ЯДЕРНАЯ ФИЗИКА, 2008, том 71, № 5, с. 863-870
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
QUANTIZATION OF A NONLINEAR OSCILLATOR AS A MODEL OF THE HARMONIC OSCILLATOR ON SPACES OF CONSTANT CURVATURE: ONE- AND TWO-DIMENSIONAL SYSTEMS
© 2008 J. F. Carinena1)*, M. F. Raniada1)**, M. Santander2)***
Received August 20, 2007
The quantum version of a nonlinear oscillator, previously analyzed at the classical level, is studied first in one dimension and then in two dimensions. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + Xx2) 1 and with a A-dependent nonpolynomial rational potential. The quantization procedure analyzes the existence of Killing vectors and makes use of an invariant measure. It is proved that this system can be considered as a model of the quantum harmonic oscillator on two-dimensional spaces of constant curvature.
PACS:05.20.Jj, 03.65.-w
1. INTRODUCTION The following nonlinear differential equation
(1 + Xx2)X — AxX2 + a2 x = 0 (1)
(A a constant),
was studied in [1, 2] as an example of a nonlinear oscillator and it was proved that it admits solutions representing nonlinear oscillations with "quasi-harmonic" form, that is, periodic trigonometric solutions but with amplitude dependence of frequency.
This particular nonlinear system has been studied from the Lagrangian and Hamiltonian viewpoint [3] and there it was proved that it can be generalized to the two-dimensional case, and even to the n-dimensional case and that these higher dimensional systems are superintegrable with 2n — 1 quadratic constants of motion. Moreover, a geometric interpretation of the higher dimensional systems was proposed in relation with the dynamics on spaces of constant curvature (two related papers are [4, 5]).
A study of the quantum version of this system, but restricted to the one-dimensional case, has been recently presented in [6, 7]. The main objective of this article is to present a report on the quantization of this nonlinear system using as an approach the theory of symmetries. The plan of the paper is as follows:
'■'Departamento de Física Teórica, Facultad de Ciencias,
Universidad de Zaragoza, Spain.
2)Departamento de Física Teórica, Facultad de Ciencias, Universidad de Valladolid, Spain. E-mail: jfc@unizar.es E-mail: mfran@unizar.es E-mail: msn@fta.uva.es
Section 2 is devoted to the study of the quantization of the one-dimensional system and Section 3 to the quantization of the two-dimensional system. Finally in Section 4 we make some final comments.
2. ONE-DIMENSIONAL NONLINEAR QUASI-HARMONIC OSCILLATOR
In Lagrangian terms, the Eq. (1) can be obtained from the following function:
1 ( v2 L(x,vx;X) = -
a
x
2 \1 + Ax2
> (2)
where for A < 0 the study of the dynamics is restricted to the interval x2 < 1/|A|, where the kinetic energy is positive definite. We see that for A < 0 the potential V(x; A) is a well with an infinite wall at x2 = 1/|A|; therefore, all the trajectories will be bounded. For A > 0 we have that V(x; A)^(1/2)(a2/A) for x ^ ^ ±(x; so for small energies the trajectories will be bounded but for E(A) > (1/2)(a2/A) the trajectories will be unbounded. It can be proved that
1. If A < 0, then the general solution of (1) is given
by
x = A sin(wi + 4), a2 = (1 + AA2)u2,
for all the values of the energy E.
2. If A > 0, then the general solution is given by
x = A sin(wi + 4), a2 = (1 + AA2)u)2,
when the energy E is smaller than the value Ea,\ = = a2/(2A), and by
x = B sinh(Qi + 4i), a2 = (AB2 — 1)Q2,
y
x
y
2
x
Fig. 1. Plot of the one-dimensional potential V(x; A) = = (1/2)(a2x2)/(1 + Ax2), as a function of x, for a = 1 and A < 0.
when the energy E is greater than this value, E >
> Ea,X.
Figures 1 and 2 show the form of the potential V(x; A) for several values of A.
Quantization and A-Dependent Schrodinger Equation
Lagrangian (2) has two interesting characteristics: the kinetic term involves a position-dependent mass and the potential has a nonpolynomial character (this system is not an harmonic oscillator perturbed by higher order terms). In the general case of a system with position-dependent effective mass the transition from the classical system to the quantum one is a difficult problem because of the ambiguities in the order of the factors (see [8—12] and references therein). Therefore, different forms of presenting the kinetic term in the Hamiltonian H, as, for example,
-ï
1
m(x)
p2 + p
2
1
r-3
1
p
y/m(x) \Jm(x)
m(x)
H
p
m(x)
p
are equivalent at the classical level but they lead to different nonequivalent Schrodinger equations. In what follows we will discuss a procedure for quantizing the system defined by Lagrangian (2) that is based on the analysis of the symmetries of the kinetic term.
The A-dependent kinetic term
1
T(A) = 5
1 +Ax2
is invariant under the action of the vector field X\ (x) given by
d
Xx(x) = y/l + Ax2
dx'
Fig. 2. The same as in Fig. 1, but for a = 1 and X> 0.
in the sense that if Xfx denotes the natural lift to the velocity phase space R x R (tangent bundle in geometric terms) of the vector field X^,
X{ = VTT^9 ■ ( Xxv* A 5
dx
+
dvx
Vl +Ax2,
then we have Xfx (T\) = 0. In differential geometric terms this property means that the vector field X^ is a Killing vector of the one-dimensional metric
1
ds2x =
1 +Ax2
dx
Proposition 1. The only measure on R that is invariant under the action of the vector field X\ is given by
dp\ = | = ) dx,
Vl +Ax2, up to a constant factor.
