Abstract In this study, a path integral approach for an isolated dynamical system which contains a moving body and its surrounding coupled to each other by a fractional dynamical friction is constructed. The dissipative system is characterized by a total Lagrangian L total = L − E d holding the energy term E d = ∫ 0 t f ( x ˙ , x , τ ) d τ which is dissipated by a dynamical fractional friction force f ( x ˙ , x , τ ) . Such a position- and time-dependent friction is in fact motivated from the Brownian motion. Our methodology aims to substitute the standard action by S = ∫ 0 t ( L ( x ˙ , x , τ ) − ∫ 0 τ f ( x ˙ , x , τ ) d μ α ( ξ ) ) d τ where [ μ α ] = − α , 0 α ≤ 1 with the particular choice d μ α ( τ ) = d τ Λ α ( τ ) for some scalar function Λ α ( ξ ) . A modified time-dependent Schrodinger equation is obtained which is characterized by a time-dependent Hamiltonian operator which is suitable to describe for quantum systems characterized by a time-dependent mass. Applications to quantum dots/zero-dimensional nanocrystals in which the size of particles is close to the exciton Bohr radius of the material are studied and explored in details by taking into account both the electron and the hole within the quantum dots besides the energy gap of the bulk. For the case of Cadmium selenide, we have obtained a decrease in the density of state in agreement with recent experimental results. Several consequences were discussed in some details.

中文翻译：

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具有动态摩擦的量子耗散系统的路径积分方法：应用于量子点/零维纳米晶体

摘要 在这项研究中，构建了一个孤立动力系统的路径积分方法，该系统包含一个运动体及其周围通过分数动力摩擦相互耦合的运动体。耗散系统的特征在于总拉格朗日 L total = L − E d 持有能量项 E d = ∫ 0 tf ( x ˙ , x , τ ) d τ 由动态分数摩擦力 f ( x ˙ , x , τ ) 。这种与位置和时间相关的摩擦实际上是由布朗运动引起的。我们的方法旨在用 S = ∫ 0 t ( L ( x ˙ , x , τ ) − ∫ 0 τ f ( x ˙ , x , τ ) d μ α ( ξ ) ) d τ 替换标准动作，其中 [ μ α ] = − α, 0 α ≤ 1 对于某些标量函数 Λ α ( ξ ) 具有特定选择 d μ α ( τ ) = d τ Λ α ( τ ) 。得到了一个修正的瞬态薛定谔方程，该方程以瞬态哈密顿算符为特征，适用于描述以瞬态质量为特征的量子系统。通过考虑量子点内的电子和空穴以及量子点的能隙，详细研究和探索粒子尺寸接近材料激子玻尔半径的量子点/零维纳米晶体的应用批量。对于硒化镉的情况，我们已经获得了与最近的实验结果一致的状态密度的降低。在一些细节中讨论了几种后果。通过考虑量子点内的电子和空穴以及量子点的能隙，详细研究和探索粒子尺寸接近材料激子玻尔半径的量子点/零维纳米晶体的应用批量。对于硒化镉的情况，我们已经获得了与最近的实验结果一致的状态密度的降低。在一些细节中讨论了几种后果。通过考虑量子点内的电子和空穴以及量子点的能隙，详细研究和探索粒子尺寸接近材料激子玻尔半径的量子点/零维纳米晶体的应用批量。对于硒化镉的情况，我们已经获得了与最近的实验结果一致的状态密度的降低。在一些细节中讨论了几种后果。