fulltext

IL NUOVO CIMENTO VOL. 112 B, N. 6 Giugno 1997 NOTE BREVI

Heisenberg equation and constants of the motion for an anharmonic

oscillator in the high-frequency limit

L. DEFALCO(1), R. MIGNANI(2) and R. SCIPIONI(3)

(1_{) Dipartimento di Fisica, I Università di Roma “La Sapienza”} P.le A. Moro 2 - I-00185 Roma, Italy

(2_{) Dipartimento di Fisica “E. Amaldi”, III Università di Roma} Via della Vasca Navale 84, I-00146 Roma, Italy

INFN, Sezione di Roma I, c/o Dipartimento di Fisica, I Università di Roma “La Sapienza” P.le A.Moro 2, I-00185 Roma, Italy

(3_{) Department of Physics, Theory Division, Lancaster University - Lancaster LA1 4YB, UK} (ricevuto il 17 Maggio 1996; approvato il 25 Marzo 1997)

Summary. — Using the best approximation of the time evolution operator of a

quartic oscillator in the high-frequency limit, we derive its equation of the motion in the Heisenberg picture and show that such an anharmonic oscillator admits a wider class of constants of the motion than the standard harmonic oscillator.

PACS 11.10 – Field theory.

Recently, we discussed [1-3] some properties of anharmonic-oscillator systems by exploiting an approximation method introduced by Burrows, Cohen and Feldman [4]. In particular, we derived by such a procedure the best approximation to the time-evolution operator of an anharmonic oscillator [2].

In this note, we want to apply such a result to discussing the equation of motion of the anharmonic oscillator in the Heisenberg picture. We shall show that, at least in the approximation considered, the anharmonic oscillator admits a wider class of constants of the motion than the usual harmonic oscillator.

Let us consider the anharmonic oscillator described by the Hamiltonian

H(a , b) 4 p 2 2 1 a 2x 2 1 bx4, (1)

where we assume both a and b to be positive (actually, the parameter a may be negative as well [4], but, in view of our interpretation of a as a squared frequency, we choose it positive).

By exploiting the approximation procedure of ref. [4], we have shown that, in the high-frequency limit (v KQ), the time-evolution operator for a system described by

L.DE FALCO,R.MIGNANIandR.SCIPIONI 930 eq. (1) is given by T(t) 4exp

k

k

l

l

We want to stress that (2) provides the best approximation to the time-evolution operator of the quartic oscillator (1) in the non-perturbative region, i.e. for large values of the parameter b.

Let us now consider an arbitrary operator A (not explicitly depending on time). In the Heisenberg picture, the time-dependent operator AH(t) reads

AH(t) 4ei[vH01 ( 2 bOv) H0 2_{] t}

Ae2i[vH01 ( 2 bOv) H02] t_. (4)

The Heisenberg equation of motion corresponding to (4) is given by 2idAH(t) dt 4 v[H0, AH(t) ] 1 2 b v [H 2 0, AH(t) ] (5)

that, after some algebraic manipulations, can be put in the form 2idAH(t)

dt 4 v[H0, AH(t) ] 1 2 b

v ]H0, [H0, AH(t) ]( ,

(6)

where ], ( denotes the anticommutator.

It is interesting to note that the equation of motion (6) involves only the harmonic-oscillator Hamiltonian H0. Moreover, it is evident from eq. (6) that all the conserved operators for the related harmonic oscillator (3) are still conserved for the system (1), but now even a non-conserved quantity for H0can have such a property for the quartic oscillator (1). Indeed, let us assume that AH(t) is not conserved for H0. Then, the commutator between AH(t) and H0 is, in general, a non-vanishing, time-dependent operator MH(t):

[H0, AH(t) ] 4MH(t) . (7)

Moreover, if MH(t) is such to satisfy vMH(t) 42

2 b

v ]H0, MH(t)( ,

(8)

it follows from eq. (6) that

dAH(t) dt 4 0 ,

i.e. AH(t) is a conserved quantity for the anharmonic oscillator (1).

Therefore, in the approximation considered, the constants of the motion of the quartic oscillator (1) include not only the quantities conserved for the usual oscillator,

HEISENBERG EQUATION AND CONSTANTS OF THE MOTION ETC. 931 but also all those operators defined by the system of equations

.

/

´

Although such a result is strictly valid only in the approximation considered, and in the high-frequency limit (in the low-frequency case, it is

T(t) 4exp [2i(2H0)4 /3t]

and therefore the two classes of conserved quantities do coincide), it suggests that anharmonic-oscillator systems are expected, in general, to admit a wider class of constants of the motion than the harmonic ones.

It is, however, worth stressing the exact meaning of the above statement. Indeed, it is well known that, for any system, the number of independent integrals of motion is actually fixed by the number of the system degrees of freedom. In particular, for one-dimensional systems, we have only two such independent integrals. However, we have a priori an infinite number of constants of the motion, namely, all possible functions of the two independent integrals. Therefore, for both harmonic and anharmonic systems, there are infinitely many possible constants of the motion. Our result apparently shows that some conserved quantities for the anharmonic oscillator (at least in the approximation considered) are not conserved for harmonic-oscillator systems. In particular, eq. (9) allows one to find out the explicit analytic form of such conserved operators. This is by no means trivial, because, as is well known, the conservation laws for a system are related to its symmetry properties, and therefore also the explicit functional form of the conserved operators plays a fundamental role in the study of the system dynamics. As an example, for a 3-dimensional system it is by no means trivial to establish the conservation of the angular momentum (although it is clearly related to the conservation of the initial position and momentum), because it implies the rotational invariance of the system.

R E F E R E N C E S

[1] DEFALCOL., MIGNANIR. and SCIPIONIR., Europhys. Lett., 29 (1995) 659. [2] DEFALCOL., MIGNANIR. and SCIPIONIR., J. Phys I, 5 (1995) 535. [3] DEFALCOL., MIGNANIR. and SCIPIONIR., Phys. Lett. A, 209 (1995) 61. [4] BURROWSB. L., COHENM. and FELDMANT., J. Math. Phys., 34 (1993) 1.