In this letter, I have considered one-dimensional quantum system with different masses $m$ and $M$ , which does not appear integrable in general. However I have found an exact two-body wave function and due to the extension of the integrable system to more general system, it was concluded that the rapidity or quasi-momentum in the integrable system should be regarded as a modification of velocity rather than that of momentum. I have also considered the three-body wave function and argued its integrable condition.In this letter, I have considered one-dimensional quantum system with different masses $m$ and $M$, which does not appear integrable in general. However I have found an exact two-body wave function and due to the extension of the integrable system to more general system, it was concluded that the rapidity or quasi-momentum in the integrable system should be regarded as a modification of velocity rather than that of momentum. I have also considered the three-body wave function and argued its integrable condition.

arXiv:solv-int/9707006Jul 9, 19970 citations

Quantum integrable systems

Miki Wadati, Taro Nagao, Kazuhiro Hikami

Physica D Nonlinear PhenomenaSep 1, 199310 citations

Quantum integrable systems in one dimension

Michael Fowler

arXiv:cond-mat/9401051Physica D Nonlinear PhenomenaSep 1, 19952 citations

Unpredictability of wave function's evolution in nonintegrable quantum systems

Ivan B. Ivanov

It is shown that evolution of wave functions in nonintegrable quantum systems is unpredictable for a long time T because of rapid growth of number of elementary computational operations $\mathcal O(T)\sim T^α$ . On the other hand, the evolution of wave functions in integrable systems can be predicted by the fast algorithms $\mathcal O(T)\sim (log_2 T)^β$ for logarithmically short time and thus there is an algorithmic "compressibility" of their dynamics. The difference between integrable and nonintegrable systems in our approach looks identically for classical and quantum systems. Therefore the minimal number of bit operations $\mathcal O(T)$ needed to predict a state of system for time interval T can be used as universal sign of chaos.It is shown that evolution of wave functions in nonintegrable quantum systems is unpredictable for a long time T because of rapid growth of number of elementary computational operations $\mathcal O(T)\sim T^α$. On the other hand, the evolution of wave functions in integrable systems can be predicted by the fast algorithms $\mathcal O(T)\sim (log_2 T)^β$ for logarithmically short time and thus there is an algorithmic "compressibility" of their dynamics. The difference between integrable and nonintegrable systems in our approach looks identically for classical and quantum systems. Therefore the minimal number of bit operations $\mathcal O(T)$ needed to predict a state of system for time interval T can be used as universal sign of chaos.

arXiv:quant-ph/0203019Mar 5, 20021 citation

Quantum theory of non-integrable systems

Tomio Yamakoshi Petrosky, I. Prigogine, S. Tasaki

Physica A Statistical Mechanics and its ApplicationsApr 1, 1991206 citations

- Path Integral Formulation

S. Rajasekar, R. Velusamy

The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to ﬁnd the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to ﬁnd the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.

CRC Press eBooksDec 10, 20141 citation

In this letter, I have considered one-dimensional quantum system with different masses

m

and

M

, which does not appear integrable in general. However I have found an exact two-body wave function and due to the extension of the integrable system to more general system, it was concluded that the rapidity or quasi-momentum in the integrable system should be regarded as a modification of velocity rather than that of momentum. I have also considered the three-body wave function and argued its integrable condition.

It is shown that evolution of wave functions in nonintegrable quantum systems is unpredictable for a long time T because of rapid growth of number of elementary computational operations

\mathcal O(T)\sim T^α

. On the other hand, the evolution of wave functions in integrable systems can be predicted by the fast algorithms

\mathcal O(T)\sim (log_2 T)^β

for logarithmically short time and thus there is an algorithmic "compressibility" of their dynamics. The difference between integrable and nonintegrable systems in our approach looks identically for classical and quantum systems. Therefore the minimal number of bit operations

\mathcal O(T)

needed to predict a state of system for time interval T can be used as universal sign of chaos.