Exact operator solutions to Euler hamiltonians

Volume 200, number4

PHYSICS LETTERSB

21 January 1988

EXACT OPERATOR S O L U T I O N S TO EULER H A M I L T O N I A N S Carl M. BENDER ~ and Gerald V. D U N N E The Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 13 October 1987

It is nearlyimpossiblein generalto solveexactlythe quantum-mechanicalHeisenbergequations of motiont~=OH/Op,i~= - OH/Oq when these operator equations are non-linear. However,in this paper we show how to obtain exact closed-formsolutions for the operators p(t) and q(t) for any Euler hamiltonian (one that can be expressed as a functionofpq).

In the study of ordinary differential equations Euler (or equidimensional) equations are those in which every derivative operator d/dx is accompanied by a factor ofx. Thus, in this paper we use the term Euler to refer to quantum-mechanical hamiltonians which can be written as functions of pq. H=pqqp, n=p3q3q-q3p 3, and H = e x p ( p q + q p ) are all examples of Euler hamiltonians. Note that every Euler hamiltonian gives a time-independent Schr6dinger equation which is of Euler type. Given a quantum-mechanical hamiltonian H we can write down the Heisenberg operator equations of motion,

?1= on/Op,

( 1)

[~= -OH/Oq .

(2)

(3)

Unfortunately, it is extraordinary difficult in general to find exact closed-form solutions to ( 1 ) - ( 2 ) . The only simple case which is commonly discussed is that of the harmonic oscillator ( H = ½p2+ ½q2) because here the Heisenberg equations are linear: q=p,

p ( t ) = - q ( 0 ) sin t + p ( O ) cos t. Henceforth we abbreviate q(0) and p(0) by qo and Po. In this paper we show how to solve exactly the large class of non-linear Heisenberg equations arising from all hamiltonians of Euler type. We begin by considering a very special system of operator equations: ?l=q(Pq)",

(4)

[a= - p ( qp)" .

(5)

q( t) =qo exp[ (poqo)"t] ,

(6)

p ( t ) =Po exp[ - (qoPo)nt] .

(7)

[This verification follows from the results that q ( t ) p ( t ) =qoPo and that p ( t ) q ( t ) =Poqo.] Next we consider a system of operator equations slightly more complicated than that in ( 4 ) - ( 5 ) where the right sides are linear combinations: ~t=q ~ a,,(pq)n ,

(8)

n

P= - P ~, o~,,(qp)" .

/~=-q.

(9)

n

On sabbatical from Department of Physics, WashingtonUniversity, St. Louis, MO 63130, USA. 520

q(t) = q ( 0 ) cos t + p ( O ) sin t,

It is easy to verify that the exact solution to the system of equations ( 4 ) - ( 5 ) satisfying (3) is

These equations determine the time evolution of the operators q(t) and p(t). They must be solved subject to the constraint that q(t) and p(t) satisfy the equaltime commutation relation [q(t),p(t)] =i.

The solution to these equations satisfying (3) is

The solution to ( 8 ) - ( 9 ) is a straightforward generalization of ( 6 ) - (7):

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q(t)=qo e x p ( ~ cG(poqo)nt) ,

(10)

p(t)=Poexp(--~a,(qoPo)"t).

(11)

In the above, the an are arbitrary numerical coefficients satisfying no restriction other than that the sum over n be convergent. Once again, it is easy to verify that (10)-(11 ) exactly solve the operator differential equations ( 8 ) - ( 9 ) and also satisfy the equal-time commutation relation (3). Thus, given a hamiltonian H for which the set of Heisenberg operator equations of motion ( 1 ) - ( 2 ) can be expressed in the form ( 8 ) - ( 9 ) by repeated use of the commutation relation (3), we can find the exact solution q(t) and p(t). In particular, this applies to all hermitian Euler hamiltonians. To illustrate the method of solution, consider the most general hermitian Euler hamiltonian of homogeneous degree 4:

H = apq2p+ bqp 2q +c(pqpq+qpqp) +d(qZp 2 +p2q2) ,

(12)

where a, b, c and d are arbitrary coefficients. The Heisenberg equations for (12) are

?l=(a+c+2d)(qZp+pq2)+2(b+c)qpq,

(13)

~b= - ( b + c + 2 d ) ( q p 2 + p Z q ) - 2 ( a + c ) p q p .

