### 物理代写|热力学代写thermodynamics代考|NEM2201

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|热力学代写thermodynamics代考|Spin-Chain Bath

A quantum bath consisting of a spin- $1 / 2$ chain with nearest-neighbor exchange interactions is defined as $X X$ chain if it is governed by the Hamiltonian
$$H_{\mathrm{B}}=\hbar J \sum_{i=1}^{N-1}\left(\hat{S}{i}^{+} \hat{S}{i+1}^{-}+\hat{S}{i}^{-} \hat{S}{i+1}^{+}\right) .$$
Here $\hat{S}^{\pm}=\hat{S}^{x} \pm i \hat{S}^{y}$ are spin-1/2 raising and lowering (excitation and de-excitation, respectively) operators and $J$ is the spin-spin coupling strength. This Hamiltonian conserves the total magnetization.

It can be transformed into the Hamiltonian of non-interacting spinless fermions that hop between neighboring sites of the form,
$$H_{\mathrm{B}}=h J \sum_{i=1}^{N-1}\left(c_{i}^{\dagger} c_{i+1}+c_{i+1}^{\dagger} c_{i}\right),$$
by using the Jordan-Wigner transformation,
$$c_{i}=e^{i \pi \sum_{j=1}^{i-1} \hat{s}{j}^{+} \hat{S}{j}^{-} \hat{S}{i}^{-} .}$$ The Hamiltonian (3.50) conserves the number of fermions. It simplifies upon performing the discrete sine transform, $$\hat{f}{q}=\sqrt{\frac{2}{N+1}} \sum_{j=1}^{N} \sin \left(\frac{j q \pi}{N+1}\right) c_{j} \equiv \sum_{j=1}^{N} U_{q j} c_{j} \quad(q=1,2, \ldots, N) .$$
The transform matrix $U=\left{U_{q j}\right}$ is real, Hermitian, and unitary, $U=U^{4}, U^{2}=I$. Correspondingly, the inverse transform of (3.52) is
$$c_{j}=\sum_{q=1}^{N} U_{j q} \hat{f}_{q} .$$

## 物理代写|热力学代写thermodynamics代考|Atom–Photon Interaction

The interaction Hamiltonian of atoms with the electromagnetic field bath is denoted here by
\begin{aligned} \epsilon \tilde{H}{\mathrm{SB}} \equiv H{\mathrm{SB}} \equiv H_{\mathrm{I}} &=\sum_{i}\left[-\frac{e_{i}}{2 m_{i} c}\left(\boldsymbol{p}{i} \cdot \boldsymbol{A}{i}+\boldsymbol{A}{i} \cdot \boldsymbol{p}{i}\right)+\frac{e_{i}^{2}}{2 m_{i} c^{2}} \boldsymbol{A}{i}^{2}\right] \ &=\sum{i}\left(-\frac{e_{i}}{m_{i} c} \boldsymbol{A}{i} \cdot \boldsymbol{p}{i}+\frac{e_{i}^{2}}{2 m_{i} c^{2}} \boldsymbol{A}{i}^{2}\right) \end{aligned} Here $\boldsymbol{A}{i}=\boldsymbol{A}\left(\boldsymbol{r}{i}\right)$ is the vector potential at the position $\boldsymbol{r}{i}$ of a particle $i$ with the charge $e_{i}$ and the mass $m_{i}$, and $p_{i}$ is the momentum canonically conjugate to the coordinate $\boldsymbol{r}{i}$. The replacement of $\boldsymbol{p}{i} \cdot \boldsymbol{A}{i}$ by $\boldsymbol{A}{i} \cdot \boldsymbol{p}{i}$ in the second line of Eq. (4.1) follows from the gauge condition $\boldsymbol{\nabla}{i} \cdot \boldsymbol{A}_{i}=0$. Equation (4.1) describes the interaction of moving charges with the electromagnetic field, but does not account for the interaction of their spin moments with magnetic fields.

The quantization of the free-atom Hamiltonian combined with the free-field Hamiltonian and the interaction Hamiltonian (4.1) is performed by subjecting the particles’ $\boldsymbol{r}{i}$ and $\boldsymbol{p}{i}$ to the standard commutation relations and quantizing the radiation field, as in Eq. (3.5). The longitudinal electric field $\boldsymbol{E}{L}$ does not provide any additional freedom in this quantization, being completely determined through the Maxwell equation $\nabla \cdot \boldsymbol{E}{L}=\rho_{\mathrm{e}}$ by the charge density $\rho_{\mathrm{e}}(\boldsymbol{x}, t)$.

The interaction $H_{\mathrm{I}}$ in Eq. (4.1) is commonly treated as a perturbation that causes transitions between the states of the free Hamiltonian. The interaction (4.1) contains a term quadratic in the vector potential that gives rise to two-photon processes within first-order perturbation theory (i.e., emission, absorption or scattering of two photons). However, as the quadratic term is typically small, it is neglected below. The first term in (4.1) is treated below in the common electrie dipole (or long-wavelength) approximation, which neglects the spatial variation of $\boldsymbol{A}(\boldsymbol{x})$. The dependence of $\boldsymbol{A}$ on $\boldsymbol{x}$ is responsible for magnetic interactions and higher-order effects that are not treated here.

## 物理代写|热力学代写thermodynamics代考|Spin-Chain Bath

$$H_{\mathrm{B}}=\hbar J \sum_{i=1}^{N-1}\left(\hat{S} i^{+} \hat{S} i+1^{-}+\hat{S} i^{-} \hat{S} i+1^{+}\right) .$$

$$H_{\mathrm{B}}=h J \sum_{i=1}^{N-1}\left(c_{i}^{\dagger} c_{i+1}+c_{i+1}^{\dagger} c_{i}\right)$$

$$c_{i}=e^{i \pi \sum_{j=1}^{i-1} \hat{s} j^{+} \hat{S} j^{-} \hat{S} i^{-}} .$$

$$\hat{f} q=\sqrt{\frac{2}{N+1}} \sum_{j=1}^{N} \sin \left(\frac{j q \pi}{N+1}\right) c_{j} \equiv \sum_{j=1}^{N} U_{q j} c_{j} \quad(q=1,2, \ldots, N) .$$

$$c_{j}=\sum_{q=1}^{N} U_{j q} \hat{f}_{q} .$$

## 物理代写|热力学代写thermodynamics代考|Atom–Photon Interaction

$$\epsilon \tilde{H} \mathrm{SB} \equiv H \mathrm{SB} \equiv H_{\mathrm{I}}=\sum_{i}\left[-\frac{e_{i}}{2 m_{i} c}(\boldsymbol{p} i \cdot \boldsymbol{A} i+\boldsymbol{A} i \cdot \boldsymbol{p} i)+\frac{e_{i}^{2}}{2 m_{i} c^{2}} \boldsymbol{A} i^{2}\right] \quad \sum i\left(-\frac{e_{i}}{m_{i} c} \boldsymbol{A} i\right.$$

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## MATLAB代写

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