Extensive studies on some analytical determinations of integrals of the associated Laguerre polynomials already exist in the literature e. In this paper, we concentrate our task on the calculations of information content arising from the radial part of the complete wave functions. This is verified because the radial wave function is required to be normalized to unity, meaning the density of radial direction. Since the radial wave functions in the above-mentioned potentials are largely controlled by the properties of the associated Laguerre polynomials, it is natural to think that the information content of the system owes a great deal to their properties.
In a previous work, we considered the relative Fisher information of the hydrogen-like atoms by setting the orbital as a reference state [ 8 ]. The relative Fisher information is one of similarity or distance measures [ 9 ], but it is so far underrepresented in utility compared to the Fisher information see e. This paper is organized in the following: In the next section, we introduce the relative Fisher information and define notations.
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In section 3 , we calculate it for systems governed by the Morse potential by setting the ground state as the reference and investigate the behavior. Then, we provide the exact expression of the information measure for the radial density of the isotropic quantum oscillator in D-dimension. The behavior as functions of the dimension and quantum numbers are investigated in section 4. In section 4. Summary and conclusion are given in the last section.
The information contained in the densities of arbitrary excited states is examined, by which we can measure how the family of excited states are similar to the ground state density. This polynomial satisfies an orthogonality relation [ 17 ]:. By the definition equation 1 , the relative Fisher information measured from the ground state density :. In the expression of the integral J 2 , a property of the associated Laguerre polynomials has invoked.
This condition includes the ones coming from the other two integrals. Although some relevant formulas to compute the integral of the product of two Laguerre polynomials with different degrees and different orders for weight functions may produce an analytical expression e. This behavior contrasts with the one of the Shannon information entropy for the quantum harmonic oscillators and hydrogen atoms in three dimensions reported in [ 7 ], where the entropy values exhibit rapid increases for small n but reach saturated values in higher excited states.
We also note that the energy eigenvalues of the states for this model are a quadratic function of n [ 15 , 16 ]; with a constant. Therefore, we do not have a positive reason for attributing the behavior in figure 1 to the dependence of energy levels on n.
Figure 1. The quantum harmonic oscillator in several dimensions has attracted much attention as a prototypical physical model of atomic systems. In this section, we focus on the relative Fisher information of these systems analytically and evaluate it numerically how we can quantify the information of excited states against the ground state. The normalization constant N n , l which depends on both k and D is given by.rieracrerothsmer.cf
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This indicates that the radial wave function R n , l r vanishes at infinity, which means that it is square integrable and it must be zero at the origin. Substituting this into the definition equation 1 , the relative Fisher information d F [ n , l : 0, 0 ] measured from the ground state radial density is thus given as. To calculate the integral I 1 , one way is to recall the normalization integral provided in the Morse's original paper equation 12 in [ 15 ] :.
Another way for the calculation of I 1 is to perform the integration by parts of I 2 by means of the property Note that the integral I 2 itself vanishes as we describe below. This leads to a relation:.
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The integral on the right-hand side can further be calculated by the orthogonality relation as. Thus one has the following expression. One can readily confirm that the former expression equation 24 is equivalent to this. On the other hand, the direct application of the orthogonality relation equation 6 readily reduces the integral I 3 to the following expression,.
In the following, we present numerically calculated behavior of the value of d F [ n , l : 0, 0 ]. It decreases monotonically as the dimension increases and approaches a constant value depending on n.
The excited states have higher information values than the lower ones for the same dimension. Figure 2. Figure 3. The dependency of the relative Fisher information d F [ n , l : 0, 0 ] of the radial wave function on the principal quantum number n for the isotropic quantum oscillator with two fixed orbital quantum number l.
For the identical n , the values tend to be small as the dimension increases. Figure 4. The dependency of the relative Fisher information d F [ n , l : 0, 0 ] of the radial wave function on the orbital quantum number l for the isotropic quantum oscillator with two fixed principal quantum number n. This section focuses on a specialization to the angular part of the wave function for the three-dimensional quantum harmonic oscillator.
As a by-product, we derive also the integral formula of the weighted product of the two associated Legendre polynomials with a little ingenuity, which is different from the previously known calculation. In the preceding studies [ 6 , 7 ] , the explicit expression of the Fisher information for the full density function both the radial and the angular parts are taken into consideration of the hydrogen-like systems i. According to the results, it is independent of the orbital quantum number l and it rapidly decreases with increasing the principal quantum number n , reflecting de-localized electronic densities in excited states.
The normalization for it is assured e. The relative Fisher information between any pairs of quantum numbers l , m and l ', m ' is by definition given as.
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To get the exact expression of the above for the arbitrary reference angular density specified by involves challenging integrations. Since the relation holds with and making use of the recurrence relation derived from the definition equation 31 :. The first and third integrals of the above can be computed respectively by means of the following useful orthogonality relations that one encounters frequently in many applications:.
We now turn to the integration of the first line of equation 35 , which we label it as K. Recall that is the solution of the associated Legendre differential equation:. Making a similar equation by replacing l with in equation 40 and adding both sides, we have. Remark: as a by-product, we obtain a formula for the integral below, which is not listed in the table of integrals, e. From equation 43 , one arrives at the expression for the relative Fisher information.
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Nonequilibrium phase diagram for the dynamics of the oscillators. A , — Published 14 August Abstract Spontaneous synchronization between coupled periodic systems occur in a wealth of classical physical setups. Research Areas. Physical Systems. Ultracold gases. Master equation Path-integral methods.
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