Studies on Phase and Coherence in Quantum Systems by K Eswaran

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STUDIES ON

PHASE AND COHERENCE IN QUANTUM

SYSTEMS

A THESIS

Submitted to the University of Madras

in Partial fulfilment of the requirement

of the Degree of

DOCTOR OF PHILOSOPHY

By

KUMAR ESWARAN1

Department of Theoretical Physics

University of Madras

Madras 6000025

INDIA

January 1973

1The support of the University Grants Commission is gratefully acknowledged



PREFACE

This thesis embodies the results of the investigations carried out by the au-thor during the period 1969-1972 in the Department of Theoretical Physics of the University of Madras, on the phase and coherence aspects of quantum sys-tems.The thesis consists of four chapters and an appendix. Chapter II embodies theresults of our studies on generalized phase operators for the quantum harmonicoscillator. In Chapter III we present an approximate solution for the energylevels of the anharmonic oscillator, which we obtain through the use of actionand phase variables. Chapter IV concerns itself with the behaviour of smallwavepackets in spherically symmetric potentials and, in particular, the questionwhether there exist any wave packets which remain coherent for all time.

The work reported in this thesis has been done in collaboration with andunder the guidance of Professor P.M. Mathews. It forms the subject matter of five papers, two of which have already been published:

1. On the Generalized Phase operators for the Quantum HarmonicOscillator, Il Nuovo Cimento 70 B, pp 1 -11 (1970)

2. On the Energy Levels of the Anharmonic Oscillator,(with P.M. Mathews), Lett. al Nuovo Cimento 5, pp 15 - 18, (1972)

and three others (see footnote)1 which are in the course of publication.

To the knowledge of the author the results of the thesis have not formed any

part of any earlier thesis.

ACKNOWLEDGEMENTS

The author is very deeply indebted to Professor P.M. Mathews withoutwhose inspiring guidance and continuous encouragement throughout the entireperiod of research, this work could never have been completed.

He is grateful to Dr. M. Seetharaman and Mr. J. Jayaraman for fruifuldiscussions. He is also grateful to his colleagues Mr. Lakshmanan, Mr. SekharRaghavan and Mr. J. Prabhakaran for their cooperation.

He wishes to acknowledge his gratitude to the University Grants Commis-sion for financial assistance in the form of a research fellowship.

Finally thanks are due to Mr. Mohamed Jawad for his excellent typing of the thesis.

1NOTE ADDED: (by Author) These three papers have subsequently been publishedand their references are as under:

3. Semi-Coherent States of the Quantum Harmonic Oscillator, (with P.M. Mathews),Il Nuovo Cimento 17 B, pp 332 - 335 (1973)

4. Simultaneous Uncertainties of the Cosine and Sine Operators, (with P.M. Mathews),Il Nuovo Cimento 19 B, pp 99 - 104 (1974)

5. On the Evolution of Wave Packets for Particles in Central Potentials,(with P.M. Mathews), Journal of Physics A, 7, pp 1547 - 1556 (1974)

1



Chapter 1

Introduction

This thesis deals with some of the problems of the quantum theory which de-spite the varied successes of the theory since it first originated [1-4], have yetremained unresolved. Broadly speaking the problems we consider have to dowith the role of the phase variable in relation to specific quantum systems (theharmonic and the anharmonic oscillators) and the question of coherence in threedimensional systems.

1. The Phase Operators for the Quantum Harmonic Oscillator

The problem of the phase variable for the quantum harmonic oscillator ariseswhen one tries to find a well behaved quantum mechanical opeator correspond-

ing to the phase or angle variable φ (canonically conjugate to the action variableJ) of the classical oscillator. The existence of a operator which is conjugate tothe number operator was postulated many years ago by Dirac [5]. Dirac pro-posed to introduce a phase operator φop through the definition

a = exp(iφop)N 1

2 , (1.1)

where N is the number operator:

N = (N +1

2), N = a† a (1.2)

a and a† being the familiar annhilation and creation operators obeying thecommutation rule

[a, a†] = 1 (1.3)

The equation (1.1) was meant to be the analogue of the classical expression

(mω

2)1

2 (x + ip/mω) = eiφJ 1

2 , (1.4)

which follows from the definition of the classical phase through

x = (2J

mω)1

2 cosφ, p = (2Jmω)1

2 sinφ, (1.5)

If exp(iφop) of (1.1) is taken to be an unitary operator, as one would expect,then the sustitution of (1.1) in (1.3) would immediately yield

exp(iφop) N exp(−iφop) = N + 1, (1.6)

3



But this cannot be: a similarity transformation of N cannot change it to N+1

since the spectrum of eigenvalues of the two operators cannot coincide. In factthe impossibility of defining an unitary operator exp(iφop) through (1.1) wasfirst pointed out by Susskind and Glogower [6]. Indeed if one assumed theexistence of such an operator satisfying the commutation relation

[N, φop ] = i (1.7)

suggested by the Poisson bracket relation {φ, J } = 1 , inconsistencies wouldimmediately arise. 1 For on sandwiching (1.7) between the number states |mand |n one obtains

(m − n)m| φop |n = i δmn, (1.8)

which would require that expectation values of the supposedly hermetian oper-ator φop should be infinite and imaginary!

Though an unique phase operator cannot be had for a quantum harmonicoscillator, one can introduce the notion of phase as is shown in the pioneeringwork of Louiselle[10 ] and the subsequent paper of Susskind and Glogower [6],by the use of a pair of hermetian operators called the cosine and sine operators.These operators 2 are intended to be the quantum equivalent of the cosine andsine of the classical phase. The commutation relations of C and S with N aresuggested by the Poisson brackets of J with cosφ and sinφ :

[C, N ] = i S and [S, N ] = −i C (1.9)

or equivalently, withU = C + iS , (1.10)

[U, N ] = U ; (1.11)

from (1.11) it is easy to see that U is a step down operator for the numbereigenstates.

Taking,U |n = (1 − δn0)|n − 1 (1.12)

Susskind and Glogower [6], have solved the eigenvalue problem for the C and Soperators, and shown that their eigenvalue spectra are continuous and fill therange (−1, 1) and that the eigenstates form a complete set.

1Varga and Aks, Ref. [7], claim to have obtained a representation of equation (1.7) in abasis related to the positive momentum eigenfunctions of a particle in a one dimensional box

(−1/2, 1/2), see also Refs. [8] and [9]. Their argument rests on the identity of the eigenvaluespectra of VP (P,Q being the momentum and position operators of the particle in the boxand V the projection operator to the states of non-negative momentum) and of the numberoperator N, from which they infer that these positive momentum states and the numbereigenstates are related by an unitary transformation. They propose then that φop be definedthrough the unitary transformation V QV → φop . However, their arguments seem untenable:while the zero eigenvalue of N is nondegenerate, that of VP is infinitely degenerate(since allnegative momentum eigenstates of P become eigenstates belonging to the eigenvalue zero of VP). Thus the assumed unitary transformation does not exist.

2A similar approach as has been employed by Louisell [10 ] is useful in dealing with theproblem of the azimuthal angle conjugate to Lz in angular momentum theory, eg. see Ref.[11], for the discussion of the phase problem in angular momentum. Carruthers and NietoRef. [12], provide an excellent review on these subjects.

4



The cosine and sine operators defined through (1.10) to (1.12) are not the

only possible operators that can be obtained. It was pointed out by Lerner [13]that equation (1.12) is not really mandatory, one could have more generally

U |n = f (n)|n − 1 ; with f (0) = 0 , (1.13)

where f (n) is arbitrary except for certain restrictions that are needed to ensurethat the C and S have eigenvalue spectra extending from -1 to +1. Lerner hasgiven examples for various forms for f (n) , following from the use of variouskinds of symmetrizations in going from the classical to the quantum variables.The eigenstates for the C and S operators for a particular choice of f (n) havebeen worked out by J. Zak [11]. Since U is not an Hermetian operator the eigen-values of U will not be necessarily real. Lerner, Huang and Walters [14] haveproved the interesting fact that all complex numbers z with |z| < 1 are eigen-

values of U. The corresponding eigenstates form an overcomplete set. Recently,some of the spectral properties of the phase operators have been obtained alsoby Ifantis [15],[16] using the observation that C and S belong to a special classof tridiagonal operators.

We have made a systematic study of the eigenvalue problem of the gener-alized phase operators C and S defined by Lerner through (1.13), (1.10) and(1.11) above. The particulars of this investigation are presented in Chapter II.We prove that the coefficients of the transformation from the number eigenstatesto the eigenstates of the C and S are quite generally orthogonal polynomials inthe eigenvalue parameter. This enables us to show that the eigenstates of thegeneralized C and S operators form a complete set. It is interesting that suchproperties are demonstrable even when f (n) is arbitrary. In the special case of f (n) = (1 − δn0) the coefficients of the expansion of the C and S eigenstates interms of the number eigenstates reduce to the familiar Gegenbauer polynomials.We also treat the problem of the minimum uncertainty state for the general-ized phase operators, following the approach of Jackiw [17] and prove that thestates which minimize [(C )(S )/i[C, S ]]2 form an overcomplete set and arelabelled by a parameter λ taking values within an ellipse in the complex λ plane.The coherent state representation of quantities of special interest involving theC and S operators is then presented. We show that it is always possible to con-struct phase operators C and S which commute within a class of coherent states|α characterized by |α| = constant. An example of such a pair of operators isgiven. This is the closest we have been able to get to the quantum mechani-cal counterpart of the Poisson bracket relation

{Cosφ,Sinφ

}= 0 reflecting the

uniqueness of phase in classical mechanics.

