在一個量子系統中，呈現如階梯般跳躍的能階，一直是物理學的一個巨大謎團。傳統量子力學採用機率詮釋，在剔除那些不滿足機率條件的波函數(又稱平方不可積分波函數)的過程中，得到不連續性的能量變化。然而，當我們使用了不完整的波函數，自然會得到不連續的能量。因此，吾人認為能階的量子化事實上是機率詮釋下不完整的產物。為了還原能階斷層之間的連續性變化，吾人引入複數力學的軌跡詮釋，將完整波函數所對應到的複數軌跡，沿著軌跡作圍線積分，即可還原完整波函數對能量連續性的貢獻。
在第一個範例中，吾人針對一個無限方井中的自由粒子進行分析。在過去，無論是傳統量子力學的機率詮釋或是波姆的實數軌跡詮釋，對井中粒子的動態皆無法給予具體的描述。在複數力學的架構之下，吾人發現井中的粒子會出現兩種複數軌跡:滿足駐波條件的波函數所對應到的封閉型軌跡以及不滿足駐波條件的波函數所對應到的非封閉型軌跡；前者的平均能量即為傳統量子力學中的特徵能量，後者的平均能量即形成了串連這些離散能階之間的連續性變化曲線。
在第二個範例中，吾人示範了簡諧振子的能階量子化，事實上來自封閉的複數軌跡所包圍的鞍點與平衡點數目的「分裂」現象。然而，若是沿著軌跡作積分，並加入量子位勢能的修正，我們一樣可以得到 E = R+1/2 的連續性能量變化，這當中也包含了傳統量子力學所允許的特徵能量 E = N+1/2 。此外，吾人也從完整的波函數，得到了在機率詮釋中認為薛丁格方程式所無法呈現的自旋與反自旋動態。種種的證據足以讓人信服，相較於量子機率詮釋所提供的不完整資訊，複數軌跡詮釋更加適合用來描述一個量子系統。
傳統的量子力學建立於「測量」得到的特徵值來描述一個量子系統。當我們對一個微觀尺度下的系統進行測量，同時系統也將塌縮至特定的特徵態，使得我們總是無法得到粒子精確的位置及完整的能量變化。哥本哈根學派將這樣的不確定性一概用機率解釋。然而，近年來的弱測量技術，已經陸續出現了機率無法詮釋的奇特現象。巧合的是，這些奇特現象與本論文利用複數軌跡得到的結果有著高程度的相似性。這些跡象彷彿皆暗示著複數的軌跡動態才是真正隱藏在量子力學機率詮釋背後的真正作用機制。

In a quantum system, the energy levels that jump like a ladder have always been a huge mystery of physics. Traditional quantum mechanics uses the probability interpretation to obtain the discontinuous changes of energy in the process of eliminating those wave functions that do not meet the probability conditions (also known as non-square integrable wave functions). However, when we only allow the existence of discontinuous wave functions, we will naturally get discontinuous energy. Therefore, we believe that the quantization of energy levels is actually an incomplete product under the probability interpretation. In order to restore the continuity between energy levels, we introduce the complex trajectory interpretation of quantum mechanics to reveal what is missing between discrete energy levels.
In the first example, we analyze a free particle moving in an infinite well. In the past, neither the probabilistic interpretation of quantum mechanics nor the Bohm’s real trajectory interpretation could give a complete description of the particle’s dynamics in a well. Under the framework of complex mechanics, we find that the complex trajectories of particle in the well have two patterns: the closed trajectory corresponding to the wave function that meets the standing-wave condition and the non-closed trajectory corresponding to the wave function that does not meet the standing wave condition. The average energy of the former is just the discrete energy in quantum mechanics, and the average energy of the latter forms a continuous curve connecting the discrete energy levels.
In the second example, we demonstrate the energy quantization of the simple harmonic oscillator in terms of the bifurcation phenomenon of the number of saddle points and equilibrium points surrounded by closed complex trajectories. If we integrate along the complex trajectory and add the quantum potential energy correction, we can obtain the continuous energy change of E = R+1/2, which also includes the eigen energy E = N+1/2 allowed by traditional quantum mechanics. In addition, from the complete wave functions including square-integrable and non-square-integrable wave functions, we obtain the spin and anti-spin dynamics that otherwise cannot be described by Schrödinger equation in the probability interpretation. It is evident that compared with the incomplete information provided by the probability interpretation of quantum mechanics, the complex trajectory interpretation is more suitable for describing a quantum system.
Traditional quantum mechanics is based on the eigenvalues obtained by measurement to describe a quantum system. When we measure a quantum system, the system will collapse to a specific eigen state, making it impossible for us to obtain the precise position of the particles and to know the transition in the collapsing process. The Copenhagen School took the probability theory to interpret such uncertainty. However, in recent years, the weak measurement technology has discovered strange phenomena that cannot be explained by the probability interpretation. Coincidentally, these peculiar phenomena are quite similar to the results obtained in this dissertation by using complex trajectories. These signs seem to imply that the dynamics of complex trajectories are the mechanism hidden behind the probability interpretation of quantum mechanics.

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