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arXiv:quant-ph/0312095 · Physical Review A

Quantum-information entropies of the eigenstates and the coherent state of the Pöschl-Teller potential

Rajneesh Atre, Anil Kumar, Nagaraja Kumar, Prasanta Kumar Panigrahi

Published May 14, 2004• 68 citations

Abstract

The position and momentum space information entropies, of the ground state of the P\"oschl-Teller potential, are exactly evaluated and are found to satisfy the bound obtained by Beckner, Bialynicki-Birula, and Mycielski. These entropies for the first excited state, for different strengths of the potential well, are then numerically obtained. Interesting features of the entropy densities, owing their origin to the excited nature of the wave functions, are graphically demonstrated. We then compute the position space entropies of the coherent state of the P\"oschl-Teller potential, which is known to show revival and fractional revival. Time evolution of the coherent state reveals many interesting patterns in the space-time flow of information entropy.

DOI: 10.1103/physreva.69.052107

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The position and momentum space information entropies, of the ground state of the P"oschl-Teller potential, are exactly evaluated and are found to satisfy the bound obtained by Beckner, Bialynicki-Birula, and Mycielski. These entropies for the first excited state, for different strengths of the potential well, are then numerically obtained. Interesting features of the entropy densities, owing their origin to the excited nature of the wave functions, are graphically demonstrated. We then compute the position space entropies of the coherent state of the P"oschl-Teller potential, which is known to show revival and fractional revival. Time evolution of the coherent state reveals many interesting patterns in the space-time flow of information entropy.

Shannon entropy for lower position and momentum eigenstates of Pöschl—Teller-like potential is evaluated. Based on the entropy densities demonstrated graphically, we note that the wave through of the position information entropy density ρ( x ) moves right when the potential parameter V 1 increases and its amplitude decreases. However, its wave through moves left with the increase in the potential parameter | V 2 |. Concerning the momentum information entropy density ρ( p ), we observe that its amplitude increases with increasing potential parameter V 1 , but its amplitude decreases with increasing | V 2 |. The Bialynicki—Birula—Mycielski (BBM) inequality has also been tested for a number of states. Moreover, there exist eigenstates that exhibit squeezing in the momentum information entropy. Finally, we note that position information entropy increases with V 1 , but decreases with | V 2 |. However, the variation of momentum information entropy is contrary to that of the position information entropy.

In this paper we calculate the position and momentum space information entropies for the quantum states associated with a particular physical system, i.e. the isotonic oscillator Hamiltonian. We present our results for its ground states, as well as for its excited states. We observe that the lower bound of the sum of the position and momentum entropies expressed by the Beckner, Bialynicki-Birula and Mycielski (BBM) inequality is satisfied. Moreover, there exist eigenstates that exhibit squeezing in the position information entropy. In fact, entropy squeezing, which occurs in position, will be compensated for by an increase in momentum entropy, such that the BBM inequality is guaranteed. To complete our study we investigate the amplitude squeezing in

x

and

p

-quadratures corresponding to the eigenstates of the isotonic oscillator and show that amplitude squeezing, again in

x

, will be revealed as expected, while the Heisenberg uncertainty relationship is also satisfied. Finally, our numerical calculations of the entropy densities will be presented graphically.

Shannon entropy for the position and momentum eigenstates of the symmetrically trigonometric Rosen–Morse potential for the lower states n = 1–4 is evaluated. The position information entropies S x for n = 1,2 are presented analytically. Some interesting features of the information entropy densities ρ s ( x ) and ρ s ( p ) are demonstrated graphically. We find that the ρ s ( p ) is inversely proportional to the range of potential a and the S x decreases with increasing the potential depth D . In particular, we note that the S x might become negative for some given parameters a and D . The Bialynicki-Birula–Mycielski inequality is also tested for a number of states and is found to generally hold well.

Abstract Shannon entropy for the position and momentum eigenstates of an asymmetric trigonometric Rosen–Morse potential for the ground and first excited states is evaluated. The position and momentum information entropies and are calculated numerically. Also, it is found that is obtained analytically and increases with the potential depth and width. Some interesting features of the information entropy densities and are demonstrated graphically. The Bialynicki‐Birula–Mycielski inequality is also tested and found to hold good.