### 物理代写|理论力学作业代写Theoretical Mechanics代考|PHY306

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Complementarity in Quantum Mechanics

Roughly speaking, complementarity can be understood as the coexistence of multiple properties in the behavior of an object that seem to be contradictory. Although it is possible to switch among different descriptions of these properties, in principle, it is impossible to view them, at the same time, despite their simultaneous coexistence. Therefore, the consideration of all these contradictory properties is absolutely necessary to provide a complete characterization of the object. In physics, complementarity represents a basic principle of quantum theory proposed by Niels Bohr $(1 ; 2)$, which is closely identified with the Copenhagen interpretation. This notion refers to effects such as the so-called wave-particle duality. In an analogous perspective as the finite character of the speed of light $c$ implies the impossibility of a sharp separation between the notions of space and time, the finite character of the quantum of action $\hbar$ implies the impossibility of a sharp separation between the behavior of a quantum system and its interaction with the measuring instruments.

In the early days of quantum mechanics, Bohr understood that complementarity cannot be a unique feature of quantum theories $(3 ; 4)$. In fact, he suggested that the thermodynamical quantities of temperature $T$ and energy $E$ should be complementary in the same way as position $q$ and momentum $p$ in quantum mechanics. According to thermodynamics, the energy $E$ and the temperature $T$ can be simultaneously defined for a thermodynamic system in equilibrium. However, a complete and different viewpoint for the energy-temperature relationship is provided in the framework of classical statistical mechanics (5). Inspired on Gibbs canonical ensemble, Bohr claimed that a definite temperature $T$ can only be attributed to the system if it is submerged into a heat bath ${ }^{1}$, in which case fluctuations of energy $E$ are unavoidable. Conversely, a definite energy $E$ can only be assigned when the system is put into energetic isolation, thus excluding the simultaneous determination of its temperature $T$.
At first glance, the above reasonings are remarkably analogous to the Bohr’s arguments that support the complementary character between the coordinates $\mathrm{q}$ and momentum $\mathbf{p}$. Dimensional analysis suggests the relevance of the following uncertainty relation (6):
$$\Delta E \Delta(1 / T) \geq k_{B},$$
where $k_{B}$ is the Boltzmann’s constant, which can play in statistical mechanics the counterpart role of the Planck’s constant $\hbar$ in quantum mechanics. Recently (7-9), we have shown that Bohr’s arguments about the complementary character between energy and temperature, as well as the inequality of Eq.(1), are not strictly correct. However, the essential idea of Bohr is relevant enough: uncertainty relations can be present in any physical theory with a statistical formulation. In fact, the notion of complementarity is intrinsically associated with the statistical nature of a given physical theory.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Complementary descriptions and complementary quantities

Quantum mechanics is a theory hallmarked by the complementarity between two descriptions that are unified in classical physics $(1 ; 2)$ :

1. Space-time description: the parametrization in terms of coordinates $q$ and time $t$;
2. Dynamical description: This description in based on the applicability of the dynamical conseroation laws, where enter dynamical quantities as the energy and the momentum.
The breakdown of classical notions as the concept of point particle trajectory $[\mathbf{q}(t), \mathbf{p}(t)]$ was clearly evidenced in Davisson and Germer experiment and other similar experiences (12). To illustrate that electrons and other microparticles undergo interference and diffraction phenomena like the ordinary waves, in Fig.1 a schematic representation of electron interference by double-slits apparatus is shown (13). According to this experience, the measurement results can only be described using classical notions compatible with its corpuscular representations, that is, in terms of the space-time description, e.g.: a spot in a photographic plate, a recoil of some movable part of the instrument, etc. Moreover, these experimental results are generally unpredictable, that is, they show an intrinsic statistical nature that is governed by the wave behavior dynamics. According to these experiments, there is no a sharp separation between the undulatory-statistical behavior of microparticles and the space-time description associated with the interaction with the measuring instruments.

Besides the existence of complementary descriptions, it is possible to talk about the notion of complementary quantities. Position $\mathbf{q}$ and momentum $\mathbf{p}$, as well as time $t$ and energy $E$, are relevant examples complementary quantities. Any experimental setup aimed to study the exchange of energy $E$ and momentum $p$ between microparticles must involve a measure in a finite region of the space-time for the definition of wave frequency $\omega$ and vector $\mathbf{k}$ entering in de Broglie’s relations (14):
$$E=\hbar \omega \text { and } \mathbf{p}=\hbar \mathbf{k}$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Complementarity in Quantum Mechanics

$$\Delta E \Delta(1 / T) \geq k_{B}$$

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Complementary descriptions and complementary quantities

1. 时空描述: 坐标参数化 $q$ 和时间 $t$;
2. 动力学描述：这种描述基于动力学conseration law的适用性，其中输入动力学量作为能量和动量。
作为点粒子轨迹概念的经典概念的分解 $[\mathbf{q}(t), \mathbf{p}(t)]$ 在戴维森和格默实验以及其他类似的经验中清楚地证明了 这一点 (12)。为了说明电子和其他微粒像普通波一样经历干涉和衍射现象，在图 1 中显示了双缝装置的电子 干涉示意图 (13) 。根据这一经验，测量结果只能用与其微粒表示相适应的经典概念来描述，即用时空描 述，例如：照相底片上的一个点，某些可移动部分的后坐力。此外，这些实验结果通常是不可预测的，也就 是说，它们显示出由波浪行为动力学控制的内在统计性质。根据这些实验，
除了存在互补描述之外，还可以讨论互补量的概念。位置 $\mathbf{q}$ 和势头 $\mathbf{p}$, 以及时间 $t$ 和能量 $E$, 是相关例子的互补量。任 何旨在研究能量交换的实验装置 $E$ 和势头 $p$ 微粒之间必须涉及时空有限区域中的测量，以定义波频率 $\omega$ 和矢量 $\mathbf{k}$ 进入 德布罗意的关系 (14) :
$$E=\hbar \omega \text { and } \mathbf{p}=\hbar \mathbf{k}$$

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