2. Action and Phase (Angle) Variables and the Energy Levels of the Anharmonic Oscillator

The problem of the anharmonic oscillator has been the subject of a greatdeal, [18] -[20], of recent work which has been motivated largely by the hopeof exploiting the analogy of the anharmonic oscillator and the nonlinear fieldsto gain some insight into the latter. The results are however of considerableinterest. Bender and Wu [19] showed that if the energy eigenvalues E n of the

5



hamiltonian

H =p2

2m + αx2 + βx4 (1.14)

are calculated by a perturbation method in powers of β the series fails toconverge for any β . They have traced the reason for this divergence to thepeculiar analytical properties of E n(β ) in the complex β plane. Despite thesingularities of E n(β ), it turns out that the values of E n for physical β can beclearly reproduced by P ade approximants [21]-[22]. Other methods of comput-ing E (eg. in terms of the roots of a Hill determinant) have also been exploredrecently[23]-[25]. However throughout this work one does not find a simple for-mula for quick and practical computation of the energy levels. We show inChapter III of this thesis that by a semiclassical method based on the use of the action and angle variables an approximate formula which serves this pur-pose can be obtained. It not only reduces to the harmonic oscillator spectrum

as β → 0 but also reproduces the known asymptotic (large n) expression forE n(β ) . We also derive a scaling relation (for the exact eigenvalues) which ismore general than any previously known in the literature [20] and show thatour approximate formula for E n(β ) does obey this scaling relation.

3. Coherence of Wave Packets in Three Dimensions

It is well known that attempts to reconcile wave-mechanics with classicalmechanics by interpreting particles as wavepackets were not completely sucess-ful because quantum mechanical wavepackets do not keep a constant size forall time [26],[27] . The size of a free wave packet increases indefinitely as timepasses. Though the rate of spreading in the case of macroscopic objects is so ex-

tremely small as to be totally unnoticable, the very possibility of spreading hasappeared to some as too disturbing to be acceptable. Louis de Broglie notablyhas suggested [28] that the Schrodinger equation should be replaced by somenonlinear equation which would preserve the main quantum mechanical struc-ture but inhibit the spread of such wavepackets3. As an alternative possibility,the use of singular solutions4 of the Schrodinger equation has been advocated,for describing point particles [32]-[34],[28] . There has also been an attempt[35] to link the problem of the spread of wavepackets with the curvature of space-time in the general theory of relativity and show that a theory that takesthis into accout will permit the motion of wavepackets along definite trajecto-ries without unlimited spreading. None of these attempts can be said to havereally made much headway, and the standard form of quantum mechanics hasthe allegiance of practically all physicists. It is surprising however that hardlyany attempt has been made within the ambit of orthodox theory, to study themotion of wavepackets even in simple realistic systems. Besides the free particle,the harmonic oscillator (in one or more dimensions) seems to be the only othersystem to have been extensively investigated. It has been long recognised thatthe latter, unlike the free particle has wavepacket states such that the size of the packet remains time independent. These are the coherent states [36]- [39]

3For a discussion of the spreading of a ‘classical wavepacket’ (i.e. the group of samples of a statistical ensemble around some point in phase space) see M.Born and D.J. Hooton, andM.Born Refs. [29]-[31].

4Such solutions have been called ‘The Double solutions’, for a discussion of this, see Ref.[28].

6



already referred to in Sec.1, and they are characterised by a minimum value for

the product of the extensions in coordinate and momentum spaces. But even inthe case of such an important system as a particle in a coulomb-type potential,nothing seems to be known about the behaviour of a small three dimensionalwavepacket as time progresses, and in particular, as to whether there exist co-herent (in the sense of non-spreading) wave packets in such a potential.

In view of this situation it seemed interesting to investigate the behaviour of wavepackets bound in spherically symmetric potentials of three dimensions. Wepresent in Chapter IV the details of such a study. Our aim is to check whetherthere exist wave packet states in which the uncertainties in the various compo-nents of position, momentum etc. are time independent. We will refer to suchstates as coherent states, for brevity, but it must be noted that the usage of thisterm in the context of the harmonic oscillator implies much more than the mere

constancy of the uncertainties. To emphasise this point as well as to introducethe method of attack in the more complicated three-dimensional case, we firstobtain the equations which govern the time variation of position and momentumuncertainties associated with wavepackes in the harmonic oscillator potential.We show that there exist a large class of states representing wavepackets whosemean positions move simple harmonically and whose sizes remain time indepen-dent but which are not necessarily coherent states in the standard terminology,we term such states as semi-coherent states. An interesting special class of suchstates is exhibited.

We then turn to study the coherence aspect of a wavepacket in an arbitrarycentral potential in three dimensions. The interlinking of different degrees of

freedom through the potential makes the problem quite complex, but when thetrajectory of the mean centre of the packet is circular the symmetry of thesystem can be taken advantage of to make the problem tractable. It turns outthat the uncertainty in the position and momentum components perpendicularto the plane of the orbit have a simple behaviour. As far as the uncertaintieswithin the plane of the orbit are concerned, it is an advantage to define themthrough the following quantities:

χR = (x − x)2R, (1.15a)

χT = (x − x)2T , (1.15b)

πR = ( p − p)2R, (1.15c)

πT = ( p − p)2T , (1.15d)

where the subscripts R,T refer to component in the radial and the tangentialdirection respectively:

(x − x)R = (x − x).xR

,

(x − x)T = (x − x). pP

, (1.16)

etc., where R, P are the radius of the circular orbit and the mean momentumin the orbit respectively.

The time derivative of the quantities (1.15) evaluated using the quantummechanical equation of motion(in the approximation of small size of the wave

7



packet) involve various cross correlations eg. (x − x)R( p − p)T . It turns

out that one can get a closed system of first order differential equations withconstant coefficients for a set of six such correlated functions together with thefour quantities (1.15). Taking these ten quantities as the elements of a tencomponent column vector u, we can write the equations in the matrix form:

du

dt= ωMu, (1.17)

where M is a 10×10 matrix in which the particular central potential enters, onlythrough a single parameter, and ω is the frequency of the orbit. The questionof coherence of the wave packet then boils down to whether (1.17) has solutionssuch that du

dt = 0. We find that formally such solutions do infact exist, sinceM happens to have zero eigenvalues (four of them); but it turns out that noneof these solutions is compatible with the essential positivity of the quantities(1.15) . This difficulty manifests itself through the non-diagonalizability of thematrix M . Positivity can be ensured only by allowing some of the uncertaintiesdefined through (1.15) to increase indefinitely. This is the case in particular forthe uncertainty in the tangential components of position: There is no choice of initial conditions by which spreading of the wavepacket in this direction can beprevented. The case of πR (which measures the uncertainty in the radial com-ponent of momentum) is similar. On the other hand the tangential componentof momentum and the radial component of position as well the components per-pendicular to the plane of motion remain well defined, their uncertainties beingconstant or varying harmonically at worst. The details of these calculations arecarried out for the case of particular interest, namely the coulomb potential.But we observe that the results hold quite generally, for central potentials, the

only exception (where true coherent states exist) being the three dimensionalharmonic oscillator. In this case the M matrix becomes diagonalizable.

8



Chapter 2

On the Generalized PhaseOperators for the QuantumHarmonic Oscillator

1. Introduction

We have noted in the last chapter, though an unique well-defined phase op-erator conjugate to the number operator N of the quantum harmonic oscillatordoes not exist, the notion of phase can be introduced through cosine and sineoperators C and S defined through the commutation relations:

[C, N ] = i S and [S, N ] = −i C (2.1)

or equivalently, with

[U, N ] = U where U = C + iS . (2.2)

Equation (2.2) implies that

U |n = f (n) |n − 1 , with f (0) = 0 , (2.3)

where the sequence of numbers f (n) has to be so chosen as to ensure that thehermetian operators C and S have the right kind of eigenvalue spectrum, namelya continuous one extending from -1 to +1 , consistent with their interpretationas cosine and sine operators. It was Lerner’s observation that the necessary and

sufficient conditions for this purpose5 are that

f (n) → 1 , as n → ∞ (2.4)

and that it be possible to express the f (n) in terms of another sequence

gn (0 ≤ gn ≤ 1) (2.5)

1

4f 2(n) = gn(1 − gn−1) . (2.6)

5See Lerner,Huang and Walters Ref. [14] for a detailed proof of the ‘chain sequence condi-tion’ (2.6) , and for the spectral properties of U .

9



Subject to these conditions one can make any of a vast variety of choices for f (n)

and hence for C and S. This freedom of choice raises interesting possibilities.One may ask, for instance, whether one can define C, S in such a way that

α|[C, S ]|α = 0 (2.7)

within some class of coherent states. Such a question is prompted by the factthat coherent states are wave packets with a classical motion and classically,the cosine and sine functions have a vanishing Poisson bracket, the phase anglebeing uniquely defined. We will consider this in section 5, but first we solve thethe eigenvalue problem for C and S in all generality. We show in section 2 thatif

C |λ = λ |λ , (2.8)

and

|λ

is expanded in terms of the number eigenstates

|n

as

|λ =

n=∞n=0

C n(λ) |n , (2.9)

then the C n(λ) form a set of orthogonal polynomials in λ . We also verify thatthe states |λ form a complete set, with the eigenvalue parameter λ rangingfrom -1 to +1 (provided the chain sequence condition holds). It is remarkablethat there are such elegant properties associated with the phase operators andthat they are demonstrable even when the f (n) is quite arbitrary. For the famil-iar special case [6],[12] when f (n) = (1−δn0) , the coefficients of transformationC n(λ) reduce to the Gegenbauer polynomials. Similar properties can b e de-duced with regard to eigenstates of the sine operator.

In section 3 we give a formal extension of Jackiw’s result regarding thenumber-phase minimum uncertainty states to the case of the generalized phaseoperators. The next section is devoted to the interesting problem of states whichminimize the uncertainty product (C )2(S )2/A2 with iA = [C, S ]. Jackiw[17] has attempted the minimization of (C )2(S )2 and concluded that theextrememum condition cannot be satisfied by any normalizable state 6. He hasexplained this in terms of the fact that (C )2(S )2 can in fact approach zero,as in the case of coherent states with average occupation numbers tending toinfinity [12],[41]. When the quantity (C )2(S )2/A2 is considered, howeverthe minimization procedure leads to a well defined class of solutions. In fact thestates for which this quantity is a minumum appear as solutions of an eigen-value problem which includes as a special case the eigenvalue problem for U .

The condition of normalizability of these states constrains the eigenvalue pa-rameter λ to lie within an ellipse in the complex λ-plane. The set of states forman overcomplete set.

In section 5 we present the coherent state representation of various quantitiesof physical interest constructed out of the phase (C and S ) operators, We thenshow that it is possible to construct phase operators such that α|[C, S ]|α = 0within a class of coherent states.

6Actually there is one normalizable state satisfying his extrememum condition, namely | 0,the ground state . However this is a state in which (C )2(S )2 has its maximum value.

10



2. The Eigenstates of the Generalized Cosine and Sine Operators

Consider now the eigenvalue problem for C, eq. (2.8), with C given in termsof U defined by (2.3), as

C =1

2( U + U † ) (2.10)

On sustituting (2.9) in (2.8) and using (2.10) and (2.3) we obtain the followingrecursion relations for the coefficients C n(λ) :

2 λ C 0(λ) = C 1(λ) f (1),

2 λ C 1(λ) = C 2(λ) f (2) + C 0(λ) f (1),

− − − − − − − − − − − − − − −,

2 λ C n(λ) = C n+1(λ) f (n + 1) + C n−1(λ) f (n), (2.11)

To solve eqs. (2.11) it is necessary to rewrite them in the form:

F n+1(λ) = 2 λ − f 2(n)

F n(λ), (2.12)

where

F n(λ) =C n(λ)

C n−1(λ)f (n), (2.13)

By virtue of ( 2.13) we can write

C n(λ) =

nr=1 F r(λ)n

r=1 f (r)C 0(λ), (2.14)

Further it is a consequence of (2.12) thet the product nr=1 F r(λ) is a polynomial

of degree n in λ. It follows from (2.14) that C n(λ) ≡ C n(λ)/C 0(λ) is also sucha polynomial. It may be verified that its explicit form is

C n(λ) ≡ C n(λ)/C 0(λ)

C n(λ) =

n

r=1

f (r)

−1 {n2}

r=0

(2λ)n−2r (−1)r D(n−1)r (f 2), (2.15)

where {n2} stands for (n − 1)/2 or n/2, according as n is odd or even, and

D(n)r (f 2) is defined by

D(n)0 (f 2) = 1 (for all n)

D(n)r (f

2

) =

f 2

(a1)f 2

(a2)....f 2

(ar) (2.16)Here the arguments a1, a2,....,ar are integers from the set 1, 2, 3,....n taken insuch away that no two of them differ by less than two and the summation in(2.16) is over all possible distinct sets of r integers which can be chosen in thismanner7. It may be noted that the definition (2.16) is only possible for those

7For example we write:

D(5)3 (f 2) = f 2(1)f 2(3)f 2(5)

D(6)3 (f 2) = f 2(1)f 2(3)f 2(5) + f 2(1)f 2(3)f 2(6) + f 2(1)f 2(4)f 2(6) + f 2(2)f 2(4)f 2(6)

andD(3)1 (f 2) = f 2(1) + f 2(2) + f 2(3)

11



values of r such that r ≤ { n+12

}. For other values of r , D(n)r (f 2) could be

defined to be zero; in any case they do not occur within the range of summationin the expansion (2.15). We observe further that the definitions (2.16) lead tothe identity

D(n−1)r (f 2) + D

(n−2)r−1 (f 2) f 2(n) = D(n)

r (f 2) , (2.17)

which is very useful in verifying explicitly that (2.15) for C n(λ) is indeed thesolution of the recursion relations (2.11).

We will now show that the polynomials C n(λ) form an orthogonal set, thatis, a weight function ρ(λ) can be found such that

+1

−1

ρ(λ) C m(λ) C n(λ) dλ = δmn (2.18)

To prove this we need the following theorem [40] :If any set of polyomials P n(x) satisfies a recursion relation of the type

P n(x) = (An x + Bn)P n−1(x) − LnP n−2(x), (2.19)

(where An, Bn and Ln are constants, with An > 0 and Ln > 0 ) and if thecoefficient of xn in P n(x) is kn then the necessary and sufficient condition forP n(x) to be orthonormal is that

An =kn

kn−1; Ln =

An

An−1=

kn kn−2

k 2n−1

(2.20)

We can write (2.11), using (2.15), in the form (2.19) with

An =2

f (n); Bn = 0 ; Ln =

f (n − 1)

f (n)and kn =

2nnr=1 f (r)

(2.21)

It is evident that they satisfy (2.20). Hence the polynomials C n(λ) form anorthonormal set, as asserted earlier. Since the spectrum of C is known [13],[14]to extend from -1 to +1, this is the domain over which the orthogonality relationholds, and this fact has been used in writing the limits of integration in (2.18).We have not been able to determine the weight function ρ(λ), however. If we choose the coefficient C 0(λ) (which is not determined by the recurrencerelations) as

ρ(λ), we can write (2.18) in view of (2.15) as

+1

−1

C m(λ) C n(λ) dλ = δmn (2.22)

It is expected that the infinite set of orthonormal functions C m(λ) form a com-plete set for normalizable functions of λ over the interval (−1, 1). In fact this canbe inferred from the orthonormality of the states |λ which are the eigenstatesof the hermitian operator C :

m

C m(λ) C m(λ) = λ|λ = δ(λ − λ) (2.23)

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whence for arbitrary f (λ),

f (λ) =

f (λ) δ(λ − λ) dλ

=

m

C m(λ)

f (λ)C m(λ) dλ

(2.24)

Thus any f (λ) can be expanded in terms of the set C m(λ) which is thereforecomplete.

While the polynomials C n(λ) cannot be identified with any known set whenthe f (n) are arbitrary, in the special case when f (n) = (1 − δn0) they reduce tothe Gegenbaur polynomials. This can be verified by inspection of (2.15). Theweight function ρ(λ) in this case is ρ(λ) =

(1 − λ2). Similar properties can

be proved for the coefficients of expansion (in terms of number eigenstates) of

the eigenstates of the S-operator. In fact it turns out that the coefficients are just inC n(λ).

3. The Number Phase Minimum Uncertainty States

Since the commutator of the number operator N and the cosine operatorC is the operator i S (and not a c-number) the equation which determines thestate for which (∆N )(∆C ) is a minimum is quite complicated as pointed outby Jackiw. But if the uncetainty product is defined to be

(∆N )2(∆C )2

S 2 ,

the condition for this to be a minimum is relatively simple,8 and followingJackiw, we will adopt this definition for our purposes. The relevant equation isthen

(N + i γ C ) |ψ = λ |ψ, (2.25)

where

λ = N + i γ C and γ = [(∆N )2/(∆C )2]1

2 , (2.25)

Jackiw has obtained the solution of this equation for the special case f (n) =(1 − δn0), and we will generalize it here for arbitrary f (n). For a state |ψ

|ψ =n=∞

n=0 an(λ) |n , (2.26)

which satisfies the minimum condition (2.25), we obtain the following recurrencerelations with the aid of equations (2.2) and (2.3):

iγ

2a1(λ) f (1) = λ a0(λ) (2.27)

− − − − − − − − − − − − − − −,

2(λ − n)

iγ an(λ) = an+1(λ) f (n + 1) + an−1(λ) f (n), (2.28)

8See Jackiw, Ref [17 ]; and Carruthers and Nieto, Ref.[12], for a further discussion of thisquestion.

13



Observing that φn(λ) , defined by

φn(λ) ≡ an(λ)

an−1(λ)f (n),

satisfies

φn+1(λ) =2 (λ − n)

iγ − f 2(n)

φn(λ), (2.29)

one finds, using the same procedure as in the solution of (2.11), that the ratio

an(λ)

a0(λ)=

nr=1 φr(λ)nr=1 f (r)

, (2.30)

is a poly nomial of the nth degree in λ. Eq. (2.26) with (2.30) gives the state

which minimises(∆N )

2

(∆C )

2

S2

.

4. Simultaneous Uncertainties in C and S

We now turn to the determination of the states which minimise :

Q ≡

(∆C )2(∆S )2

i[C, S ]2

.

Jackiw has pointed out the relevance of this kind of uncertainty product whendealing with a pair of operators whose commutator is not a c-number. From histreatment it follows that the states which minimise Q must satisfy the equation:

C

∆C − i S

∆S

C

∆C + i S

∆S

|ψ = 0 ; (2.31)

whereC = C − C and S = S − S . (2.32)

This would be ensured of course if C

∆C + i

S

∆S

|ψ = 0 , (2.33)

and we now proceed to solve this more restrictive equation. We first rewrite itas

(1 + γ )U + (1 − γ )U † |ψ = 2 λ |ψ , (2.34)

where

γ =

(∆C )2

(∆S )2

1

2

, (2.35)

andλ = C + iγ S . (2.36)

If we now assume an expansion

|ψ =

n=∞n=0

an |n , (2.37)

14



On letting n → ∞, remembering that f (n) → 1, we get

α2 − 2 x α + 1 = 0. (2.49)

The relevant9 root isα = x +

x2 − 1 (2.50)

Equations (2.50) and (2.46) lead to the condition

|x +

x2 − 1 |2 <

1

q

(2.51)

for normalizability. On using definitions (2.41) and (2.39) of x and q, thisbecomes

|λ + λ2 + γ 2

−1

|2 < (1 + γ )2 (2.52)

Note that γ is, by definition, positive. The boundary of the allowed domain of λ is thus characterized by the equality

|λ +

λ2 + γ 2 − 1 |2 = (1 + γ )2. (2.53)

This equation describes an ellipse in the λ-plane. we now proceed to prove thisassertion. There are two cases to be considered : γ 2 > 1 and γ 2 < 1. If γ 2 > 1,we write

λ =

γ 2 − 1 sinh θ, (2.54)

where θ is in general complex,

θ = u + iv.

Then eq. (2.53) reduces to

eu =1 + γ γ 2 − 1

. (2.55)

Feeding this back into eq. (2.54) we have

λ =1 + γ

2eiv − γ 2 − 1

2(1 + γ )e−iv ≡ µ + iν (say) (2.56)

It is easily verified now that

µ2 +ν 2

γ 2= 1. (2.57)

When γ 2 < 1 , a similar calculation with λ written as

(1 − γ 2) sin θ enablesus to show once again that the allowed region of values of λ is once again theregion of an ellipse (2.57). It is proved therefore that the domain of convergenceof the series in the denominator of (2.44) is an ellipse and hence the states |ψdefined by (2.33) are normalizable for all λ within this ellipse. It may be notedthat for the special case when (∆C )2 = (∆S )2, i.e. γ = 1, the states |ψ areeigenstates of the operator U , see (2.34). In this case the ellipse (2.57) becomesa circle and the minimum uncertainty states with ∆C = ∆S coincide with the

9Since bn+1(x)/bn(x) is a ratio of polynomials of degree n +1 and n respectively, it shouldbehave like x as x → ∞. Equation (2.50) is consistent with this; the choice α = x − √

x2 − 1would violate this requirement.

16



eigenstates of U found by Lerner at. al. [14]. It has also been shown by Lerner

et. al. that the eigenstates of U form a complete (or rather an overcomplete)set. By using a similar argument as that used by Cahill [42], we can assert thatfor general values of γ , the set of all states |ψ characterized by λ lying withinthe ellipse of convergence form an over complete set.

5. The C and S operators and the Coherent stateIt is well known that if any oscillator is at some time a coherent state |α thenthe subsequent time variation of the state is given completely by a change inargument in arg α : α = ρeiφ → α(t) = ρei(φ−ωt). Further the expectationvalue of position varies simple harmonically with phase (φ − ωt). Therefore itis natural that one should think of the coherent states in connection with thephase operators. Though the coherent states are eigenstates of the annihilationoperator a rather than of C and S , it may nevertheless be expected that the

expection values of of C and S in coherent states should show an intimaterelation to cos φ and sin φ ( φ = arg α). Actually Carruthers and Nieto[12],[41] have considered these and various related quantities in the special casef (n) = (1 − δn0), and found that apart from factors depending on ρ = |α|, theexpected correspondence does exist. Here we cary out the simple extension tothe case of general f (n), and verify that the same type of relations continue tohold. Following this we raise the interesting question whether it is possible tochoose f (n) in such a way that the commutator [C, S ] has vanishing expectationvalues within the coherent states. We show by an example that this is indeedpossible for all the coherent states with a fixed value of |α|, though the exampleitself is rather artificial.

Let us define the following relations10

ψ1(N ) =

∞n=0

N n

n!√

n + 1f (n + 1) ,

ψ2(N ) =

∞n=0

N n

n!

f (n + 1)f (n + 2)

{(n + 1)(n + 2)}1/2,

ξ1(N ) =∞

n=1

N n

n!{f 2(n + 1) + f 2(n)} , (2.58)

ξ2(N ) =

∞n=1

N n

n!{f 2(n + 1) − f 2(n)} .

By using the representation [37]

|α = exp

−1

2|α|2

∞n=0

αn

{n!}1/2|n (2.59)

we defineα = N

1

2 eiφ

10We have defined these functions using the same notation as in Ref. [12], except that heref (n) is quite general.

17



for coherent states the following expressions can be easily obtained :

α|C |α = N 12 exp(−N ) cos φ ψ1(N ), (2.60a)

α|S |α = N 1

2 exp(−N ) sin φ ψ1(N ), (2.60b)

α|C 2|α = exp(−N ) [1

4f 2(1) +

1

4ξ1(N ) +

N

2cos 2φ ψ2(N )], (2.60c)

α|S 2|α = exp(−N ) [1

4f 2(1) +

1

4ξ1(N ) − N

2cos 2φ ψ2(N )], (2.60d)

α|C 2 + S 2|α = exp(−N ) [1

2f 2(1) +

1

2ξ1(N )], (2.60e)

α|C 2 − S 2|α = N exp(−N ) cos 2φ ψ2(N ), (2.60f)

α|CS + SC |α = N exp(−N ) sin 2φ ψ2(N ), (2.60g)

α|CS − SC |α =

i

2 exp(−N ) [ξ2(N ) + f 2

(1)]. (2.60h)

The quantities:

P (N, φ) = (∆N )2(∆C )2/S 2, (2.61a)

R(N, φ) = (∆N )2(∆S )2/C 2, (2.61b)

which measure the degree of incompatibility of the number phase operators canbe evaluated with the aid of eqs (2.60). Note that for the coherent states

(∆N )2 = N 2 − N 2 = N

We will now consider the following question:

It is well known that the cosine and sine operators cannot commute, thisbeing a reflection of the impossibility of defining an unique phase operator. Butin view of the fact that a phase φ can be associated with a coherent state inthe sense indicated above, one may ask whether the commutator of C and Scannot be made to vanish atleast weakly, in the sense of the vanishing of itsexpectation value with respect to the coherent states. A quick check with theaid of eq (2.60h) shows that the familiar C and S operators (ie. with f (n) = 1for all n ≥ 1) as well as others considered explicitly in Ref. [13] do not havethis property. However with the freedom of choice of the f (n), available to usin turns out that we can define C and S in such away that within a class of coherent states α|[C, S ]|α = 0. In fact if we choose

f 2

(0) = 0, f 2

(1) =

N

N − 1

f 2(r) = 1 (for r > 1), (2.62)

then we see that equation (2.60h) is satisfied because

N [f 2(1) − f 2(2)] = f 2(1) (2.63)

We must verify that f (n) defined in (2.62) do follow the chain sequence condi-

18



tion: 11

14

f 2(n) = gn(1 − gn−1), (2.64a)

with g0 = 0 and 0 ≤ gn ≤ 1. (2.64b)

Now since1

4f 2(1) = g1(1 − g0), (2.65)

from eq.(2.62) we have

g1 =1

4

N

N − 1

(2.66)

Since

g2(1 − g1) =1

4 f 2(2)

=1

4

we have

g2 =1

4/(1 − g1).

gn =1/4

1 − gn−1.

Hence,

gn

=1/4

1 − 1/4

..... − 1/4

1 − g1

(2.67)

Now using the method of continuous fractions [44] and eq(2.66) it turns outthat one can sum up eq. (2.67) , to obtain

gn =1

2

nN − 2(n − 1)

(n + 1)N − 2n(2.68)

We see that for all n > 1,

gn(1 − gn−1) =1

4. (2.69)

Hence the f (n) given in (2.62) obey eq (2.64a). Now all the gn’s must obey theinequality (2.64b) i.e.

0 ≤ gn ≤ 1 . (2.70)

Now if gn+1 ≤ 1 . (2.71)

11The choice of f (n) given in our paper [43] , is incorrect for, though they satisfy the chainsequence conditions (2.64) they do not have the property Limit: f (n) = 1 as n → ∞. Itis for this reason, the spectra of C and S will not span the entire region (-1,1). We aregrateful to Professor Lerner for pointing this out (Private Communication). The choice of (2.62) overcomes this difficulty.

19



i.e.1

2

N (n + 1)−

2n

N (n + 2) − 2(n + 1) ≤ 1 .

i.e.N n − N − 2n ≤ 2nN + 4N − 4n − 4,

whence2(n + 2)

n + 3≤ N. (2.72)

The maximum value of the l.h.s. of (2.72) is attained when n → ∞, which is 2.Therefore if

N ≥ 2, (2.73)

equation (2.71) is satisfied. Since we must also have

gn+1 ≥ 0, (2.74)

i.e.,

N ≥ 2

n

n + 1

(2.75)

The maximum value of r.h.s of (2.75) is 2. Therefore we see that (2.73) ensuresthat (2.70) holds. We have shown that there exist phase operators such thatα|[C, S ]|α = 0 for all coherent states having |α| = N with N ≥ 2. Sincethe chain sequence conditions with the requirement limn→∞ f (n) = 1 are sat-isfied the spectra of these operators will span the entire region (-1,1). It isapparant from eq. (2.60h) that there could be other phase operators which aremore complicated than the simple choice given in (2.62) which will satisfy these

conditions.

20



Chapter 3

The Action and Phase(Angle) Variables and theEnergy Levels of theAnharmonic Oscillator

1. Introduction

In this chapter we treat the problem of the anharmonic oscillator in theaction and phase variables and show how an approximate formula for the energy

levels can be obtained. We demonstrate that this formula satisfies a generalscaling law which must be obeyed by the exact eigenvalues.

In Sec. 2 of this Chapter we derive this general scaling law. In Sec 3 weintroduce the action and angle variables through the use of Jacobi’s ellipticfunctions. The Hamiltonian is dependent only on the action variable J , and isindependent of the phase (angle) variable. By a semiclassical treatment basedon the Bohr-Sommerfeld quantum rules as modified in the W.K.B. approach,we obtain in Sec 4, an expression for the energy levles E n of the anharmonicoscillator. This formula exhibits the correct limits for small anharmonicity andgives the correct behaviour (corresponding to that of a pure quartic potential)for large n or alternatively for large anharmonicities.

2. A general Scaling Law for the Anharmonic Oscillator

The hamiltonian for the anharmonic oscillator with a quartic term is

H =p2

2m+ αx2 + βx4 (3.1)

If the nth energy level of (3.1) is denoted by E n(m,α,β ) , then a general scalinglaw can easily be established by writing m = µm0 and x = ξ y (ξ = constant)in the eigenvalue equation

− 2

2m

∂ 2ψ

∂x2+ (αx2 + βx4) ψ = E n(m,α,β ) ψ,

21



to obtain

− 2

2m0

∂ 2ψ∂y2

+ (α µ ξ4 y2 + β µ ξ6 y4) ψ = ξ2 µ E n(m,α,β ) ψ .

Comparing these two equations, we discover that

E n(m,α,β ) = f E n(m0, α0, β 0)(3.2a)

where f is given by

f =

a

µ

1/2

=

b

µ2

1/3

=a2

b; (3.2b)

in terms of the ratios

µ =m

m0, a =

α

α0and b =

β

β 0; (3.2c)

which have to be so chosen that

µ a3 = b2. (3.2d)

Our scaling law is a generalization of that found in Symanzik [20]. The lattercorresponds to setting µ = 1 in our equation (3.2).

A familiar special case, ( Simon Ref. [20]) is obtained when µ = 1,b = β and α = 1,

E n(m0, 1, β ) = β 1/3 E n(m0, β −2

3 , 1) (3.3)

which exhibits the β 1/3 type of behaviour of the energy levels and the cube roottype of singularity at β = 0.

3. Action and Phase (Angle) Variables

The identification of the action angle variables is facilated by the knowledgethat in the solution of the classical anharmonic oscillator problem one encounterselliptic functions in place of the trignometric functions of the harmonic oscillatorcase.12 Thus we write

x = A cn(φ, k) , (3.4a)

p = mx = − (2mα)1

2 A

1 +2βA2

α 1

2

sn(φ, k) dn(φ, k) , (3.4b)

where the fact that ddu cn(φ, k) = −sn(u, k) dn(u, k) and φ has been written as a

constant since φ is an angle variable. The value of the constant and the relationbetween A and the modulus k of the elliptic functions are so chosen that H isfree of φ. We have

k2 =βA2

α + 2βA2or A2 =

αk2

β (1 − 2k2)(3.5)

12For definitions of the elliptic functions used inthis Chapter and some of their elementaryproperties, see Ref. [45].

22



On using (3.4) and (3.5) H reduces to

H = A2(α + βA2), (3.6)

where the standard properties of the elliptic functions such as

cn2(φ, k) + sn2(φ, k) = 1

dn2(φ, k) + k2sn2(φ, k) = 1, (3.7)

are taken into account,The action variable J conjugate to φ may now be determined as a function

of A by requiring that{x, p}J,φ = 1. (3.8)

We have {x, p}J,φ = {x, p}A,φ∂A∂J

. (3.9)

Because of the dependence of the modulus k on A, {x, p}A,φ cannot be evaluatedthrough a straight-forward use of eqs. (3.4). However by differentiating theidentity

p2

2m+ αx2 + βx4 = A2

α + βA2

(3.10)

with respect to A and φ respectively:

p

m

∂p

∂A+

2αx + 4βx3 ∂x

∂A= 2αA

1 +

2βA2

α

, (3.11a)

pm

∂p∂φ

+

2αx + 4βx3

∂x∂φ

= 0 , (3.11b)

and multiplying (3.11a) by ∂x∂φ and (3.11b) by ∂x

∂A , subtracting and using eqs.

(3.4) we have

{x, p}A,φ = (2mα)1

2 A

1 +

2βA2

α

1

2

. (3.12)

Hence from (3.8) and (3.9) we obtain

∂J

∂A= (2m)

1

2 A

α + 2βA2 12

J = A

0(2m)

1

2 A

α + 2βA2 1

2 dA

=

2mα3

12

6β

1 +

2βA2

α

3

2

− 1

. (3.13)

Thus:

H =α2

4β

1 +

2

mα3

1

2

3βJ

4

3

− 1

. (3.14)

23



4. The Energy Levels

The quantized energy levels will now be obtained by requiring that J dφ = (n +

1

2) h, n = 0, 1, 2,... (3.15)

This is of course a heuristic procedure which is based on the Bohr-Sommerfeldapproach of the old quantum theory, but with the important modification (re-placement of n by n + 1

2 on the right hand side) suggested by the W.K.B.treatment. Since the elliptic functions introduced in eqs. (3.4) are periodic inφ with period 4K (k) , where K (k) is the complete elliptic function of the firstkind

K (k) = 1

0

dt

(1 − t

2

) (1 − k2

t2

)

(3.16)

eq. (3.15) requires that

4 K (k) J = (n +1

2) h, n = 0, 1, 2,... (3.17)

On substituting the solutions J n of this equation for J in (3.14) we get theenergy levels of the quantum anharmonic oscillator.

Eq(3.17) is a transcendental equation , since the modulus of the elliptic func-tion, is a function of J, c.f. (3.13) and (3.5). However K (k) is quite insensitiveto the parameters involved. The extreme limits are:

βA2

α →

0, k→

0, K (0) =π

2(3.18a)

βA2

α

→ ∞, k → 1√

2, K

1√

2

=

1

4√

π

Γ

1

4

2≈ 1.18 K (0)

(3.18b)Thus as the anharmonicity increases from a vanishing value to infinity, K (k)changes no more than 18 % . In the limit (3.18a) the energy levels are expectedto tend to those of the harmonic oscillator and indeed they do. For (3.18a) and(3.17) give J n = (n+ 1

2), and when this is introduced in (3.14) the leading term

in the binomial expansion is taken, we get E n = (n + 12)ω0 with ω0 =

2αm

1

2 .At the opposite extreme (large anharmonicity or high energy) (3.18b) appliesand then one gets from (3.17)

J n =2π

3

2Γ14

2 (n +1

2). (3.19)

On substituting this in eq. (3.14) and noting that the second term in curlybrackets now dominates we get:

E n ≈

18π3 1

2

n + 1

2

Γ14

2

4

3 β 4

4m2

1

3

(3.20)

24



This expression coincides 13 with the asymptotic formula given in Titch-

marsh [46] for the eigenvalues in the case of the pure x4

potential. It may benoted that the energy increases as n4/3 β 1/3 for large n. Finally it is a trivialmatter to verify that our approximate formula obeys the scaling law eqs. (3.2)satisfied by the exact energy levels. Actual numerical computation shows thateven for low lying energy levels the formula reproduces the energy values withina few percent, the error being of the same order as the W.K.B. evaluation [47]which is more complicated.

13The apparent missing of a factor of 2 within the curly brackets is because the range of xin our case is −∞ to +∞ and not from 0 to ∞ as is the case in Ref. [46].

25



Chapter 4

Coherence in QuantumMechanical Systems

1. Introduction

In this Chapter we study the behaviour of small wavepackets in sphericallysymmetric potentials in three dimensions. In particular we investigate whetherany coherent states exist in such potentials, i.e. whether it is possible to havewavepackets which move along classical orbits without spreading. Before pro-ceeding to this question we consider, in Sec. 2, the familiar problem of theharmonic oscillator in one dimension, and obtain a simple criterion for the con-stancy for the position and momentum uncertianties. Among the variety of

states which satisfy this criterion (besides the coherent states) we identify aninteresting class of semicoherent states. It is conceivable that such states mightexist in more general cases and might show up if one demands coherence only inthe loose sense of constancy in position and momentum uncertainties. Actuallyour investigations in Secs. 3-5, show that no small wavepackets which are co-herent even in this loose sense can exist14 in any central poential other than thethree dimensional harmonic oscillator potential. We define a set of quantitieswhich measure the uncertainties in position and momentum in various direc-tions, and solve the set of coupled equations governing their time variation. Weshow that the initial conditions required to ensure the constancy of the var-ious uncertainties are unphysical, so that the wavepackets necessarily spreadindefinitely. Particulars of derivation of the equations employed in Sec. 3, are

indicated in the Appendix.

2. The Wavepacket in the Harmonic Oscillator PortentialIt is a simple matter to obtain the class of harmonic oscillator states char-

acterised by constant values of ∆x and ∆ p. Defining:

χ = (∆x)2 = x2 − x2,

π = (∆ p)2 = p2 − p2, (4.1)

14Strictly speaking our proof is only for wavepackets moving in circular orbits, but it ismost unlikely that the case of other types of orbits would be very different.

26



and calculating the time derivatives using the quantum equation of motion:

dA

dt=

1

i[A, H ] +

∂A

∂t, (4.2)

with

H =p2

2m+

1

2mω2x2 , (4.3)

one finds from the expressions χ and π involve also xp + p x, but

m2ω2χ(t) + π(t) = 0 , (4.4)

as can be seen from an application of (4.2). A dot denotes differentiation withrespect to time. The second time derivatives yield the equations15

χ(t) = −2ω2χ(t) + 2m2

π(t) , (4.5a)

π(t) = 2m2ω4χ(t) − 2ω2π(t) , (4.5b)

which may be solved immediately by rewriting them as

π(t) + m2ω2χ(t) = 0 , (4.6a)π(t) − m2ω2χ(t)

+ 4ω2

π(t) − m2ω2χ(t)

= 0 , (4.6b)

We obtain

χ(t) = 2m2ω2

−1

m2ω2χ(0) + π(0) + m2ω2χ(0) − π(0) cos2ωt+

2m2ω2−1 1

2ω

m2ω2χ(0) − π(0)

sin2ωt

(4.7a)

π(t) =

π(0) + m2ω2χ(0)− m2ω2χ(t) . (4.7b)

Actually the general solution contains another term

m2ω2χ(0) + π(0)

t,whichhowever vanishes identically for any state according to eq.(4.4). Thus both χand π are harmonically varying in general with an angular frequency 2ω. If theyare time independent we must have

m2ω2χ(0) − π(0) = 0 , (4.8a)

m2ω2χ(0) − π(0) = 0 , (4.8b)

These can be expressed more simply in terms of the creation and annihilationoperators. On setting

x =

2mω

1

2 a + a†

p =

mω

2

1

2 a − a†

(4.9)

and simplifying, the conditions (4.8) become

a2(0) = a(0)2 , (4.10a)

15For slightly different forms of equations (4.5) see Yu E. Murakhver, Ref. [48] and A.Messaih Ref. [49].

27



a†2

(0) = a†(0)2 , (4.10b)

These are necessary and sufficient conditions for a harmonic oscillator wavepacketto maintain a constant size or extension. They are of course trivially satisfiedby the coherent states |α which are eigenstates of a. At the other extreme wehave the energy eigenstates which being stationary states have constant valuesof χ and π. In between there is a whole class of semicoherent states for whicheqs (4.10) are valid. An interesting example is provided by states of the type

|ψ = N [ |α − |β β |α ] , (4.11)

where |α and |β are coherent states and N is a normalization factor, N =1 − |β |α|2− 1

2 . It may be easily verified that the equations (4.10) do hold goodin these states. Note that (4.11) defines |ψ as the projection of a coherent state

|α orthogonal to another coherent state |β . Therefore the ordinary coherentstates |α are not obtainable as the limit of |ψ when the two superposed states|α and |β are made identical : in the limit α → β, |ψ does not exist. We willrefer to the states (4.11) as semicoherent states. it is a curious fact that in thesemicoherent state (4.11), the mean position or (momentum) of the oscillatoris exactly the coherent state |α i.e. ψ|x(t)|ψ = xα(t) ≡ α|x(t)|α. Theposition probability density |ψ(x, t)|2 in the state (4.11) is describable in termsof two gaussian packets of constant amplitude and width (oscillating simpleharmonically with phases characterised by arg α and arg β ) together with aninterference term which ensures that χ and π remain time independent. Theirvalues are

χ =

2mω1 +2 |α|β |2 |α − β |2

(1

− |α

|β

|2) , (4.12a)

π =

mω

2

1 +

2 |α|β |2 |α − β |2(1 − |α|β |2)

. (4.12b)

Clearly the product of these exceed the minimum value 2/4 unlike the case of

true coherent states.This example illustrates the wide variety of states which can satisfy the con-

dition of constant values for the uncertainties ∆x and ∆ p. Of these the truecoherent states have the property of being small wavepackets around the meanposition. In the search for coherent states in three dimensions, in the remainingsections of this chapter, we use the constancy of uncertainties of various quanti-ties as a criterion. But we also demand small size, so that semi-coherent states(if at all they exist) are excluded from consideration.

3. Behaviour of a Three Dimensional Wavepacket in aSpherically Symmetric Potential

We now consider a wavepacket representing a particle in a central potentialin three dimensons and determine how various parameters representing the ex-tension of the wavepacket (in the configuration and momentum spaces) changewith time. To avoid nonessential complications we will assume that the meanposition moves in a circular orbit of radius R. Taking advantage of the geome-try of the system, we consider the uncertainties in the components of position

28



and momentum in the radial and tangential directions as well as the directions

perpendicular to the plane of the orbit. We define the quantities

χR =

xR

· (x − x)

2

, (4.13a)

χT =1

m2ω2

pR

· (x − x)

2

, (4.13b)

χz ={z − z}2

, (4.13c)

πR =1

m2ω2

xR

· ( p − p)

2

, (4.13d)

πT =1

m4ω4

pR

· ( p − p)

2

, (4.13e)

πz =1

m2ω2

{ pz − pz}2

, (4.13f)

The subscript R indicates the radial component, T the component tangentialto the orbit (and hence parallel to p ) and z the component normal to theplane of the orbit which is taken to be the x − y plane. The factors of mωare included to reduce the dimensions of all the quantities to the dimension of (length2). Here m is the mass of the particle and ω the angular frequency of its motion in the circular orbit. Of course by the radius of the orbit we mean

|x|

. It is evident that

mω2R =

dV

dr

r=R

≡ V (R) . (4.14)

On calculating the rates of change of these quantities as dictated by the Hamil-tonian

H =p 2

2m+ V (r) , (4.15)

with the assumption that the wavepacket is very small (χR, χT , χz << R2)one obtains a set of coupled differential equations which involve not only thequantities defined above in (4.13), but also various cross correlations betweenposition and momentum components. However, the equations for χz and πz

decouple from those for uncertainties in the plane of the orbit and can be solvedeasily. We will now obtain the relevant equations (relegating the proof to theAppendix) and solve them. In particular the question whether time independentsolutions (corresponding to the coherent wavepacket) exist will be considered.

3.1 Spread of a wavepacket normal to the plane of the orbit

The equations for χz and πz can be shown to be reducible to the followingclosed system:

χz = 2ω2 πz − 2

mV 33 χz , (4.16a)

29



πz = − 1

m2ω2V 33 χz , (4.16b)

where V 33 is defined by

V 33 =

∂ 2V

∂z2

0

=

V

z

r

2− V

z2

r3+

V

r

0

=V (R)

R= m ω2 . (4.17)

The subscript 0 indicates that the quantity concerned is to be evaluated at theposition of the orbit; the fact that the orbit has been chosen to be in the x-yplane (z = 0) has been used in the above reduction. Equations of exactlythe same form16 as (4.16), with V 33 = m ω2, hold also for a one dimensionaloscillator.On solving these equations we obtain

χz(t) =

1

2 (χz(0) + πz(0)) +

1

2 (χz(0) − πz(0)) cos2ωt

+1

2ωχz(0) sin2ωt (4.18a)

πz(t) = (χz(0) + πz(0)) − χz(t) . (4.18b)

Clearly the extension of the wavepacket perpendicular to the orbital plane re-mains bounded; infact, initial conditions may be so chosen as to make χz andπz time independent.

3.2 Spread of wavepacket in the orbital plane

Expressions for the first time derivative of χR, χT , πR and πT involve alsothe following quantities:

χRT =1

mω

1

R2{x · (x − x)} { p · (x − x)}

, (4.19a)

πRT =1

m3ω3

1

R2{x · ( p − p)} { p · ( p − p)}

, (4.19b)

µRR =1

mω

1

R2{ x · (x − x)} {x · ( p − p)}

+1

mω

1

R2{ x · ( p − p)} {x · (x − x)}

(4.19c)

µRT =1

m2ω2

1

R2{x · (x − x)} { p · ( p − p)}

, (4.19d)

µT R = 1m2ω2

1

R2{ p · (x − x)} {x · ( p − p)}

, (4.19e)

µT T =1

m3ω3

1

R2{ p · (x − x)} { p · ( p − p)}

+1

m3ω3

1

R2{ p · ( p − p)} { p · (x − x)}

(4.19f)

16Infact Yu. E. Murakhver Ref.[48], has obtained the same set of equations for an arbitrarypotential in one dimension. While the equations are exact for the harmonic oscillator, thederivation of these equations for other potentials involves the neglect of third and higher orderderivatives of V, in a Taylor’s expansion around the point x, the ‘centre’ of the wavepacket,a procedure which is valid for ‘small’ wavepackets (i.e. χz << R2).

30



These are the correlations 17 along various compnents of (x − x) and

( p − p). The µ’s contain one component of (x − x) and one of ( p − p) andthe convention regarding the subscripts in this case is that the first subscriptindicates which component of (x − x) is involved while the second subscriptrefers to the component of ( p− p). The equations for the ten coupled quantitieswritten in matrix form are as follows:

χR

χT

χRT

πR

πT

πRT

µRR

µRT µT R

µT T

= ω

0 0 2 0 0 0 1 0 0 00 0 −2 0 0 0 0 0 0 1

−1 1 0 0 0 0 0 1 1 00 0 0 0 0 2 η 0 0 00 0 0 0 0 −2 0 0 0 −10 0 0 −1 1 0 0 η −1 0

2η 0 0 2 0 0 0 2 2 0

0 0 −1 0 0 1 −1/2 0 0 1/20 0 η 0 0 1 −1/2 0 0 1/20 −2 0 0 2 0 0 −2 −2 0

χR

χT

χRT

πR

πT

πRT

µRR

µRT µT R

µT T

(4.20)or

du

dt= ω M u , (4.21)

Where M and u stand for the square matrix and the column vector whichappear in the r.h.s. of eq.(4.20). The parameter η is defined as

η = −

1 +r

ω2

∂ω2

∂r

r=R

which by (4.14) can also be written as

η = −

rV

V

r=R

. (4.22)

One is familiar with solutions of the form

u =

i

u(i)eλiωt (4.23)

for equations of the above type, where λi and u(i) are respectively the eigenvaluesand column eigenvectors of M . In the present case however, the solutions arenot of this kind, because it so happens that M is not a diagonalizable matrix (it

does not have ten independent eigenvectors) except for the special case η = −1.The characteristic equation for M can be verified to be

λ4 (λ2 + 3 − η)2 (λ2 + 12 − 4η) = 0 , (4.24)

so that the eigenvalues are 0 (4 times) , ±(η −3)1

2 (twice each ) and ± 2(η −3)1

2 .If M were diagonalizable it would satisfy the equationM (M 2 + 3 − η)(M 2 + 12 − 4η) = 0. Actual evaluation shows thatM (M 2 + 3 − η)(M 2 + 12 − 4η) becomes (η + 1) times a non- zero matrix, so

17It may be mentioned that the quantities in eqs. (4.19) which do not appear to symmetrizedare symmetric for circular orbits (see Appendix).

