_{1}

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We review the semiclassical method proposed in [1], a generalization of this method for n-dimensional system is presented. Using the cited method, we present an analytical method of obtain the semiclassical Husimi Function. The validity of the method is tested using Harmonic Oscillator, Morse Potential and Dikie’s Model as example, we found a good accuracy in the classical limit.

Since early times of quantum theory, some quantization difficulties of non integrable systems were pointed by Einstein [2,3]. Recently, due to the pioneer discoveries of classically chaotic systems, the subject has yielded many interesting and important results both from the point of view of numerical models and (not as many) analytical proofs [4-6]. Also in this direction, the phenomena of scar [7-11] drew much attention. They showed that the Hamiltonians eigenfunctions of chaotic systems exhibit “scars” around unstable periodic orbit. An question that appears from those analyses is related to chaotic manifestation of classical chaos over the eigenfunctions in terms of quantities that are base independent [12-15]. In opposition, it has been reported that scars can exist in regions where there are no periodic orbits [

In the present contribution we begin by generalizing the semiclassical expansion [

This paper is organized as follows. In Sections 2 and 3 we present the method, we closely follow ref. [

Let us consider a classical one degree of freedom Hamiltonian of the form

where p stands for the particle momentum and q for its position. We make a change of variables

where

The Hamiltonian can then be rewritten as

with.

We can write as a Taylor expansion,

where.

The classical equations of motions are

We choose the quantum Hamiltonian in order to have, if is a coherent field state.

We make our semiclassical expansion around a quantum operator. The difference

will be considered as a perturbation. We choose the semiclassical Hamiltonian, , in a way that for a coherent initial state, all expectation values of point classical observables will be precisely reproduced.

The semiclassical Hamiltonian which satisfies this condition is [

We can write the semiclassical evolution operator for an one degree of freedom, observing that

where is the well known displacement operator

and is given by 5, 6 and

Thus, for the N dimensional case we have

(9)

where, and is the semiclassical evolution operator related to the k-th degree of freedom, note that it depends solely on but in general we have

The phase is given by

where is the classical Lagrangian of the (independent) systems. In equation (9) we chose = 0^{1}, what can be done choosing a specific form of the semiclassical Hamiltonian, see [

where

In general is a function of all. In the next sections we use the fact that the labels of coherent states follow the classical trajectories.

We consider a two degrees of freedom system, which the complete Hamiltonian is given by

where represent the autonomous dynamics of the degree of freedom 1 (2) and is their interaction. The semiclassical Hamiltonian has the following form^{2}

and by definition we have

As discussed in section II we rewrite the Hamiltonian (12) in the following form where . We make a perturbation expansion about Using Schrödinger’s equation, where we will always use as initial state and are coherent states. Thus, after some straightforward algebraic manipulations [

where The general problem of convergence of the serie (14) is an open problem. The convergence of the method was demonstrated for the quartic oscilator [

The Q-function or Husimi’s function is, see refs. [4,19], defined by:

is a density operator, and is the harmonic coherent state according to the definitions:

q and p are position and momentum operator respectively, and the mean is calculated in the coherent state From this definition, we are able to write the Husimi function as

is the system Hamiltonian eigenfunction, and is the harmonic coherent state in three dimensions , it is given by

where, , is the momentum related with, and

For the simplest case of the Harmonic Oscillator, using equation [

In terms of Q and P, we have

The Morse potential is used to model diatomic molecules, it is defined as :

where

The values are the equilibrium position of the center of mass, is the reduced mass of the two atoms and r is the distance between the atoms. The constant D defines the minimum value o the potential wich is. The constant determines the potential range. The Hamiltonian that describes the center of mass can be written as:

where L is the angular momentum. The time independent Schrödinger equation is:

We can write the wavefunction as

where is the spherical harmonics:

For L = 0 case we find the eigenvalues:

where and

(31)

and for the eigenfunctions:

A_{1} is fixed by normalization, is the gamma function.

Following the definition (20), we obtain the Husimi the Function [

where. The exact Husimi function is obtained by numerically integration of (33).

The semiclassical expansion, as defined above, gives us the time evolution of a quantum state as a perturbative expansion. An eigenstate has only a time dependent phase as its dynamics. The nearest semiclassical scenario we can build is to choose a coherent state with the same energy. The time dependence can be eliminated by a time integration, i.e., a mean in time. This integration can be justified noting that as we are dealing with eigenstates we have not time precision. Under this considerations we may write the semiclassical Husimi function as

The states and are coherent states of the harmonic oscillator. is defined as

andwhere x and are parameters of the Husimi Function, and correspond to the classical canonical conjugate pairs redefined as in Equations (16) and (17). In case of classical mixed dynamics we must perform a mean considering all possible initial condition with the same specific energy. We should also use adequate coherent state base for each algebra. This semiclassical Husimi function is calculated by taking a long time mean, formally we write it as

As we known, classically chaotic systems stay longer times [

The semiclassical Husimi distributions is determined by the classical trajectory solely. In general the semiclassical Husimi function is obtained by numerical methods, it is the case of all chaotic model.

For the Morse potential, with L = 0, we obtain the classical trajectory [

We also have, and we can choose and choosing the energy as into (36) to obtain the semiclassical Husimi function. The mean (35) is obtained by a numerical integration. In

Now consider the Harmonic potential with a natural frequency, its classical dynamics is given by

and

We redefined Q and P in a such way that the Hamiltonian can be written as

Substituting (38) and (39) into (36) we obtain a semiclassical Husimi Function for the Harmonic oscillator with an energy. Without any loss of generality we can use, and. In

In the Figures 1 to 7 we have used classical integrable models, although that our approach is also useful for non-integrable thus let us take a look in Dikie model [

where

is the harmonic oscillator natural frequency, is precession frequency, G and are coupling constants.

In the harmonic term of (42) and are bosonic anihilation and creation operators of harmonic oscillator, is the angular momentum operator in k direction and. In order to obtain the Semiclassical Husimi function for the Dickie model we have to integrate numerically the corresponding classical equations of motion and calculate the mean (35).

In ^{3} function that can be found in ref. [

Now we quantify the quality of the approximation using the function, which is defined as

where H is the exact Husimi function and H_{sc} is the semiclassical Husimi function. Due to the symmetry we have chosen p = 0. As we increase the principal quantum number (n), in the classical limit, we hope we have. In order to see the classical limit, let us define the function, which is

Suppose we have. It means that quantum description of the state, in the Husimi’s representation, is almost contained in the semiclassical one. In spite of that we can say that the quantum classical difference becomes smaller, as expected. Of course it does not mean that we have no quantum features^{4}, it only means that Husimi is not a good observable for this situation [

We show that the semiclassical Husimi function reproduces the major features of the quantum one, in the particular harmonic case, we show that the first semiclassical term is able to reproduce the Husimi function with a increasing accuracy as we increase the principal quantum number n. We must remark that there is no demonstration that would suggest an existence of the limit procedure which turns quantum corrections less important in terms of the proposed semiclassical expansion. The building blocks of semiclassical Husimi Function are the classical trajectories, then we can conclude that classical periodic orbits (stable or unstable) contribute with higher weight. The fidelity decay rate has a gaussian regime that is only perturbation potential dependent, although its validity is determined by.

The author is grateful to Fapesb for partial financial support. The author also acknowledge, A. R. Bosco de Magalhaes, M. C. Nemes, Fernanda Alves de Oliveira for helpful comments.