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arXiv:0803.2968 · Journal of Physics A Mathematical and Theoretical

Some features of the statistical complexity, Fisher–Shannon information and Bohr-like orbits in the quantum isotropic harmonic oscillator

Jaime Sañudo A, Ricardo López‐Ruiz

Published Jun 11, 2008• 34 citations
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Abstract

The Fisher-Shannon information and a statistical measure of complexity are calculated in the position and momentum spaces for the wave functions of the quantum isotropic harmonic oscillator. We show that these magnitudes are independent of the strength of the harmonic potential. Moreover, for each level of energy, it is found that these two indicators take their minimum values on the orbitals that correspond to the classical (circular) orbits in the Bohr-like quantum image, just those with the highest orbital angular momentum.

DOI
10.1088/1751-8113/41/26/265303

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The Fisher-Shannon information and a statistical measure of complexity are calculated in the position and momentum spaces for the wave functions of the quantum isotropic harmonic oscillator. We show that these magnitudes are independent of the strength of the harmonic potential. Moreover, for each level of energy, it is found that these two indicators take their minimum values on the orbitals that correspond to the classical (circular) orbits in the Bohr-like quantum image, just those with the highest orbital angular momentum.
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Abstract We first present the analytical solutions to the Schrödinger equation with the parity-restricted harmonic oscillator V ( x ) = 1 2 m ω 2 x 2 ( x > 0 ) and then calculate the Shannon information entropies S x n and S p n both in position space and in momentum space. It is interesting to find that the variation of the Shannon information entropy S p n in momentum space is different from our previous studies, i.e. the S p n first increases with the quantum number n and then decreases with the number n . This is a new and abnormal phenomenon and may be explained by the parity-restricted system. The BBM inequality is verified to be saturated, but the sum of the entropies first increases with the number n and then decreases with it. The entropy densities ρ s ( x ) and ρ s ( p ) are also demonstrated. We find that the Fisher entropy is exactly given by I F = 8 n + 6 , n = 0 , 1 , 2 , … .
Information‐based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length ( x c ). In the symmetric case, a wide range of x c has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary‐time propagation scheme. For asymmetric confinement, a variation‐induced exact diagonalization procedure is also introduced, which produces very high quality results. One finds that, in symmetric confinement, after a certain characteristic x c , all these properties converge to respective values of a free harmonic oscillator. In the asymmetric situation, excited‐state energies always pass through a maximum. For this potential, the classical turning point decreases, whereas well depth increases with the strength of asymmetry. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter. image