . (37)

In this case, by use of (30) and (3), Eqs. (29.a) and (29.b) take the f o r n a s

• + c o

2~-i ,, . . . . o 6 ( , _ ~,,) (38) ( - ' t o A d ' ) : L~ ~ " J ~ ° - ° ))'~ : 0 . '

~lj = -- OO

27ri 27ri --/+~ - , 2 . . 0 dc~'a'e L ( i - n k ) ° ' - ,t~njnk ~

L (,~jlool/h) = -D- ~_~ On s

which, provided the orders of differentiation and summation or (om~;~ ferention aim integration cml be interchanged, are consistent with the derivatives with respect to a t mad nj respectively of Eqs. (34) and (33).

For Eq. (38) to be valid the summation on its left hmld side needs to be identical to tlm derivative of the delta-flmction on its right hand side. This requires the summation to vanish for a ~ ~ a"

2zri L 2

--~oo

Z njeL~(a,_o,,),g ) t j = - - O O

4 r r ~ (2rr , ) = L- 7 n /s in ~-(c~ - c~')nj

,l i = 0

= o , ( . ' # . " ) ,

(40)

and to have ill addition tile prol)erty of the derivative of a 6 flmction expressed by the equation

f f(a)~'(c~ - = ~ 1~,=,,'. (41) df(~)

a')da da

with 6'(c~ - ct') - ~gS(c~ - cd) and f ( a ) an arbitrary function of a. Adopting a definition of convergence ill the case of a summation over

O1)strvnl)le ~ilh l)iscrt'le Eillenvnlues 431

an oscillatory flmction with an argument tending to infinity emalo- gous to that adopted in the case of an integral with an oscillatory dependence on an infinite limit, Eq. (40) can be established as a valid identity, and it is left to check Eq. (41) with 6n(c~ - cd) replaced by the summation in Eq. (38).

It is sufficient in Eq. (41) to choose f(c~) to equal or, in which case the condition that (41) imposes on the summation in Eq. (38) can be expressed as the relation

~t-oo / ~+a , ,2~i ~-~ (~ , ' - o " ) , , ~ ' ; ° - s = - 1 . (42)

Evaluation of the integration on its left h~md side allows this relation to be rewritten in the form

+ o o s i l l 2 t rA 7~ • L ,)

7r n j n i ~ - - o o

2 7 r A 9 2 7 r A ] sin - F - n j + • _A cos - ~ - n i ] - - 1 ,

(41)'

which by re-ext)ression of the bracketed quantity in terms of expo- nentials emd use of Eq. (34.b) is reduced to the valid identity

2wA ?l 1_ sin - V - ~ _ 1. (35 ) ' 7r nj

n i = --C~

Equation (38) is therefore shown to be a correct equation. On the other hand, by a similar analysis, Eq. (39) can be

shown to be incorrect. In particular, after the integration on the left ha~ld side of Eq. (39) is performed for n i ¢ n~,, that equation is expressible as the relation

c o s ( n s - ,,.k)~ sin(, ,~ - - k ) ~ co~(n j - , ,k )~ - = = 0 n i ~ nk,

( h i - n k ) ~ ( n j - , ~ k ) ? ( ,~ j - , ~ k ) (39)'

where the zero on the right skte of the equality results from the defin- ing property of the Kroneeker-delta symbol. But, since the cosine of an integral multiple of ~r is non-zero, Eq. (39) ~ is clearly invalid. In fact Eq. (39) can be rescued only if L is infinite, in which case the discrete observable/3 becomes a conthmum observable aald equation (39) is replaced by Eq. (36).

432 Deck and ()zlfirk

Because Eq. (39) is not valid for finite L, it follows that the expression (24.1)) cannot reprcsent an entirely consistent solution of Eq. (23.b); and, since it is not apparent that any consistent solutions of Eqs. (23) exist, it cm~ be inferred that these equations asld the ba- sic commutat ion relation (2) which underlies them, must be invMid for any pair of obscrvaMes where one member of the pair has discrete eigenvalues. If it is then accepted, as assumed in the case of con- t immm observables, that the quaa~tmn formalism, and in particular Eq. (2), is sufficient to allow for a complete expression of the basic uncertainty relation (1), it can be concluded that no two observables caal be conjugate in the exact sense of Eq. (1) in any case where one of the obserwd)les has discrete eigenvalues (a surprising result)! To obtain a valid representation of the conmmtation relation (2), both observables a and /3 need to have an infinite range of eigenvahms; but where this is true, no representation of the commutation relation /~ / exists ill which one of the observal)les is discrete. That relation

requires both obserwtbles to have an infinite range of eigenvMues is in fiwt a reflection of the underlying uncertainty relation (1) ms a consequence of which the uncertainty in the value of one member of a pair of conjugate observal)les must approach infinity as the uncer- tainty in the value of the other member of the pair approaches zero. For the uncertainty in the value of all observable to approach infinity, that observable needs to have aal infinite remge of eigenvaiues.

