Quantum mechanics in finite-dimensional Hilbert spaceA. C. de la Torrea) and D. GoyenecheDepartamento de Fı´sica, Universidad Nacional de Mar del Plata, Funes 3350, 7600 Mar del Plata,Argentina

~Received 28 May 2002; accepted 21 August 2002!

The quantum mechanical formalism for the position and momentum of a particle on aone-dimensional lattice is developed. Some mathematical features characteristic offinite-dimensional Hilbert spaces are compared with the infinite-dimensional case. The constructionof an unbiased basis for state determination is discussed. ©2003 American Association of Physics Teachers.

@DOI: 10.1119/1.1514208#

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I. INTRODUCTION

The quantum mechanical description of a physical sysfor which the observables have a continuous set of varequires a continuous set of states that belong to an infindimensional Hilbert space. However, there are many physsystems for which the observables take on a discrete sevalues and their corresponding states belong to a discretefinite set, whose formal description is a finite-dimensionHilbert space. The best known example of such an obsable is angular momentum: a physical system with angmomentumJ5Aj ( j 11) is described by a Hilbert space odimension 2j 11. A less well-known example, which we widiscuss in this paper, involves the description of the positand momentum observables in a finite-dimensional Hilbspace, where the position and momentum observables dotake on a continuum of values, but instead are restrictevalues on a lattice.

The possibility of formalizing quantum mechanics infinite-dimensional Hilbert space was recognized very earl1,2

and has found renewed interest in many applications in,example, quantum cryptography, quantum computers,quantum optics. One important practical motivation fstudying such systems is that a computer simulation of ption and momentum will necessarily involve a finite numbof sites. A highly speculative motivation is that the existenof a fundamental length scale, that is, a measure of lenbelow which the concepts of distance and localizationcome meaningless, would make a discrete quantum mecics more appropriate than a continuous one.

To make this discussion more useful from the didacpoint of view, the formalism of quantum mechanics infinite-dimensional Hilbert space will be presented in a costructive way, where all the steps are logically connectThis work may therefore be a useful complement to a tebook where quantum mechanics in an infinite-dimensioHilbert space is developed. Another didactic feature of twork is that finite-dimensional quantum mechanics requmany interesting mathematical tools such as finite sumsthe discrete Fourier transform, which are not usually psented at the undergraduate level. Furthermore, the impodifferences between finite- and infinite-dimensional Hilbspaces are emphasized, and the limit of infinite dimensioconsidered.

II. NOTATION AND DEFINITIONS

We will consider a particle in a one-dimensional periodlattice with N sites and lattice constanta. The quantum me-

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chanical treatment of this system requires anN-dimensionalHilbert spaceH. Given any two elements of this spaceF andC, we will denote their internal product byF,C&. We willuse operators of the formA5C^F,•&, where the dot indi-cates a space holder to be occupied by the Hilbert spelement on which the operator acts. The corresponding Hmitian conjugate isA†5F^C,•&.

Although we do not need to choose any particular repsentation for the abstract Hilbert spaceH, it may be conve-nient for didactic reasons to specialize the formalism tothree- or four-dimensional Hilbert space whose elementsCare column vectors of complex numbers.~It is better to avoidtwo-dimensional spaces because they have some peccharacteristics that do not generalize to higher dimensio!In this caseC,•& represents a complex conjugate row vecand operators are square matrices. This special representis recommended for clarity, but it is important to emphasto students that the formalism of quantum mechanics candeveloped in an abstract Hilbert space and a particularresentation is never required. The mathematical beautyquantum mechanics is, indeed, most apparent in the absformulation.

Any basis$wk% in the Hilbert space will haveN elementslabeled by an indexk with the values 2 j ,2 j 11,2 j12,...,j 21,j , with j 5 1

2,1,32,2,52,..., corresponding to the di-mensionsN52 j 1152,3,4,5,... This choice of labels hasome advantages and disadvantages. The main virtue ofsymmetric labeling is that it corresponds to the physiproperty that the position and momentum can have posiand negative values. Its main shortcoming is that theremany summations and results that are usually given in tbooks with integer indices running from 0 toN21. To re-duce this shortcoming, we give in Appendix A some of thesums with a symmetric index. For the choice of symmetindices, we must keep in mind that, for evenN, j takeshalf-odd-integer values and we need to be careful with ninteger exponents.

