# 物理代写|量子光学代写Quantum Optics代考|PHYS4055

## 物理代写|量子光学代写Quantum Optics代考|Optical Tweezers

The above principle can be generalized for trapping particles in all three spatial dimensions. The basic principle is depicted in Fig. $4.4$ where a dielectric sphere is located in the maximum region of a tightly focused laser beam. Whenever the sphere moves out of the focus, the light exerted by the scattered light pushes it back. Note that for sufficiently large spheres the equilibrium position corresponds to a slightly displaced sphere center, such that the net force of the scattered light sums up to zero.

Optical tweezers have received enormous interest in life sciences within the last decades. In particular with the rapid laser developments and the use of spatial light modulators it is nowadays possible to generate complex optical trapping potentials, with one or several minima that can be displaced, rotated, and deformed at will. For a detailed discussion the interested reader is referred to the literature, see for instance [9-11,13] and references therein.

Figure $4.5$ shows one of the numerous beautiful examples for light trapping [13]. A dielectric sphere is attached to a kinesin motor that moves along a microtubule. By positioning the sphere in an optical trap and following its position, one obtains information about the propagation details of the motor biomolecule. One can also exert a force on the system by pulling away the dielectric sphere through displacement of the trap minimum. The blue and red lines in the figure report the propagation in absence and presence of such a load.

## 物理代写|量子光学代写Quantum Optics代考|Continuity Equation

A prototypical example for conservation laws in physics is given by the continuity equation, which we briefly discuss in the following because it will serve us as a blueprint for other conservation laws. We start from Ampere’s law, Eq. (2.15d), and take the divergence on both sides of the equation
$$\nabla \cdot \nabla \times \frac{1}{\mu_0} \boldsymbol{B}=\nabla \cdot \boldsymbol{J}+\varepsilon_0 \nabla \cdot \frac{\partial \boldsymbol{E}}{\partial t}=0 .$$
Note that the expression has to be zero since the divergence of a curl field is always zero. Together with Gauss’ law $\varepsilon_0 \nabla \cdot \boldsymbol{E}=\rho$ we are thus led to the continuity equation
Continuity Equation
$$\frac{\partial \rho}{\partial t}=-\nabla \cdot \boldsymbol{J} .$$
To understand this expression better, it is instructive to integrate it over some volume $\Omega$ and to employ Gauss’ theorem on the right-hand side,
$$\int_{\Omega} \frac{\partial \rho}{\partial t} d^3 r=\frac{d}{d t} \int_{\Omega} \rho d^3 r=-\int_{\Omega} \nabla \cdot \boldsymbol{J} d^3 r=-\oint_{\partial \Omega} \boldsymbol{J} \cdot d \boldsymbol{S} .$$

We are thus led to the integral form of Eq. (4.11) which reads
$$\frac{d Q_{\Omega}}{d t}=-\oint_{\partial \Omega} J \cdot d \boldsymbol{S}$$
with $Q_{\Omega}$ being the charge enclosed in the volume $\Omega$. The implication of this equation can be summarized as follows:

Global Charge Conservation. If we extend the volume $\Omega \rightarrow \infty$ to infinity the term on the right-hand side becomes zero, and we find that charge is a conserved quantity.
Local Charge Conservation. However, the continuity equation does not only imply global charge conservation, which could still mean that charge is annihilated in some region of space and instantaneously created somewhere else. Such spooky action at a distance is forbidden by Eq. (4.12) which states that whenever the total charge changes in a volume (left-hand side of equation) it must be transported into or out of the volume through a local current density $\boldsymbol{J}$ (right-hand side of equation).

For this reason, electrodynamics is sometimes called a “local field theory.” Here “local” means that all quantities, such as fields or charge, change locally and have to be transported by some means to another position. This renders this approach compatible with the theory of relativity, which states that no information can propagate faster than the speed of light. Equations (4.11), (4.12) will serve us as blueprints for other conservation laws.

## 物理代写|量子光学代写量子光学代考|连续性方程

$$\nabla \cdot \nabla \times \frac{1}{\mu_0} \boldsymbol{B}=\nabla \cdot \boldsymbol{J}+\varepsilon_0 \nabla \cdot \frac{\partial \boldsymbol{E}}{\partial t}=0 .$$

$$\frac{\partial \rho}{\partial t}=-\nabla \cdot \boldsymbol{J} .$$

$$\int_{\Omega} \frac{\partial \rho}{\partial t} d^3 r=\frac{d}{d t} \int_{\Omega} \rho d^3 r=-\int_{\Omega} \nabla \cdot \boldsymbol{J} d^3 r=-\oint_{\partial \Omega} \boldsymbol{J} \cdot d \boldsymbol{S} .$$ 是有指导意义的

$$\frac{d Q_{\Omega}}{d t}=-\oint_{\partial \Omega} J \cdot d \boldsymbol{S}$$
，其中$Q_{\Omega}$是卷$\Omega$中包含的费用。这个等式的含义可以总结为:

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: