论文标题
在非同时量子力学的时空对称延伸中，分数整合差异方程和（反）时间的墓治时间
Fractional integrodifferential equations and (anti-)hermiticity of time in a spacetime-symmetric extension of nonrelativistic Quantum Mechanics
论文作者
论文摘要
在现代时代，时间仍然是一个有趣的物理财产。一方面，我们具有经典和相对论的时间概念，其中空间和时间具有相同的层次结构，这对于描述时空中的事件至关重要。另一方面，在量子力学中，时间显示为经典参数，这意味着它与其规范共轭没有不确定性关系。在这项工作中，我们使用最近提出的时空对称形式主义〜\ href {https://doi.org/10.1103/physreva.95.032133} {[phys.i.扩展了通常的希尔伯特空间。时间参数$ t $和一个子空间中的位置运算符$ \ hat {x} $，以及另一个子空间中的位置参数$ x $和时间操作员$ \ m athbb {t} $。作为操作员的时间更适合描述隧道过程。然后，我们为受强度和弱电势限制的粒子解决新颖的$ 1/2 $分解整合方程，并通过矩形屏障获得隧道时间的分析表达。我们与以前的作品进行了比较，获得了在屏障以下的能量的纯粹想象时间，以及在屏障上方的能量的快速付费零件，表明隧道时间的时间操作员的抗热度。我们还表明，隧道问题的预期到达时间具有经典到达时间的能量平均值以及量子贡献。
Time continues to be an intriguing physical property in the modern era. On the one hand, we have the Classical and Relativistic notion of time, where space and time have the same hierarchy, which is essential in describing events in spacetime. On the other hand, in Quantum Mechanics, time appears as a classical parameter, meaning that it does not have an uncertainty relation with its canonical conjugate. In this work, we use a recent proposed spacetime-symmetric formalism~\href{https://doi.org/10.1103/PhysRevA.95.032133}{[Phys.~Rev.~A {\bf 95}, 032133 (2017)]} that tries to solve the unbalance in nonrelativistic Quantum Mechanics by extending the usual Hilbert space. The time parameter $t$ and the position operator $\hat{X}$ in one subspace, and the position parameter $x$ and time operator $\mathbb{T}$ in the other subspace. Time as an operator is better suitable for describing tunnelling processes. We then solve the novel $1/2$-fractional integrodifferential equation for a particle subjected to strong and weak potential limits and obtain an analytical expression for the tunnelling time through a rectangular barrier. We compare to previous works, obtaining pure imaginary times for energies below the barrier and a fast-decaying imaginary part for energies above the barrier, indicating the anti-hermiticity of the time operator for tunnelling times. We also show that the expected time of arrival in the tunnelling problem has the form of an energy average of the classical times of arrival plus a quantum contribution.