We will adopt a useful notation for the principalNth rootof the identity, defined by

v5ei ~2p/N!. ~1!

Integer powers ofv build a cyclic group with the importanproperty that

15vNn5v (2 j 11)n, ;n50,61,62,.... ~2!

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III. POSITION AND MOMENTUM

The position of the particle in the lattice can take avalue~eigenvalue! ax, where the discrete numberx can takeany value in the set$2 j ,2 j 11,...,j 21,j %. The state of theparticle in each position is represented by a Hilbert spelementwx , and the set$wx% is a basis inH. We can writethe position operatorX as

X5 (x52 j

j

axwx^wx ,•&, ~3!

which clearly satisfiesXwx5axwx . We can now construct atranslation operator Twith the property that

Twx5H wx11 ~xÞ j !

~21!N21w2 j5vN jw2 j ~x5 j !.~4!

We will later explain the reason for defining the operatorT tobe periodic for odd dimensions, butantiperiodic for evendimensions. This operator is given by

T5 (x52 j

j 21

wx11^wx ,•&1~21!N21w2 j^w j ,•&, ~5!

with its Hermitian conjugate

T†5 (x52 j

j 21

wx^wx11 ,•&1~21!N21w j^w2 j ,•&. ~6!

It is straightforward to check that this operator is unitaTT†5T†T51, and therefore its eigenvalues are compnumbers of unit modulus and their eigenvectors form a b~see Appendix B!. Now let $fp% and $lp% be the eigenvec-tors and eigenvalues ofT for p52 j ,2 j 11,...,j 21,j . Inorder to determine them, we expandfp in terms of$wx% andconsider

T (x52 j

j

^wx ,fp&wx5lp (x52 j

j

^wx ,fp&wx . ~7!

From Eq.~4! we obtain

^wx21 ,fp&5lp^wx ,fp&, for xÞ2 j ~8a!

^w j ,fp&vN j5lp^w2 j ,fp&. ~8b!

Up to an arbitrary phase that can be absorbed in the detion of fp , the solution of the set of Eqs.~8! is

^wx ,fp&51

ANvpx and lp5v2p, ~9!

where we have normalizedfp . The two bases$wx% and$fp% are then related by

wx51

AN(

p52 j

j

v2pxfp , ~10!

and

fp51

AN(

x52 j

j

vpxwx . ~11!

Except for the symmetric index and a different factor, E~10! and~11! are the discrete Fourier transform. Note thawe had not defined the translation operator to be antisymm

50 Am. J. Phys., Vol. 71, No. 1, January 2003

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ric for N even, we would have not obtained such a simrelation as in Eq.~9!, and we would have obtained differenexpressions for even and oddN. In other words, we chooseto define the translation operator so that we would obtaisimple relation between the bases.

With the eigenvalue given in Eq.~9!, the eigenvalue equation for T becomes

Tfp5v2pfp , ~12!

or, equivalently,

T5 (p52 j

j

v2pfp^fp ,•&. ~13!

We can now construct a Hermitian operatorP as a superpo-sition of projectors in the basis$fp%,

P5 (p52 j

j

gpfp^fp ,•&, ~14!

whereg is a real constant to be determined later. Clearly,operatorP is Hermitian and satisfies the eigenvalue equatPfp5gpfp . From this eigenvalue equation, together wEq. ~12!, and the power series expansion of the exponenwe can show that

T5expS 2 i2p

N

P

g D . ~15!

We can now give a physical interpretation to the operatorP.Equation~15!, together with Eq.~4!, implies thatP is thegenerator of translations in the position observable. We idtify this observableP with the momentum, as is done iclassical mechanics. If the position observable takes onues on a lattice with lattice constanta, then the momentumobservable must also assume values on a lattice with laconstantg. In Sec. IV we will see that these values mustrelated byga52p/N, that is, the momentum lattice is threciprocal lattice of the position.

In an manner identical to what was done before, we cnow construct a unitary operatorB that ‘‘boosts’’ the mo-mentum states,

Bfp5H fp11 , pÞ j ,

~21!N21f2 j5vN jf2 j , p5 j ,~16!

and show that

Bwx5vxwx , ~17!

B5 (x52 j

j

vxwx^wx ,•&, ~18!

and

B5expS i2p

N

X

a D . ~19!

