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In this paper we introduce a simple procedure for computing the macroscopic quantum behaviour of periodic quantum systems in the high energy regime. The macroscopic quantum coherence is ascribed to a one-particle state, not to a condensate of a many-particle system; and we are referring to a system of high energy but with few degrees of freedom. We show that, in the first order of approximation, the quantum probability distributions converge to its classical counterparts in a clear fashion, and that the interference effects are strongly suppressed. The harmonic oscillator provides a testing ground for these ideas and yields excellent results.

The superposition principle lies at the heart of quantum mechanics, and it is one of its features that most distinctly marks the departure from classical concepts [

At the macroscopic scale, all happen as if, by a conspiracy of nature, the naked quantum is hidden, leaving us with an apparent world described consistently in a classical language [

In this paper we introduce a simple procedure to compute the high energy regime of a general density matrix for periodic quantum systems. We show that, in the first order of approximation, the quantum probability distribution converges to its classical counterpart in a clear fashion, and that the interference effects are strongly suppressed. The harmonic oscillator provides a testing ground for these ideas as we will illustrate. This problem has been considered before by Cabrera and Kiwi [

The remainder of the paper is organized as follows. In Section 2, we introduce the general procedure. The results for the harmonic oscillator are presented in Section 3. Finally, some conclusions and remarks are given in Section 4.

It is generally accepted that the classical and quantum probability density functions for periodic systems approach each other in a locally averaged sense when the principal quantum number becomes large [8-11], i.e.

where the interval

where

Note that Planck’s constant keep a finite value, so

Now we focus on the general problem. Let us consider a physical system described completely by the Hamiltonian

The corresponding density operator reads

The terms

with

To explain our procedure, let us consider the coordinate representation of the matrix density. Its components are given by

with

It is well known that for periodic quantum systems, nodes are always present in the density matrix (by means of

where

We now study the temporal behaviour. We observe that Equation (7) is an almost periodic function of the time (except for the diagonal terms), even for arbitrarily high energies. It is clear that the off-diagonal terms are a rapidly oscillating functions of the time when the energies differ significantly, and a slowly varying functions if the energies are close each other. This means that in the macroscopic regime becomes important only the interference between states with high energies

where we used the standard definition

We conclude this section with the final expression for the macroscopic density matrix in coordinate representation, i.e.,

where we have included the spatial and temporal analysis described above. We observe that (10) is the Fourier expansion of a classical function, which can be immediately identified with the classical probability distribution

The harmonic oscillator provides a testing ground for these ideas as we now illustrate. We first consider again the spatial behaviour. The energy eigenfunctions and eigenvalues of the corresponding Schrödinger equation are well known. The quantized energies are given by

with

where

We first calculate the Fourier coefficients appearing in Equation (8). The corresponding inverse Fourier transform is reported in many handbooks of mathematical functions [17,18]:

where

The asymptotic behaviour of

where

Szegö also shows that the iteration terms are strongly suppressed compared with the first order of approximation in the limit

with

In Equation (16) we have used the approximation

Finally the macroscopic behaviour of

(17)

where

(17) to

We observe that the diagonal terms in (17) are simply

with

The exact classical limit requires the off-diagonal terms equal to zero (means no interference), however according to our result (17) this is never attained. In figure 2 we present the asymptotic behaviour of

Regarding to the temporal behaviour, from Equation (7) it follows that the off-diagonal temporal terms in the matrix density are simply

To complete this section we write an asymptotic expression for the expectation value of an arbitrary observable

where

It is well known that the concepts of classical and macroscopic systems are distinct, as the existence of macroscopic quantum phenomena (such as superconductivity) demonstrates, but the behaviour of most macroscopic systems can be described by classical theories [

quantum states. It would be interesting to test these effects with real quantum systems approaching the microscopic-macroscopic boundary, taking Rydberg atoms or neutron interferometry for example. At higher energies these macroscopic quantum effects are so strongly suppressed that it is impossible to detect, leaving us with an apparent world described consistently in a classical language. With the appropriate experimental devices such effects should be observed even in our real-world experience, however nowadays it is impossible.

Technical difficulties in the calculation of the Kepler problem are greater than in the simple case which we have treated here, however we can definitely foresee that our procedure gives its correct macroscopic behaviour. Even though our approach gives the correct classical results for periodic quantum systems, it is far from the general solution to the classical limit problem. Several other questions remain to be resolved as the study of unbound systems and entanglement, but it is not clear in the framework adopted here. The environment induced decoherence approach successfully resolves these problems. Based on our results, in our future research we concentrate on the definition of the classical regime, which is of considerable importance in the study of quantum chaos.