複合系の量子マスタ方程式 Born‐Markov近似は本当に有効か?

Gen Kimura, Kazuya Yuasa, Kentaro Imafuku

Yi Cao

In SunQM-4, to build a full-QM deduced Solar system’s 3D probability density map with time-dependent orbital movement, we developed a non-Born probability (NBP) method. In the current paper, we showed that NBP is directly proportional to the wave function (rather than Born probability’s conjugated-squared wave function). For a cos(x) wave function, its NBP is simply to lift-up wave function by one to make its min =0, and then divided by two to make its max =1, so its NBP = [1 + cos(x)]/2. The trigonometric formula [1 + cos(x)]/2 = [cos(x/2)}^2 revealed that for a planet doing orbital movement in Solar system, its NBP ground state is n=1, and its Born probability ground state is n = 1/2. We showed that NBP is valid not only for {N,n} QM’s nLL QM state, but also for the 1D infinity deep square potential well QM, for the circular 1D QM, and for the plane wave QM. NBP demonstrates more direct and intuitive physical meaning than that of Born probability. In contrast to that Born probability is only for the standing wave function, NBP is for a uni-directional traveling wave function (which naturally includes the linearly combined two opposite-directional traveling waves, or a standing wave). Therefore, Born probability is possible merely a special case of (the more generalized) NBP. With NBP, we may explain the flute sound wave (an air mass density vibration) mechanics directly as the 1D quantum mechanics! This may significantly change our view on QM and its application on our daily-life-world’s (Newtonian) physics. The planet formation through accretion is also discussed by using NBP.

元 木村, 一哉 湯浅, Kentaro Imafuku

Artur F. Izmaylov, Ignacio Franco

The role of electron-nuclear entanglement on the validity of the Born-Oppenheimer (BO) approximation is investigated. Although nonadiabatic couplings generally lead to entanglement and to a failure of the BO approximation, surprisingly the degree of electron-nuclear entanglement is found to be uncorrelated with the degree of validity of the BO approximation. This is because while the degree of entanglement of BO states is determined by their deviation from the corresponding states in the crude BO approximation, the accuracy of the BO approximation is dictated, instead, by the deviation of the BO states from the exact electron-nuclear states. In fact, in the context of a minimal avoided crossing model, extreme cases are identified where an adequate BO state is seen to be maximally entangled, and where the BO approximation fails but the associated BO state remains approximately unentangled. Further, the BO states are found to not preserve the entanglement properties of the exact electron-nuclear eigenstates, and to be completely unentangled only in the limit in which the BO approximation becomes exact.

Giulio Gasbarri, Luca Ferialdi

Artur F. Izmaylov, Ignacio Franco

The role of electron-nuclear entanglement on the validity of the Born-Oppenheimer (BO) approximation is investigated. While nonadiabatic couplings generally lead to entanglement and to a failure of the BO approximation, surprisingly the degree of electron-nuclear entanglement is found to be uncorrelated with the degree of validity of the BO approximation. This is because while the degree of entanglement of BO states is determined by their deviation from the corresponding states in the crude BO approximation, the accuracy of the BO approximation is dictated, instead, by the deviation of the BO states from the exact electron-nuclear states. In fact, in the context of a minimal avoided crossing model, extreme cases are identified where an adequate BO state is seen to be maximally entangled, and where the BO approximation fails but the associated BO state remains approximately unentangled. Further, the BO states are found to not preserve the entanglement properties of the exact electron-nuclear eigenstates, and to be completely unentangled only in the limit in which the BO approximation becomes exact.