31



that M is diagonalizable if η = −1 but not for any other value of η. However

one has for any η,

M 3(M 2 + 3 − η)2(M 2 + 12 − 4η) = 0 , (4.25)

which shows that in general M has only two (instead of four) independent eigen-vectors belonging to the eigenvalue 0 and only one eigenvector each (instead of

two) corresponding to the eigenvalues ±(η − 3)1

2 . This means that the Jordancanonical form [50] of M is

M c = SM S −1 =

0 1 . . . . . . . .. 0 1 . . . . . . .. . 0 . . . . . . .. . . 0 . . . . . .

. . . . β 1 . . . .

. . . . . β . . . .

. . . . . . −β 1 . .

. . . . . . . −β . .

. . . . . . . . 2β .

. . . . . . . . . −2β

(4.26)

Where we have defined, for convenience,

β ≡ (η − 3)1

2

The consequences of the nondiagonalizability of M will be explored in the par-ticular case of the coulomb potential in the next section. Here we remark that

the general solution of eq (4.20) has sinusoidal or exponential parts according asη < 3 or η > 3 for the potential considered. But the basic question which we areinterested in is whether initial conditions can be so presented that the solutionsof the equations become time independent. We now proceed to analyse thisproblem in the context of the coulomb potential, which is of course physicallythe most important case.

4. Solution of Eq. (4.21) for the Coulomb Potential

WhenV (r) =

α

r(4.27)

η = 2 , (4.28)

the eigenvalues of M are0, ±i , ±2i . (4.29)

Rewriting eq. (4.21) as

dv

dt= ω M c v , v = Su ; (4.30)

we can readily obtain the general solution for this set of coupled equations for

32



the components of

v =

c1 + c2 ωt + c3 (ωt)2

v(1) + (c2 + 2 c3 (ωt)) v(2)

+ 2 c3 v(3) + c4 v(4) + (c5 + c6 (ωt)) eiωtv(5)

+ c6 eiωt v(6) + (c7 + c8 (ωt)) e−iωt v(7) + c8 e−iωt v(8)

+ c9 e2iωt v(9) + c10 e−2iωt v(10) (4.31)

where the v(i) are column vectors with the elements

v(i)

j = δij . (4.32)

With this definition, it is easy to see from the form of M , eq.(4.26), with η = 2,that

M c v(1) = 0 ; M c v(2) = v(1) ; M c v(3) = v(2) ; M c v(4) = 0 ;

M c v(5) = i v(5) ; M c v(6) = i v(6) + v(5) ;

M c v(7) = −i v(7) ; M c v(8) = −i v(8) + v(7) ;

M c v(9) = 2i v(9) ; M c v(10) = −2i v(10) ; (4.33)

The general solution for u is now obtained from (4.31) as

u = S −1 v

=

c1 + c2 ωt + c3 (ωt)2

u(1) + (c2 + 2 c3 (ωt)) u(2)

+ 2 c3 u(3) + c4 u(4) + (c5 + c6 (ωt)) eiωtu(5)

+ c6 eiωt u(6) + (c7 + c8 (ωt)) e−iωt u(7) + c8 e−iωt u(8)

+ c9 e2iωt u(9) + c10 e−2iωt u(10) (4.34)

where u(i) = S −1 v(i). The u(i) evidently satisfy equations (4.33) with M creplaced by M . It can easily be verified that since M is a real matrix

u(5)∗ = u(7) ; u(6)∗ = u(8) ; u(9)∗ = u(10) ; (4.35)

and hence for the reality of the solution (which is required in view of the natureof the components of u ) we must choose

c∗5 = c7 ; c∗6 = c8 ; c∗9 = c10 . (4.36)

The vectors u(i) can be explicitly shown to be given by:18

18The u(i) in (4.37) are not normalized and u(1), u(4), u(5), and u(9) are arbitrary up toa factor. However such a factor can be absorbed into the integation constants c1, c2,.... etc.,and one can easily check by differentiation that (4.34) with (4.37) is the most general solutionof (4.21), with η = 2.

33



u(1) u(2) u(3) u(4) u(5) u(6) u(9)

01010000

−10

00

−1/300

−1/62/3

00

1/3

2/9000

1/1800

−1/900

1300100

−1−10

02

−i/210

−i/2i0

−3/2i

2i/32i/3

−1/2i/3i/3

01/3

−i/2−i/2

1/3

1−42i−11i

−2i−12

−4i

(4.37)

4.1 Spreading of wavepacket in a Coulomb PotentialThe question of constancy of the size of the wavepacket now resolves into

the following: Is it possible to choose initial conditions on the packet in such away that all coefficients ci except c1 and c4 vanish? Even if this is not possiblecan we atleast arrange to make c2 = c3 = c6 = c8 = 0 so that the size of thewavepacket has at worst an oscillating behaviour but no indefinite increase? Theanswer turns out to be in the negative. To see this let us consider the constantc3 which appears as the coefficient of t2. It can be expressed in terms of theinitial values of u as

1

9c3 = χR(0) + πT (0) + 2 µRT (0) (4.38)

The vanishing of this quantity is a necessary condition for the constancy of thesize of the wave packet. However this condition cannot be met as we will nowshow.

We note first in view of the definitions (4.13) and (4.19) of χR, πT and µRT ,the r.h.s. of the above quantity is equal to

{xR

· (x − x)}2t=0 +1

m4ω4{ p

R· ( p − p)}2t=0 +

1

m2ω2

{xR

· (x − x)}{ pR

· ( p − p)} + { pR

· ( p − p)}{xR

· (x − x)}t=0

=

xR

· (x − x) + p

m2ω2R· ( p − p)

2

t=0

.

Since the quantity in curly brackets is evidently a Hermetian operator the aboveexpectation value can vanish only if x

R· (x − x) +

pm2ω2R

· ( p − p)

t=0

|ψ = 0 (4.39)

Now let us suppose that the x-axis is chosen so as to pass through the meanposition at t = 0, so that x = R, y = z = 0 at t = 0 and px = py = 0,and pz = mωR. Then eq.(4.39) reduces to

(x − R) +

imω

∂

∂y− R

ψ(x,y,z) = 0 , (4.40)

34



ie.1

ψ

∂ψ

∂y = i

mω

(2R − x) . (4.41)

Henceψ =

exp

i

mω

(2R − x) y

f (x, z). (4.42)

where f (x, z) is any arbitrary function of x and z. It is evident that the wave-function merely oscillates and does not fall off in the y direction. Thus theredoes not exist any compact wavepacket which does not spread indefinitely.

It is to be noted however that χR (the uncertainty in the radial component of position) and πT ( the uncertainty in the tangential component of momentum)either remain constant or vary utmost periodically. This is because, as may beseen from the explicit expressions (4.37) for the u(i), the columns u(1), u(2) andu(5) do not contribute to χR and πT and these are the columns which appear

with factors of t or t2 in (4.34). On the other hand u(1) and u(5) contribute toχT and this quantity increases indefinitely i.e. the wavepacket (even if it is welldefined initially) spreads gradually in the tangential direction.

5. Wavepacket in an Arbitrary Spherically Symmetric Potential

As we have already noted M is diagonalizable only for η = −1. The onlypotential for which η ≡ − (rV /V ) is identically equal to −1 is the parabolicpotential (i.e. the harmonic oscillator in three dimensions) as is seen by solvingthe equation rV = V . In other potentials there may be some particular radiusR at which η = −1, and if this happens, a wavepacket moving in a circular orbitat that particular radius can retain its size indefinitely.19 In all other cases

the wavepacket must necessarily spread, and the spreading is in the directiontangential to the orbit. This can be verified by following the procedure adoptedin the coulomb case, or more directly by observing that the postulate du

dt = 0,i.e., M u = 0, leads to the same constraint eq. (4.39) as in the coulomb case,which cannot be satisfied by a well defined wavepacket.

19This statement is of course for ‘small’ wavepackets, subject to the approximaton of ne-glecting third and higher derivatives of V in a Taylor’s series expansion around x. In the caseof the harmonic oscillator of course all such derivatives vanish identically and the nonspreadingis a rigorous property.

35



Chapter 5

Appendix

We will now derive the equations of spread (4.20), for a wavepacket moving ina circular orbit in an arbitrary central potential V (r).

The Hamiltonian for such a system is given by

H =3

i=1

p 2i

2m+ V (r) (5.1)

For our derivation we will have occasion to use the equation of motion (4.2) andthe commutation relations

[ pi, f (x)] = −i∂f (x)

∂xi(5.2)

together with the Taylor’s expansion of V (x), about the ‘centre’ x of thewavepacket:

V (x) = V (x) +3

i=1

∂V

∂xi

x= x

. (xi − xi)

+1

2!