On the other htul(1, since Eqs. (24) are consistent with all equations derived fi'om the orthogonaiity and completeness relations except Eq. (39), the equations (24) can be said to represent "nearly consistent" solutions of Eqs. (23); and from this it follows that the comnmtation relation (2) can be at least approximately valid in a case of two conjugate observables which do not both have continumn eigenvMues, provided that the eigenvalues of the discrete observable are comlected to the range of eigenvalues of the conjugate eontinmun observable via Eq. (3). Therefore there exists the possibility that a continuum observable with an inherently limited range of eigenvatues c,'m be "nearly conjugate" in the sense of Eq. (1) with a second observable which has discrete eigenvalues quantized in accord with Eq. (3).

Of course, a continuum observable with an inherently limited raslge of eigenvalues is represented by the aslgle variable that specifies the position of an object in spherical coordinates; the limitation on the range of the eigenvMues of the contimmm observable in this case being imposed by the fact that the physically distinct values of the angle are confined to the interval hetween -re and +rr (or 0 and 2~r). Consider, in particular, the continmml observable ~0 represented by the azimuthal angle coordinate of asl object in a plaale perpendicular to a given axis z, aalct assume that there exists an observable denoted g~ which is "nearly conjugate" to ~0 in the sense of Eq. (1). In this

OI;set"vable with I)lscrete Elgrnvnlues 433

case the commutation relation

lO~,, Or,] = ih (43)

exists as a "nearly consistent" equation, provided that the eigenval- ues of t?~ are restricted by Eq. (3) to the allowed values

2rrh (g~)J = 2--'~ni = njh, n i = 0,-t-1,-1-2,.... (44)

The result correctly predicts the quasltized eigenvalues of the z-th coinponent of the orbital angular momentum. But in addition the analysis above indicates that both the coInnmtation relation (43) and the uncertainty relation to which it connects,

ACAg= >_ hi2, (45)

are not entirely consistent with tile a.ssmned completeness relations for the eigenkets of ¢.

The "inappropriateness" of the relation (45) has been noted for a long time.[3] a Here we empha.size that, for the underlying com- nmtation relation between O~, and Or, to be a completely consistent equation, the angle observable ~ needs to have an infirfite range of eigenvalues. But there is a mathematical sense in which any ,'ingle observable casa be said to have an infinite range of eigenvalues, pro- vided only the angles which are greater than 7r or less then - r r arc interpreted to define the total angular rotation of a position vector, counter-clockwise or clockwise respectively, as measured from a given axis. In this nlathematical sense, the uncertainty in the value of an angle observable in m~ uncertainty relation of the fornl (45) casl be said to have. an upper limit of infilfity, and the limits of integration in tim completeness relation with respect to the angle observable can be taken to be mimls and plus infilfity. Given this change in the limits of integration only in the completeness relation for the angle ol, erator O~ in Eq. ~:)X the conmmtation relation between O~, and Oe, casl be said to valid equation. And of course the results derived from the commutation relation (44) are in agreelnent with observations.

(c) Case W h e r e B o t h c~ and fl H a v e D i s c r e t e E i g e n v a l u e s

Finally, by way of an analysis similar to the above it is pos- sible to investigate the implications of the commutat ion relation of Eq. (2) in the case that both observables have discrete eigenvalues. In this case, we find that it is not possible to find any representation of Eq. (2) which is consistent with the assumed orthogonality and

434 Deck u,,d (-)zllirk

completeness relations for the eigenkets of the operators 0~, and 0 B. The result demonstrates the singular role of the uncertainty principle (by itself) in deternfining the entire structure of the physical theory. Because this principle is ahnost entirely restrictive in its content, its role is primarily to restrict the mathematical formalism in terms of which the physicM theory caJ1 be expressed. Given the stated result, and a.ssunfing that the comnmtation relation (2) provides for a com- plete expression of the uncertainty relation (1), it can be concluded that no two observables caJa be conjugate in the sense of relation (1) whenever both observables have discrete eigenvalues (a conclusion which seems to be consistent with observations).