IV. THE COMMUTATION RELATION AND THELIMIT N\`

Quantum mechanics textbooks emphasize that the posand the momentum operators satisfy the commutation rtion @X,P#5 i ~we use units such that\51). However, inmost cases it is not mentioned that this commutation relais false in a finite-dimensional Hilbert space. It becom

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clear that@X,P#Þ i , because it can be shown that the comutation relation@X,P#5 i implies that the operatorsX andP are unbounded. However, in a finite-dimensional Hilbspace,all operators are bounded; therefore such a commtion relation is impossible in a finite-dimensional Hilbespace. Indeed, if the Hilbert space is finite, the basis musfinite, and a finite basis, in turn, implies that the operatmust have finite-dimensional matrix representations. Butfinite-dimensional matrices Tr(XP)5Tr(PX) always, soTr@X,P#50, showing that@X,P#5 i leads to a contradictionin a finite-dimensional Hilbert space.

An explicit calculation of the commutator in the positiorepresentation, that is, in terms of the basis$wx%, results in

@X,P#5ag (k52 j

j

(s52 j

j

(r 52 j

j

k~s2r !1

N

3expS i2p

Nk~s2r ! Dws^w r ,•&. ~20!

The sum overk can be performed, but it is advantageousleave it as is. We can now see that in the continuous limwhereN→`, a→0, andg→0, but Nag→const, the abovecommutator approaches the valuei , provided thatagN52p. In this limit, the sums over discrete indicesk, s, andr become integrals over continuous variablesk, s, and r,according to

A2p

Nk→k, A2p

Ns→s, A2p

Nr→r ~21a!

A2p

Nws→w~s!, A2p

Nw r→w~r!, (

2 j

j

→E2`

`

.

~21b!

The continuous limit is then given by

@X,P#→ iagN

2p E2`

`

dsE2`

`

dr21

2p E2`

`

dk ik~s2r!

3exp„ik~s2r!…w~s!^w~r!,•&. ~22!

The sum overk, which was left unperformed, reduces tosimple form in the continuous limit. Indeed, the integral ovk is a well-known representation of Dirac’s delta functioTherefore, we have

@X,P#→ iagN

2p E2`

`

dsE2`

`

dr d~s2r!w~s!^w~r!,•&

5 iagN

2p E2`

`

ds w~s!^w~s!,•&5 iagN

2p1, ~23!

where the last equality in Eq.~23! is the completeness relation. Hence, the usual commutation relation for the contious case is recovered, provided that

agN52p. ~24!

V. STATE AND TIME EVOLUTION

At any instant of time, the state of the particle will bdetermined by a Hilbert space elementC. We can representhis state in the position or momentum representation, thain the bases$wx% or $fp%:

51 Am. J. Phys., Vol. 71, No. 1, January 2003

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is,

C5 (x52 j

j

cxwx5 (p52 j

j

dpfp , ~25!

where the complex coefficientscx anddp are related by thediscrete Fourier transformation,

dp51

AN(

x52 j

j

v2pxcx , cx51

AN(

p52 j

j

vpxdp . ~26!

Their absolute values squareducxu2 and udpu2 represent theprobability distributions for position and momentum, respetively. Let C(t0) be the state of the system at timet0 , whichwe choose to bet050. In the Schro¨dinger picture, this statewill evolve according to the time evolution unitary operatgiven in terms of the HamiltonianH as

Ut5exp~2 iHt !. ~27!

This description of the time evolution is equivalent to Sch¨-dinger’s equation if time is represented by a continuous vable. However, in some cases it may be convenient tosume that time also takes discrete values and Eq.~27! cannotbe transformed in to a Schro¨dinger equation. If the state igiven in the position or the momentum representation,coefficients of Eq.~25! will become functions of time,cx(t)anddp(t).

Let us consider a free particle with HamiltonianH5P2/2m. In the momentum representation the coefficieare simply given by

dp~ t !5dp~0!expS 2 ig2p2

2mt D5dp~0!v2p2t/t, ~28!

where we have introduced a time scalet, defined by

t52ma

g. ~29!

In the position representation we have

cr~ t !5 (x52 j

j

cx~0!1

N (p52 j

j

v (p(r 2x)2p2t/t). ~30!