3i=1

3j=1

∂ 2V

∂xi ∂xj

x=x

. (xi − xi) (xj − xj)

+ ... (5.3)

where for a central potential

V i ≡

∂V ∂xi

x=x

= xi

1r

∂V ∂r

r=R

(5.4a)

V ij ≡

∂ 2V

∂xi∂xj

x=x

=

δij

1

r

∂V

∂r+

xi xjr

∂

∂r

1

r

∂V

∂r

r=R

(5.4b)

Equations (4.20) have been derived for a bound wavepacket moving in a circularorbit. For such an orbit we have

p 2 = m2 ω2 R2 (5.5a)

p . x = 0 , (5.5b)

36



and

m ω2

=

1

R

∂V

∂R . (5.5c)

When these results are used, some of the expressions (4.13) and (4.19) simplifysomewhat. For example

χR =1

R2xi xj xi xj − R2 (5.6a)

χRT =1

mω

1

R2 pi xj xi xj

, (5.6b)

πR =1

m2ω2

1

R2xi xj pi pj

, (5.6c)

πRT =

1

m3ω3 1

R2 xi pj pi pj , (5.6d)

µRR =1

mω

1

R2xi xj pi xj + xi pj

. (5.6e)

Here and in the sequel, we use the summation convention with respect to re-peated indices, which range from 1 to 3.

To derive the equation corresponding to the first row in the matrix equation(4.20), we note from (5.6a) that

idχR

dt=

1

R2xi xj [xi xj , H ] + i

1

R2xi xj d

dt(xi xj) , (5.7)

where we have used the quantum equation of motion to evaluate d

dt xi xj

. On

evaluating the commutator and substituting ddt

xi = p i/m, we get

dχR

dt=

xi xjmR2

xi pj + pi xj +xi xjmR2

{xi pj + pi xj} ,

in view of (5.6b) and (5.6e) this reduces to

1

ω

dχR

dt= µRR + 2 χRT , (5.8)

which corrsponds to the first row in the matrix equation (4.20). This equationis exact.

We will now derive the equation for πR corresponding to the 4th row in

(4.20), this however involves an approximation depending on the small size of the wave packet. From (5.6c) we have

dπR

dt=

1

m2ω2

xi xjR2

d

dt pi pj +

pi pjR2

d

dt{xi xj}

=1

m2ω2

−xi xj

R2

∂V

∂xi pj

+

p i

∂V

∂xj

+1

m2ω2

pi pjmR2

{xi pj + pi xj}

, (5.9)

37



where the quantum equation of motion has been used to evaluate ddt pi pj. To

proceed further we expand ∂V/∂xi in a Taylor’s series about the position x :∂V

∂xi pj

= V i pj + V it ( xt − xt) pj

=1

R

∂V

∂Rxi pj + {xt pj − xt pj} V it . (5.10)

V i and V it are given in eqs. (5.4). We have dropped the third and higher orderderivatives in the Taylor’s expansion. This is a good approximation for mostpotentials of interest provided the size of the wave packet is much less than theradius of the orbit. From (5.10) we have

−xi xjR2

∂V

∂xi pj = −xi xj

R2{xt pj − xt pj} V it . (5.11)

Note that the contribution to (5.11) from the first term in (5.10) is zero becauseof (5.5b). Similarly the second term in (5.11) is zero, hence

−xi xjR2

∂V

∂xi pj

= −xi xj

R2xt pj V it .

= −xi xjR2

xt pj

δit1

R

∂V

∂R+

xi xtR

∂

∂R

1

R

∂V

∂R

= −xt xjR2

xt pj

1

R

∂V

∂R+ R

∂

∂R

1

R

∂V

∂R

= −xt xjR2

xt pj mω2

1 +

R

ω2

∂ ω2

∂R

= mω2

R2xt xj xt pj η (5.12)

where we define

η = −

1 +R

ω2

∂ ω2

∂R

= −

r V

V

r=R

(5.13)

Putting (5.12) into (5.9) and using (5.6d) and (5.6e) we have

1

ω

dπR

dt= 2 πRT + η µRR , (5.14)

which is the desired equation. Similarly all the other equations in (4.20) can bederived.

Before we close the appendix we make a final remark, namely that all thequantities which are defined in (4.19) and which do not appear to be properlysymmetized are symmetric for circular orbits. Consider for example µT R, eq.(4.19c) :

µT R =1

m2ω2

1

R2{ p · (x − x)} {x · ( p − p)}

,

=1

m2ω2

pi xjR2

{xi pj − xi pj} (5.15)

38



In the case of circular orbits this reduces to

µT R = 1m2ω2

pi xjR2

xi pj

=1

m2ω2

pi xjR2

pj xi + i δij

=1

m2ω2

pi xjR2

pj xi (5.16)

confirming that the order of factors in the product whose expectation value istaken is irrelevant.

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REFERENCES

1. W. Heisenberg : Z. Physik 33, 879 (1925)2. M. Born and P. Jordon : Z. Phyzik 34, 858 (1925)3. E. Schrodinger : Ann. Physik 79, 361, 489, 734 (1926)4. Van der Waerden : “ Sources of Quantum Mechanics”, North Holland Publ.

Amsterdam (1967)5. P.A.M. Dirac : Proc. of the Royal Soc. A114, 243 (1927)6. L. Susskind and J. Glogower : Physics 1, 49 (1964)

7. B.B. Varga and S. Aks : Phys. Lett. 31A, 40 (1970)8. J. Garrison and J. Wong : Journ. of Math. Phys. 11, 2242 (1970)9. S.G. Eckstein, B.B. Varga and S. Aks : Nuovo Cimento 8B, 451, (1972)10. W.H. Louisell : Phys. Lett. 7, 60 (1963)11. J. Zak : Phys. Rev. 187, 1803 196912. P. Carruthers and M.M. Nieto : Rev. of Mod. Phys. 40, 411 (1968)13. E.C. Lerner : Nuovo Cimento 56B, 183 (1968); erratum 57B, 251(1968)14. E.C. Lerner and H.W. Huang and G.E. Walters : Journ. of Math. Phys.

11, 1679 (1970)15. E.K. Ifantis : Journ. of Math. Phys. 11, 3138 (1970)16. E.K. Ifantis : Journ. of Math. Phys. 12, 1021, 2512 (1971)17. R. Jackiw : Journ. of Math. Phys. 9, 339 (1968)18. C.M. Bender and T.T. Wu : Phys. Rev. Lett. 21, 406 (1968)19. C.M. Bender and T.T. Wu : Phys. Rev. 184, 1231 (1969)20. B. Simon : Annals of Phys. 58, 76 (1970)21. J.J. Loeffel, A. Martin, B. Simon : Phys Lett. 30B, 656 (1969)22. S. Graffi, V. Grecchi and B. Simon : Phys. Lett. 32B, 631 (1970)23. S.N. Biswas, K. Datta, R.P. Saxena, P.K. Srivastava and V.S. Varma :

Phys. Rev. D4, 3617 (1971)24. S.N. Biswas, K. Datta, R.P. Saxena, P.K. Srivastava and V.S. Varma :

Pre-print Delhi Univ., (1971)25. For a variation calculation for the anharmonic oscillator, see : N. Bazley

and D. Fox : Phys. Rev. 124, 483 (1961)26. C.G. Darwin : Proc. of the Royal Soc. A117, 258 (1927)

39



27. E.H. Kennard : Z. Physik 44, 326 (1927)

28. Louis de Broglie : “Nonlinear Wave Mechanics”, Elsever Publ.,New York (1960)29. M. Born and D.J. Hooton : Z. Physik 142, 201 (1955)30. M. Born and D.J. Hooton : Proc. Camb. Phil. Soc. 52, 287 (1956)31. M. Born : Z. Physik 153, 372 (1958)32. G. Petiau : Comptes Rendus 238, 998, 1568 (1954)33. G. Petiau : Comptes Rendus 239, 344, 792, 1158 (1954)34. G. Petiau : Comptes Rendus 240, 2491 (1955)35. F. Karolhazy : Nuovo Cimento 42A, 390 (1966)36. E. Schrodinger : Z. Physik 14, 664 (1926)37. R.J. Glauber : Phys. Rev. 131, 2766 (1963)38. P. Carruthers and M.M. Nieto : Amer. Journ. of Phys. 33, 537 (1965)39. E.C.G. Sudarshan : Phys. Rev. Lett. 10, 277 (1963)

40. G. Szego : “ Orthonormal Polynomials”, American Mathematical SocietyColloquium Publ. Providence R.I. Vol. 23 (1965)

41. P. Carruthers and M.M. Nieto : Phys. Rev. Lett. 14, 387 (1965)42. K.E. Cahill : Phys. Rev. 138, B 1566 (1965)43. K. Eswaran : Nuovo Cimento 70B, 1 (1970)44. H. Barnard and J.M. Child : “ Higher Algebra” , Macmillan and Co. (1959)45. E.T. Whittaker and G.N. Watson : “ A Course of Modern Analysis”,

Cambridge Univ. Press, Cambridge (1952) Chapter XXXI.46. E.C. Titchmarsh : “ Eigenfunction Expansion”, Oxford Univ. Press,

Oxford (1962) Chapter VII, p 144.47. Pao Lu : Journ. of Chem. Phys. 47, 815 (1967)48. Yu. E. Murakhver : Soviet Phys. Doklady 10, 1079 (1966).

(Translated from Doklady Akad. Nauk. SSSR 165, 5261 (1965))49. A. Messiah : “Quantum Mechanics”, North Holland Publ.

Amsterdam (1966) Vol I, Chapter VI.50. F.R. Gantmacher : “ The Theory of Matrices ”, Chelsea Publ. New York

(1960) Vol I, Chapter VII.

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Studies on Phase and Coherence in Quantum Systems by K Eswaran

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Transcript of Studies on Phase and Coherence in Quantum Systems by K Eswaran