Proof: This property is proved by direct computation.
This proposition indicates that the operator H, representing the quantum Hamiltonian, must be self-adjoint not in the standard space L2 (R), but in the space L2 (R,d/^\). The classical Hamiltonian of the A-dependent oscillator is given by
1
1
H 2mPx + 29 \1 + Xx2
x
(3)
Px = y/l + A x2px
where g denotes g = ma2 + Aha and Px denotes the Noether momentum determined by X\, that is, a constant of motion for the geodesic motion. In this way the transition from the classical system to the quantum one is given by defining the operator
d
Px = -ihyj 1 +Ax2 —, dx
1
1
HflEPHAH OM3MKÂ tom 71 № 5 2008
QUANTIZATION OF A NONLINEAR OSCILLATOR so that we obtain the following correspondence
(1 + Ax2)p2x -h2 (Vl + Ax2^j
\/l + Xx2-^-
dx
in such a way that the quantum version of the Hamiltonian becomes
H - - — (I + Xx2) — - —Xx— + 2 m dx2 2 m dx
1
+ - Q 2
x
1 + Xx2
At this point it is convenient to simplify this operator by introducing adimensional variables (x, A) defined by
x
'JL
ma
, fma\
E = (ha)e.
Then Hamiltonian H takes the following form:
H
(4)
(h
a
and the Schrodinger equation H^ = E^ reduces to the following adimensional form
(1 + Ax2)-^^ + Aa;jL^ -
dxdx
(5)
- (1+A)
x
1 + Ax2
tf + (2e)tf = 0.
It is known that in the A = 0 case the asymptotic behavior at the infinity suggests a factorization for the wave function. The idea is that a similar procedure can be applied to this A-dependent equation. Let us first denote by ^^ the following function ^^ = (1 + + Ax2)-1/(2A) that vanishes in the limit x2 — ^ (in the case A > 0) or in the limit x2 — — 1/A (in the case A < 0). We have
d2
d
dx2
dx
- (1 + A)
x
1+Ax2
^(x; A) = h(x; A)(1 + AX2)"1/(2A),
(6)
and then the new function h(x; A) must satisfy the differential equation
(1 + Ax2)h" + (A — 2)xh' + (2e — 1)h = 0, (7) h = h(x; A),
that represents a A deformation of the Hermite equation.
It is clear that the limit when A - 0 is well defined and we obtain
lim A) = he-(1/2)x, h = h(x, 0),
as well as the Hermite equation
h" — 2xh' + (2e — 1)h = 0.
That is, all the characteristics of the standard quantum harmonic oscillator are recovered.
3. TWO-DIMENSIONAL NONLINEAR QUASI-HARMONIC OSCILLATOR
The following two-dimensional Lagrangian:
m = I (T^TI.) X (8)
2 V 1 + Xr2
Thus, ^^ is the exact solution in the very particular case e = 1/2 and represents the asymptotic behavior of the solution in the general case. Consequently, this property suggests the following factorization:
ay2 i r2 \
x [V2 + v2y + X{xvy - yvx)2] - — (^1 + Ar2j ,
2 2,2 r = x + y ,
was studied in [3] and it was proved that it represents, at the classical level, the appropriate two-dimensional generalization of Lagrangian (2). In fact, the general solution of the Euler—Lagrange equations which are given by
(1 + Xr2)x — A[i2 + y2 + X(xy — yx)2] x + a2x = 0, (1 + Xr2)y — X [x2 + y2 + X(xy — yx)2] y + a2 y = 0 is:
1. If X < 0, then the general solution is given by x = A sin(wi + 01), y = B sin(wi + 42),
for all the values of the energy E.
2. If X > 0, then the general solution is given by
x = A sin(wi + 01), y = B sin(wi + 42),
when the energy E is smaller than a certain value Ea,x, and by
x = A sinh(Qi + 01), y = B sinh(Qi + 02), when the energy E is greater than this value, E >
> Ea,\.
The coefficients A and B are not arbitrary but related with a and u or with a and Q for the oscillatory and unbounded motions, respectively. So the general solution represents "quasi-harmonic" nonlinear oscillations in the case of bounded motions and high-energy scattering solutions when X > 0.
5 MEPHA^ OH3HKA tom71 № 5 2008
x
x
3.1. The Harmonic Oscillator on Spaces of Constant Curvature: Three Different Approaches
In differential geometric terms, the three spaces with constant curvature, sphere S2, Euclidean plane E2, and hyperbolic plane H2, can be considered as three different situations inside a family of Rieman-nian manifolds M2 = (S1, E2, H) with the curvature k as a parameter k e R. In order to obtain mathematical expressions valid for all the values of k, it is convenient to make use of the following k-trigonometric functions:
CK (x) =
Sk (x) =
[cos y/Hx if k > 0,
1 if k = 0,
[ cosh y/^nx if k < 0,
-7= sin J~KX if k > 0,
y/K v
x if k = 0,
sinh yJ—KX if k < 0,
1
L(K) = 0 « + SKRH) - ^2T2(R), (9)
tk (r) =
sk(r) ck(r) '
In this way, the harmonic oscillator on the unit sphe
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