(14)

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solving the Heisenberg equations ( 1 ) - (2) to that of rewriting them in the form ( 8 ) - ( 9 ) . For any given Euler hamiltonian this reordering of operators can be accomplished by explicit computation. However, it is remarkable that for any Weyl-ordered Euler hamiltonian this reordering can be done in closedform in terms of an elegant, unusual, and recently discussed set of orthogonal polynomials called continuous Hahn polynomials [ 1-5] ~ Weyl-ordering W is defined as follows [ 6 ]: Given the classical function p'q", its Weyl-ordered quantum equivalent is n

1"/

W(pmq~)=lk~=o(k)qkpmqn-k.

(15)

Weyl-ordering is identical to totally-symmetric ordering in which we sum over all possible arrangements of m operators p and n operators q, and normalize by dividing by the total number of terms in this sum, (n + m)!/(n!m!). In a previous paper [4], the unnormalized sum over all possible arrangements was denoted by Tin,,. For example, if we Weylorder p2q2 we obtain

W(p2q2) = ¼(p2q2 + 2qp2q+q2p2) = ~T2,2 = _~(p2q2 +qpqp+pqpq+qppq+pqqp+q2p2) . In refs. [4,5] it is shown that 0

opT,,,,=(m+n)Tm-,,,,

Using the equal-time commutator (3), (13)-(14) can be rewritten in the form ( 8 ) - ( 9 ) :

0 oqT,,~,n=(m+n)Tm,n-I ,

?1=aqpq ,

which means that if the hamiltonian is in totally symmetric (or, equivalently, Weyl-ordered) form, then the operator Heisenberg equations are also in totally symmetric form. An Euler hamiltonian in totally symmetric form is a linear combination of the

/~= - apqp , where a = ( 2 a + 2 b + 4 c + 4 d ) . Thus, the solution to (13)-(14) is

q( t) =qo exp( apoqot) ,

Tn,n: H= Z 7.T.,..

p ( t ) =Po exp( - aqoPot) • Notice that this solution depends on only one parameter a. If we had been given a hermitian Euler hamiltonian of homogeneous degree greater than 4, the solution would have depended on more parameters. It is clear that we have reduced the problem of

(16)

n

The Heisenberg equations for this hamiltonian are

Cl= Z 2nT,,T,,_,,,, ,

(17)

n

~ For a complete list of referenceson discrete Hahn polynomials and associated functions see refs. [2,3]. 521

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2n?.T~,~_,.

p= - ~

(18)

21 January 1988

Ug(X) = 92 378 (x 9 - 60x 7 q- 903x 5 - 3590x 3

n

We have discovered that T._ ~.. = qU._ ~(pq) and that T,,,._ ~=pUn_ ~(qp), where U.(x) is a polynomial in x o f degree n. The polynomials U,,(x) have the following properties: (i) The U,,(x) satisfy a recurrence relation:

(n+ 1 ) U~(x) = 2 x ( 2 n + 1 ) Un_, (x) - ( n - 1 ) ( 2 n + 1 ) ( 2 n - 1) U._2(x).

(19)

(ii) The U.(x) have a simple generating function:

Un(x)(n+l)t" .=o

(2n+l)!!

=

e x p ( 2 x arctan t) l+t 2

Ulo(X) = 352 716(X I° .~ 4 1 8 7 7 X 2

"A,165 ~ - ..8

"1-±1848X 6 - -

2513--'"~5X4

14 175"~ 4 I •

Note that in U,,(x) the coefficient o f x n is 2 n ( 2 n + 1 )!!/(n+ 1 )!. (vii) The U,,(x) are special cases o f a very general four-parameter class of orthogonal polynomials described in ref. [3] and which are known as continuous Hahn polynomials. A closed-form expression for the nth polynomials can be given in terms of a generalized hypergeometric function

(20)

Un(x) = ( - i ) n ( 2 n + 1 )!! (iii) The U.(x) have a simple expression in terms o f the H a h n polynomials S. (x) o f refs. [ 4, 5 ]: (2n+l)!! Un(x)[S.(2x+i)+Sn(Zx-i)]. 2(n+l)

(21)

(iv) The U.(x) are orthogonal on the interval (-~, ~ ) with respect to the weight function w(x) = 2 x / s i n h ( n x ) :

× 3 F 2 ( - n , n + 2 , ½-½ix; 1, 3; 1 ) .