The second summation in Eq.~30! is a discrete Fourier transform that becomes, in the continuous limit, a Fourier integtransform of a Gaussian function with a well-known resuIn the discrete case, the algebra is much more difficult athe summation cannot be evaluated in general. It took Gafour years working ‘‘with all efforts’’3 to evaluate a similarsummation~the ‘‘Gauss sum’’! for some special values of thparameters. In any case we can see that the state is perC(t1T)5C(t) with period T5Nt if N is odd andT54Nt if N is even.

VI. STATE DETERMINATION AND UNBIASEDBASES

At an early stage in the development of quantum mechics, Pauli4 raised the question of whether knowledge of tprobability density functions for position and momentuwas sufficient to determine the state of a particle. Becaposition and momentum are all the~classically! independentvariables of the system, it was conjectured that this queswould have an affirmative answer. However, different stawith identical probability distributions were found. A reviewof these issues with references to the original papers ca

51A. C. de la Torre and D. Goyeneche

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found in Refs. 5–7. If we consider the similar problemclassical statistics, that is, find the combined distributgiven the marginal distributions, we should not be surpristo find out that the Pauli question does not have a posianswer. The marginal probability distribution functionstwo random variables uniquely determine the combinedtribution functiononly if they are uncorrelated, that is, independent random variables. Position and momentumhowever, always correlated; that is the essence of Heiberg’s uncertainty principle, and therefore we should notpect that their distributions uniquely determine the quantstate.

Explicitly stated, the Pauli question is, can we find theof N complex numbers$cx% that determine the state in Eq~25! given the knowledge of the sets$ucxu2% and $udpu2% re-lated by Eq.~26!? Let us notice that the state has an arbitrphase and is normalized; therefore, we only need to2N22 real numbers to determine the state. The numb$ucxu2% and$udpu2% are not independent, because the numbof each set are probabilities and should add to 1. We thfore have 2N22 equations at our disposal in order to fin2N22 unknowns. However, the equations arenot linearandare insufficient to determine the state unambiguously. This another very important feature in these 2N22 equations.We will see that not every set of data$ucxu2% and$udpu2% arecompatible. The 2N22 equations have solutions only if thposition and momentum data satisfy a number of relatioThis constraint on the data is just Heisenberg’s uncertaprinciple and is a consequence of the relations in Eq.~26!.These ideas are clarified by an example withN52.

Let $w2 ,w1% and$f2 ,f1% be the position and momentum bases in a two-dimensional Hilbert space. Their interproduct is given by Eq.~9!. An arbitrary state, normalizedand with a fixed phase, is determined by 2N2252 numbers0<%<1 and 0<a<2p:

c5%eiaw21A12%2w1 . ~31!

The independent data on position isu^w2 ,c&u25%2, whichdirectly determines% and the independent data on mometum is u^f2 ,c&u25Ã2. With this last data we must detemine a. Using that^f2 ,w6&5exp(6ip/4)/A2, we obtainafter some algebra that

sina5Ã221/2

%A12%2. ~32!

Equation~32! can have a solution only if

U Ã221/2

%A12%2U<1. ~33!

If we square both sides of Eq.~33! and rearrange the termswe find

~Ã221/2!21~%221/2!2<~1/2!2. ~34!

This relation is indeed the uncertainty principle:%250 or 1,that is, exact localization inw1 or w2 , implies Ã251/2,that is, maximal spread in momentum and, vice versa, emomentum (Ã250 or 1) implies maximal spread in postion (%251/2). Even if the data is consistent with the unc

52 Am. J. Phys., Vol. 71, No. 1, January 2003

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tainty principle, there is an ambiguity in the solution of E~32! because ifa is a solution, thenp2a is also a solution.This ambiguitycannot be solved with the given data anrequires more experimental information. We will next cosider what observables we can measure to determine thewithout ambiguity.

From the previous example it is clear that we need furtinformation besides the distribution of position and of mmentum in order to determine the state of the particle. Tadditional information must involve an observable depeing on both positionandmomentum, because any observabdepending on only one of them will not bring new indepedent information. Some candidates might beX1P or thecorrelationXP1PX or any functionF(X,P) symmetric orantisymmetric under the exchangeX↔P. Perhaps the beschoice of an observable that provides information onsystem that is not available from the knowledge of the potion and momentum distributions, is an observable whassociated basis isunbiasedwith respect to the position anto the momentum bases. Two bases in a Hilbert spaceunbiased when they are as different as possible in the sthat any element of one basis has the same ‘‘projection’’all elements of the other basis. More precisely, the moduof their internal product is a constant for all pairs. Unbiasbases are candidates for the quantum mechanical descriof classically independent variables such as position andmentum; indeed, we have from Eq.~9! that u^wx ,fp&u51/AN,;x,p. The problem of state determination leadsthen to the search of a basis$hs% unbiased to$wx% and withrespect to$fp%.

The importance of unbiased bases associated with ncommuting observables was recognized by Schwinger8 longago, but the existence and explicit construction of maximsets of mutually unbiased bases for any dimension is stilopen problem. When the dimensionN is a prime number, aset ofN11 mutually unbiased bases was presented,9,10 andthe same could be achieved ifN is a power of a primenumber.11 To find unbiased bases we will follow the methogiven in Ref. 12. We have seen that the eigenvectors$fp% ofthe operatorT that produces a translation or shift on the ba$wx% build an unbiased basis to$wx%. This result is general-ized in Ref. 12, where it is shown that, ifN is prime, theeigenvectors of the unitary operatorsT, B, TB,TB2,...,TBN21 build a set ofN11 mutually unbiased baseswhereT andB are the translation operators for position amomentum defined in Eqs.~4! and~16!. In our case we wantto find a set of only three unbiased bases, and thereforeconsider only the first three operators that provide unbiabases for anyN ~prime or not!. The first two operators provide the bases$fp% and $wx%, which are related by the discrete Fourier transform and are clearly unbiased. Weeasily prove using Eqs.~4!, ~12!, ~16!, and~17! that the op-eratorTB is a shift operator forboth bases$fp% and $wx%,and therefore its eigenvectors$hs% build a basis unbiased toboth of them. We have indeed

TBwx5H vxwx11 , xÞ j ,

v22 j 2w2 j , x5 j ,

~35!

and

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TBfp5H v2(p11)fp11 , pÞ j ,

v22 j 2f2 j , p5 j .

~36!

The eigenvectors ofTB in the position representation arfound by expandinghs in the basis$wx%, and using Eq.~35!and the relation

TBhs5vshs . ~37!

In this calculation we must use with care the modular maematics defined in Eq.~2!, resulting in the eigenvectors

hs51

AN(

x52 j

j

v~1/2! x22(s1 1/2)xwx . ~38!

A similar calculation yields the eigenvectors ofTB in themomentum representation,

hs51

AN(

p52 j

j

v2 ~1/2! p22(s1 1/2)pfp . ~39!

Clearly $hs% is unbiased to$wx% and to$fp%. The analyticalcalculation of discrete Fourier transforms is much more dficult than the Fourier integral transform, and therefore itof interest that, from Eqs.~38! and ~39! we can obtain thediscrete Fourier transform for a family of sequences. Ifequate Eqs.~38! and ~39! and expandwx in terms of$fp%,we obtain

1

AN(

x52 j

j

v~1/2! x22bxv2px

5v2 ~1/2! p22bpH b50,61,62,... ~N even!

b561/2,63/2,... ~N odd!.~40!

We rewrite this result in terms of the asymmetric indicmore common in the mathematical literature as

1

AN(n50

N21

~21!nv~1/2! n22bnv2mn

5~21!mv2 ~1/2! m22bm, ;b integer. ~41!

The unbiased basis that was found is of course not uniquwas generated by the operatorTB, but there are many otheoperators whose eigenvectors build an unbiased basis to$wx%and to $fp%. Indeed, any combination,TnBm or BmTn,wheren andm are not divisors ofN, could do the job.

We will now find the physical meaning for the basis$hs%.That is, we would like to find a Hermitian operatorS(X,P)with $hs% as eigenvectors. That is,

TB5exp„iS~X,P!…. ~42!

In terms of the operatorsX and P, and using the relationagN52p, we have

exp„iS~X,P!…5exp~2 iaP!exp~ igX!. ~43!

Note that here wecannot use the Baker–Campbell–Hausdorff relationePeX5eP1X2 i /2 that is valid forN→`,where the commutator@X,P# is a constant. If we could usit, then the operatorS would be simply equal togX2aP. Itis possible to find the eigenvectors of the operatorgX2aP, but the basis so obtained is not unbiased to either$wx%or $fp%; however, it becomes unbiased in the infinit

53 Am. J. Phys., Vol. 71, No. 1, January 2003

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dimensional limit.13 The relation ofS to X and P is notsimple, but at least we can prove thatS is antisymmetricunder the exchangeP↔X anda↔g. Indeed, from the Her-mitian conjugate of Eq.~43!, we have

exp„2 iS~gX,aP!…5exp~2 igX!exp~ iaP!

5exp„iS~aP,gX!…. ~44!

VII. CONCLUSION

We have presented the quantum mechanical formalismthe position and momentum of a particle in a ondimensional periodic lattice in a way that complementsdiscussion of the infinite-dimensional case presentedquantum mechanics textbooks. Several mathematical suties related to the difference between infinite- and finidimensional Hilbert spaces and of modular mathemawere noted. We discussed the physical and mathematicaevance of unbiased bases and, as a consequence fromconstruction of such a basis, the discrete Fourier transffor a family of sequences is given.

It is a strongly recommended that the reader reproducethe results in terms of the asymmetric indices running fromto N21. The calculations of the eigenvectors of the operaTB are useful exercises for modular mathematics.

ACKNOWLEDGMENTS

This work received partial support from ‘‘Consejo Nacinal de Investigaciones Cientı´ficas y Tecnicas’’ ~CONICET!,Argentina.

APPENDIX A: FINITE SUMS WITH SYMMETRICINDICES

All sums appearing in this appendix can be derived frothe fundamental expression

(k50

N21

zk512zN

12z, ~A1!

valid for any complex numberz. For the symmetric indexthe sum is

(k52 j

j

zk5zN/22z2N/2

z1/22z21/2 for H j 5 12 ,1,32 ,2,...,

N52 j 1152,3,4,5,...,z complex.

~A2!

If z takes the special valuesz5exp„i (2p/N) r …5v r with ran arbitrary number, then

(k52 j

j

vkr5sin~pr !

sinS p

Nr D . ~A3!

In our case,r will often assume integer or half-odd-integevalues. For these cases we have

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s,’’

(k52 j

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vkr

5H ~21!n(N21)N for r 5nN, n50,61,62,...,

0, for r 561,62,63,...ÞnN,

2v r /2

12v r , for r 561

2,6

3

2,6

5

2,... .

~A4!

The first two cases correspond, with the asymmetric indexthe well-known result(k50

N21vkr5Nd r ,nN , and we see thathis choice leads to simpler mathematics. The third cabove must be handled with care in numerical evaluatibecause the numerator is the fourth root of exp(i „2p/N) u…,with u an odd integer. Therefore it has four possible numecal results. The denominator also has two possible resultformal derivative of Eq.~A3! with respect to the parameterrleads to the result

(k52 j

j

kvkr5i

2

sin~pr !cosS p

Nr D2N cos~pr !sinS p

Nr D

sin2S p

Nr D .

~A5!

By taking more derivatives, again with respect tor , we canobtain other summations involving higher powers ofk.

APPENDIX B: BASES RELATED TO UNITARYOPERATORS

In most textbooks it is proven that the nondegenerategenvalues of a Hermitian operator are real and their eigvectors are orthogonal. We give here the corresponding pfor unitary operators.

Let T be a unitary operator andlk andfk the nondegen-erate eigenvalues and normalized eigenvectors. Then,will prove that ulku251 and ^f r ,fk&5d rk . From Tfk

5lkfk andT†T51, it follows that

54 Am. J. Phys., Vol. 71, No. 1, January 2003

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ulku25^Tfk ,Tfk&5^fk ,T†Tfk&51. ~B1!

In order to prove orthogonality, consider

Tfk5lkfk→^f r ,Tfk&5lk^f r ,fk&, ~B2!

T†f r5l r* f r→^T†f r ,fk&5l r^f r ,fk&. ~B3!

If we subtract these equations from each other, we obta5(lk2l r)^f r ,fk&. Because the eigenvalues are nondegerate, the productf r ,fk& must vanish forkÞr .

BecauseT is bounded, the completeness of the eigenvtors can be proved in the usual way, and therefore$fk% is abasis.

a!Electronic mail: dltorre@mdp.edu.ar1H. Weyl, Theory of Groups and Quantum Mechanics~Dover, New York,1931!.

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54A. C. de la Torre and D. Goyeneche