到目前為止，關於量子力學的軌跡詮釋有二種版本，一種是基於波姆力學的實數軌跡，另一種是基於複數力學的複數軌跡。將量子軌跡的定義擴展到複數平面的必要性，近年來已有許多文獻討論過，本論文將根據波爾的相對應原理(correspondence principle)，提出一個新的證據顯示量子軌跡複數化的必要性。相對應原理提到，當量子數(能階)逐漸增加時，量子系統的行為應該逐漸趨近於古典系統的行為。本論文以相對應原理來測試量子力學目前的三種詮釋:哥本哈根的機率詮釋，波姆力學的實數軌跡詮釋，以及複數力學的複數軌跡詮釋。接受測試的是二個標準量子系統:簡諧振子以及無限深量子井內的粒子運動。測試結果顯示只有複數軌跡詮釋能通過相對應原理的考驗。長久以來大家都認為量子力學的機率詮釋滿足波爾的相對應原理，然則本論文指出這一看法實際上並不全然正確。量子力學機率論所描述的粒子平均行為確實滿足相對應原理，但是機率論所描述的粒子隨機行為則不然。亦即當量子數逐漸增加時，描述粒子隨機運動的量子機率密度並沒有收斂到描述粒子巨觀運動的古典機率密度。
除了證明複數軌跡滿足相對應原理之外，本論文同時找到了複數軌跡詮釋與量子機率論間之關聯性。在進行複數軌跡落點的機率統計時，如果只選取與實數軸相交的軌跡點進行統計，吾人發現所得到的機率分布曲線即是量子機率密度曲線。亦即量子機率原來是源自純實數軌跡點的統計，而忽略了帶有虛部軌跡點的統計，正是由於這樣的忽略使得量子機率論無法滿足相對應原理。唯有加入虛部動態的考慮，微觀的量子力學才能藉由量子數的增加完美收斂到巨觀的古典力學。

Till now, there are two versions of trajectory interpretation of quantum mechanics: one is the real-trajectory version based on Bohmian mechanics and the other is the complex-trajectory version based on complex mechanics. The necessity of extending quantum trajectories from real domain to complex domain has been studied recently by many researches. Based on Bohr’s correspondence principle, this thesis finds a new evidence for the necessity of the complexification of quantum trajectories. The correspondence principle states that as the quantum number increases to infinity, the behavior of a quantum system converges to the behavior of its corresponding classical system. The thesis tests the validity of the correspondence principle for the three interpretations of quantum mechanics: Copenhagen’s probability interpretation, Bohm’s real-trajectory interpretation and complex-trajectory interpretation. Two benchmark quantum models, namely, the harmonic oscillator and the infinite square well, are analyzed in the test by the three quantum interpretations, and the outcomes show that only the complex-trajectory interpretation satisfies the correspondence principle. While it is a common belief that the probability interpretation of quantum mechanics satisfies the correspondence principle, the test indicates that this long-standing claim is not necessarily true. The average behavior of a quantum system does satisfy the correspondence principle, but the actual random behavior of the quantum system does not. In other words, as the quantum number increases, the quantum probability density does not converge smoothly to the classical probability density.
Apart from proving the satisfaction of the correspondence principle by complex trajectories, this thesis also clarifies the connection between the quantum probability density and the complex trajectories. In the process of computing the statistical point-wise distribution of the complex trajectories, if we only consider the crossovers of the complex trajectories with the real axis and neglect all the remaining points on the trajectories, then the statistical distribution of such crossovers turns out to be the probability density solved from the Schrödinger equation. In other words, it can be said that the quantum probability density only accounts for the pure real points on the trajectories and neglect all the points with nonzero imaginary parts. It is this neglect that causes the deviation of the evolution of the probability density from the correspondence principle. Once all the points with nonzero imaginary parts are taken into account, probability interpretation of quantum mechanics recovers the Bohr’s correspondence principle.

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