4 squeeze angle as shown in Eqs. (13) for the free particle and (18) for the harmonic oscillator. Then, the random-phase entropy is Even for r = 0, the damping effect contributes to the lower bound of entropy by the amount of the last term, which is related with the generalized minimum uncer- tainty [14]. For time-dependent Hamiltonians (1), we may find a useful lower and upper bound. Using the identity, ln(2 m ( t ) | u$_{0}$ ( t ) ˙ u$_{0}$ ( t ) | ) = ln(1 + ( m ( t ) d | u$_{0}$ ( t ) | $^{2}$/dt ) $^{2}$) / 2, we find the lower bound for the random-phase entropy Similarly, from 〈 H ( t ) 〉 Ψ 0 ≥ ℏ ω ( t ) m ( t ) | u$_{0}$ ( t ) ˙ u$_{0}$ | , with re- spect to the Gaussian wave packet Ψ$_{0}$ obtained using u$_{0}$ ( t ), we find the upper bound We have investigated the Leipnik’s joint entropy of the general Gaussian wave packets for quadratic Hamiltoni- ans. The Gaussian wave packets (2) have four integration constants: two constants for the classical position and momentum, and two variances for the position and mo- mentum. The first two constants are equivalent to a com- plex parameter for the coherent state and the second two constants to the squeeze parameter and the squeeze an- gle. The joint entropy (8) is expressed entirely in terms of the uncertainty relation, which is independent of the cen- troid but depends on the position and momentum vari- ances. We computed the joint entropy for a free particle, a harmonic oscillator and the Caldirola-Kanai Hamilto- nian with an exponentially varying mass. The joint en- tropy depends only on the squeeze parameter and the SPK appreciates APCTP for the warm hospitality, where this paper was finished. This work has been sup- ported by KORDI research program (PG47100) through KRCF(Korea Research Council of Fundamental Science and Technology) of Ministry of Education, Science and Technology. In this paper, we have proposed that the random- phase entropy with respect to the squeeze angle leads to monotonously increasing entropy for the general Gaus- sian wave packets, which is shown explicitly for a free particle, a harmonic oscillator and the Caldirola-Kanai Hamiltonian. For a quadratic Hamiltonian with time- dependent mass or frequency, the joint entropy satis- fies the lower bound (26) and the upper bound (27). It remains an open question whether the random-phase entropy monotonously increases for all time-dependent quadratic Hamiltonians. The general Gaussian wave packets have the initial en- tropy (14) that depends on the squeeze parameter and the squeeze angle. It is shown that the Leipnik’s joint entropy satisfies the lower bound S ( t ) ≥ ln( e/ 2) and monotonously increases for the range of squeeze angle π ≥ ϑ ≥ 0, while for the remaining range 2 π > ϑ > π it first decreases until the lower bound, ln( e/ 2), is reached and then it monotonously increases. This counter- example implies that the joint entropy monotonously in- creases only for the initially maximally classical states that have the minimum uncertainty. Similarly, the joint entropy of the squeezed vacuum states of the initially maximally classical state for the harmonic oscillator os- cillates between the maximum and the lower bound, ln( e/ 2). S = ln ( e 2 ) + ln ( cosh(2 r ) + 1 2 ) + 1 2 ln ( 1 + γ 2 4 ω 2 ) . (25) S ( t ) ≥ ln ( e 2 ) + ln ( cosh(2 r ) + 1 ) . (26) ln ( e 2 ) + ln ( cosh(2 r ) + 1 ) + 1 2 ln ( 〈 H ( t ) 〉 Ψ 0 ℏ ω ( t ) / 2 ) ≥ S ( t ) . (27) Acknowledgments IV. CONCLUSION [1] J. Dunkel and S. A. Trigger, Phys. Rev. A 71 , 052102 (2005). [2] P. Garbaczewski, Phys. Rev. A 72 , 056101 (2005); V. Majern´ık and T. Opatrn´ y, J. Phys. A: Math. Gen. 29 , 2187 (1996). [3] ¨ O. ¨ Ozcan, E. Akt¨ uurk, and R. Sever, Int. J. Theor. Phys. 47 , 3207 (2008); E. Akt¨ uurk, ¨ O. ¨ Ozcan, and R. Sever, Int. J. Mod. Phys. B 23 , 2449 (2009). [4] S. P. Kim, J. Korean Phys. Soc. 49 , 464 (2006) [arXiv:quant-ph/0511073]. [5] P. Caldirola, Nuovo Cimento 18 , 393 (1941); 77B , 241 (1983); E. Kanai, Prog. Theor. Phys. 3 , 440 (1948). [6] S. P. Kim and D. N. Page, Phys. Rev. A 64 , 012104 (2001). [7] H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 10 , 1458 (1969). [8] J-Y. Ji, J. K. Kim, and S. P. Kim, Phys. Rev. A 51 , 4268 (1995); J.-Y. Ji, J. K. Kim, S. P. Kim, and K-S. Soh, Phys. Rev. A 52 , 3352 (1995). [9] I. A. Malkin, V. I. Man’ko, and D. A. Trifnov, Phys. Rev. D 2 , 1371 (1970). [10] J. K. Kim and S. P. Kim, J. Phys. A 32 , 2711 (1999). [11] S. P. Kim and C. H. Lee, Phys. Rev. D 62 , 125020 (2000). [12] M. M. Nieto and D. R. Truax, New J. Phys. 2 , 18 (2000). [13] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, San Diego, 1994), p 560, formula 4.225. [14] S. P. Kim, J. Phys. A: Math. Gen. 36 , 12089 (2003).