3. C O N C L U S I O N

The conclusions arrived at here with respect to the eigem values of conjugate observables follow strictly from the discrete or continuous character of the spectra of the eigenvalues. In particular Eq. (3) is not specific to the ca.se of an angle observable. On the other hand the inconsistency in Eq. (45) resulting from the finite range of the emgle variable has been recognized by many others. (See, for exmnple, Refs. 2-7.)

The standard approach to elimination of the inconsistency chooses to retain the conmmtation relation of Eq. (43) but eliminate the connection between the commutation relation mad the uncer- tainty relation (45) on the basis of the non-setf-adjointness of the operator Or, in the space of the non-periodic tingle observable. The argument suggests replacement of the aalgle observable 4, by an ob- servable which is periodic in ¢, most simply either sin4, or cos 4', and the subsequent replacement of the uncertainty relation (45) by relations of the form 4

h (A sin 4,)Ae~ _> 5 < cos ¢ >,

h (A cos 4,)Ae. > - < s i n 4 , > .

(46)

The non-self-adjointness (in the conventional sense) of the operator Or, with respect to eigenkets of 4' can be used in addition to inval- idate Eq. (23.b) in the c~e 0~, = O~o by preventing the operator Or, from acting to the left in the bra-ket product < g,jl(Ot, O~ - O,Ot, )[g.,~ >. Instead we choose to retain this basic property of an ohservable operator expressed in the second of Eqs. (4) s and inter- pret the singular equation which derives from it,

(g~j -gzk) < g, i [O~lg:,) = -ih~ik, (47)

Ol)~e~tthle ~*ith Discrete ElgenvMue~'i 435

as the anMogue of the even more singular equation (23.a). The commutation relation (43) is a consequence of the defini-

tion of Oe, as the generator of an infinitesimal rotation; and, because the qualitative content of Eq. (45) is not questioned [5}, we choose to ~ssert the general validity of the connection between relations (43) mad (45). On the other hand we emphasize that a conmmtation relation of the form (2) which involves an operator with a discrete spectrum can be conmstent with the orthogonality and completeness relations for the eigenkets of the conjugate operator only if the latter operator has a continuous spectrum in a limited range; and this leads to a difficulty in the interpretation of the uncertainty in the value of the continuum observable in the case where the uncertainty in the discrete obser-atble vasfishes.

R E F E R E N C E S

1. H. P. Robertson, Phys. Rev. 34, 163 (1929). 2. P. JordmL Z. Phys. 44, 1 (1927). 3. D. Judge, Phy.~. Left. 5, 189 (1964); Nuovo Cimento 31, 332

(1964). 4. R. Jackiw, J. Math. Phy.q. 9, 339, (1968). 5. P. Carruthers and M. M. Nieto, Rev. Mod. Phys. 40,411 (1968). 6. D. Deutsch, Phy.~. Rev. Let¢. 50,631 (1983); M. H. Partovi, Phy.~.

Rev. Lett. 50, 1883 (1983). 7. E. Breitenberger, Found. Phys. 15, 353 (1985). 8. P. A. M. Dirac, The Principte.~ of Quantum Mechanic.% 4th edn.

(Cl~endon, Oxford, 1958).

NOTES

1. However, it is significant that the result in Eq. (3) is obtained without reference to either asl angle or an aalgular monlentum variable.

2. Since Eq. (16) is equivalent to a derivative of Eq. (15) only if the order of differentiation and integration can be interchanged in the derivative of (15) (as already assmned in the derivation of Eqs. (9)), it is advisable to check the separate equations inde- I)endentiy.

3. The maximmn uncertainty in ¢ is repre~nted by the root-mean- square value of the angle in the case where the probal)ility of a particular vMue of ¢ is uniform throughout the range -rr to rr; and because this maximmn uncertainty equals 7r/3, an alterna- tive to the uncertainty relation (45) has been proposed in (Ref. 3) in the form Ae~A¢/[1 -- 3(ZX¢)2/,~1 > h/2.

436 Deck and Oztfirk

4. The uncertainty in the value of an observable is conventionally equated to the root-mean-square deviation from its expectation value. However the perceived problematic form of Eq. (45) has inspired several alternative definitions (in Ref. 3-7).

5. A more complete justification of this approach requires more space thaJa casl be allotted here; but the reader can be assured that the subject ha.s been carefillly investigated.