(24)

Having summarized the properties o f the polynomials Un(x) we return to ( 1 7 ) - ( 1 8 ) and use the Un to rewrite them in the form o f ( 8 ) - ( 9 ) :

?l=q ~ 2n~,,U,,_~(pq) ,

(25)

n

P= - P Z 2nT.U._~(qp) .

(26)

n

i d x w ( x ) U,,(x)U,,,(x)=J,,m [ ( 2(nn++ l1)!!]2 )

(22)

-~zc

(v) The m o m e n t s o f the weight function w(x) are expressible in terms of the Bernoulli numbers B. [ 7 ]:

i dxxZ"w(x)= ( - 1 ) " B 2 ~ + 2 2 ( 2 2 " + 2 - 1 )

Finally, from ( 1 0 ) - ( 1 1 ) the solution to ( 2 5 ) - ( 2 6 ) is

q(t)=qo e x p ( ~ 2nT. Un_,(poqo)t) ,

(27)

P( t ) = Po e x p ( - ~

(28)

(23)

(n+l)

2n, n U~_ l ( qoPo) t ) .

(vi) The first few polynomials U.(x) are

U3(X) = 35(x 3 - 2 x ) ,

This is the main result o f this paper. It is the exact solution to a huge class of non-linear time-evolution operator differential equations. Here is an example to illustrate the procedure we have outlined. Consider the quantum-mechanical hamiltonian

U4(x) = 126(x 4 - -

H=exp[c(pq+qp)]

Uo(x) = 1 , U~(x) = 3x, U2(X)

= 10(X 2 __1)

,

5X 2 + 3 ) ,

U5 (x) = 462 (x 5 - 10x 3 + ~ x ) , U6(x) = 1716 (x 6 - ~ x 4 + 49x 2 - ~ ) , U7 (x) = 6435 (x 7 - 28x 5 + 154x 3 - 132x) , U8 (x) = 24 310 (x 8 - 42x 6 + 399x 4 - 818x 2 + ~ ) ,

522

(Jcl<½n).

(29)

In ref. [ 4 ] it is shown how to express H as a sum o f totally symmetrized Euler terms: H = s e c ( c ) ~ (tanc)~ n=o ( 2 n _ 1)!! Tn,.

(Icl <½n) •

(30)

Observe that this is a hamiltonian of the form (16).

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Thus, from ( 2 7 ) - ( 2 8 ) we know that the solution to the time-evolution operator equations is

21 January 1988

?l=q Z nfl.Q._,(pq), p= - p Z nil. Q._~ (qp) ,

q(t) =qo exp(2 sec(c)

I1

from which the exact solution is X ~ n(tan c)" ) .=~(2n-l)!!U~-l(P°q°)t ,

(31)

q(t)=qo e x p ( ~ nfl.Q~_l(poqo)t) , p(t) =Po e x p ( - 2 see(c)

P(t)=Po e x p ( - ~ nfl.Qn_,(qopo)t) . × ~ n(tan c)" ) .=j ( 2 n - 1 ) ! ! U n - ' ( q ° p ° ) t •

(32)

These equations can be simplified using the generating function for the U.(x) in (20):

We prefer not to use this operator ordering because the polynomials Q11(x) are not orthogonal and are consequently more clumsy, The first five such polynomials are

q( t) =qo exp[2t sin c exp( 2poqoc) ] ,

(33)

Qo(x) = 2 ,

p(t) =Po exp[ - 2 t sin c exp(2qoPoC)] .

(34)

Ql(x)=4x,

This gives the exact time-dependence of the operators in the quantum-mechanical theory described by the hamiltonian (29). Before concluding, we reemphasize that the general solution in ( 2 7 ) - ( 2 8 ) was obtained by reordering the operators in the hamiltonian to reexpress it as a sum of totally symmetric terms T.... This allowed us to express the operator equations of motion in terms of a set of orthogonal polynomials which are very convenient to use. It is not crucial that we choose a totally symmetric ordering of the hamiltonian. However, most other hermitian orderings give polynomials which are not orthogonal and are more cumbersome. We illustrate this point by considering two other previously studied orderings [ 6]. Suppose instead of (16) we expand the Euler hamiltonian as a linear combination of terms of the form (pnq.+q.p.); that is,

H= Z f l . ( p " q n + q n p " ) .

(35)

Q2(x) = 6(x 2 - 2) , Q3(x) =8(x 3 - 1Ix) , Q 4 ( x ) = 1 0 ( x 4 - 3 5 x 2 -I- 24) .

Had we used a Born-Jordan ordering [ 6 ], in which the classical function pmqn has the quantum form BJ(pmq ") = - 1 y" okono m-k n+l ~o . . . . . then the Heisenberg equations would also be in Born-Jordan form. The solution proceeds as above, and the relevant polynomials R.(x), the first five of which are

Ro(x) = 1 , Ri(x)=2x , R2(X)--3(x 2 - 1) , R3(X) = 4 ( x 3 -- 5x) ,

n

Ra(X)=5(x

The equations of motion then have the form

?1= Z n f l . ( p " - ' q " + q " p " - ' ) ,

(36)

P= - Z nfl.(p"q"-' +q"-~p").

(37)

tl

Solving these equations of motion requires that they be rewritten in the form

4 -

15x2 + 8 )

,

are again not orthogonal. In this paper we have made a first step in solving non-linear operator differential equations. Although Euler hamiltonians are a rather special class of hamiltonians, we hope that the method of expressing operator derivatives in terms of expansions of continuous Hahn polynomials will enable us to solve 523

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the o p e r a t o r d i f f e r e n t i a l e q u a t i o n s o f o t h e r q u a n t u m t h e o r i e s ,2,3.

O n e o f us ( C . M . B . ) wishes to t h a n k the T h e o r e t ical Physics G r o u p at I m p e r i a l College for its hospitality a n d s u p p o r t a n d the U S D e p a r t m e n t o f E n e r g y for partial f i n a n c i a l s u p p o r t . T h e o t h e r ( G . V . D . ) wishes to a c k n o w l e d g e the s u p p o r t o f a C o m m o n w e a l t h S c h o l a r s h i p f r o m the British C o u n cil a n d an O R S A w a r d . ~2 Of particular interest are some hamiltonians arising in the study of quantum mechanics in curved space. See for example ref. [8]. ~3 The connection between operator orderings and polynomials, and in particular Hahn polynomials, is very extensive. This connection will be elaborated elsewhere [9].

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References [ 1 ] W. Hahn, Math. Nachr. 2 (1949) 4. [2 ] N.M. Atakishiyev and S.K. Suslov, J. Phys. A 18 (1985) 1583. [3] R. Askey, J. Phys. A 18 (1985) L1017. [4] C.M. Bender, L.R. Mead and S.S. Pinsky, Phys. Rev. Len. 56 (1986) 2445. [5] C.M. Bender, L.R. Mead and S.S. Pinsky, J. Math. Phys. 28 (1987) 509. [ 6 ] F. Langouche, D. Roekaerts and E. Tirapegui, Functional integration and semiclassical expansions (Reidel, Dordrecht, 1982), Chap. II. [7] A. Erd61yi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher transcendental functions, Vol. I (McGraw-Hill, New York, 1953) p. 39. [8] B.S. De Witt, Phys. Rev. 85 (1952) 653; H. Leschke, A.C. Hirschfeld and T. Suzuki, Phys. Rev. D 18 (1978 ) 2834, and references therein. [9] C.M. Bender and G.V. Dunne, in preparation.