### 物理代写|原子物理代写Atomic and Molecular Physics代考|PHYS144

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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|原子物理代写Atomic and Molecular Physics代考|TIME IN QUANTUM MECHANICS

Time is introduced into the quantum mechanics simply by incorporating a “clock” as a part of the physical system in addition to the parts that constitute our main interest. This can be done in a general way, demonstrating that the details of the clock make no difference in final result [17]. Alternatively, one can use the simplest possible “clock” to introduce time since the exact nature of the timepiece is of peripheral importance. To that end we will suppose that time is measured by a time of flight technique [16]. That is, we suppose that if the mass and energy, or velocity of a particle is known, then its position $R=v t$ defines the time $t$. For simplicity, the coordinates of the clock are just the particle’s position $\varsigma=v t$ on the z-axis. If $H$ is the Hamiltonian for the system without the clock then the Schrödinger equation with the clock is,
$$\left(-\frac{\hbar^{2}}{2 M} \frac{\partial^{2}}{\partial \varsigma^{2}}+H\right) \Psi=E \Psi$$
where $M$ is the mass of the “clock” particle. Setting $\Psi=\exp [i K \zeta] \psi$ with $E=\frac{\hbar^{2} K^{2}}{2 M}$ in Eq. (2.1) gives the equivalent equation.
$$\left(-\frac{\hbar^{2}}{2 M} \frac{\partial^{2}}{\partial \zeta^{2}}-i \hbar^{2} \frac{K}{M} \frac{\partial}{\partial \zeta}+H\right) \psi=0$$
Eq. (2.2) is fully equivalent to Eq. (2.1) but has a rather different appearance owing to the absence of the energy $E$ on the right hand side. This is compensated for by the presence of the first derivative term on the left hand side. Using that $\hbar K / M=\mathrm{v}$, setting $\varsigma=v t$, and taking the limit that $M \rightarrow \infty$ gives the timedependent Schrödinger equation;
$$H \psi=i \hbar \frac{\partial \psi}{\partial t}$$
where the Hamiltonian $H$ may or may not depend explicitly upon the time variable $t$. In either case, the time-dependent Schrödinger equation emerges from the time-independent equation when a macroscopic clock is explicitly introduced. Note that the limit $M \rightarrow \infty$ with $v$ held constant is considered as a macroscopic limit in this construction.

## 物理代写|原子物理代写Atomic and Molecular Physics代考|BASIS SET METHODS

In those cases where the Hamiltonian $H$ is time-independent the Schrödinger equation Eq. (2.3) has solutions with a simple phase factor.

$$\psi({\Gamma}, \mathrm{t})=\phi({\mathrm{r}}) \exp [-\mathrm{i} E \mathrm{t} / h]$$
so that we recover the time-independent Schrödinger equation without the “clock” degrees of freedom.
$$\mathrm{H}{\phi}({\mathrm{r}})=\mathrm{E} \phi({\mathrm{r}}) .$$ Here $E$ is energy different from the essentially infinite value of $E=\lim {M \rightarrow \infty} \hbar^{2} M v^{2} \rightarrow \infty$, appearing in Eq. (2.1). Solutions of Eq. (2.3) include bound states $\phi_{m}$ and continuum states $\phi_{c}$. It will be assumed that the center of mass motion is factored out and the remaining particle coordinates ${\boldsymbol{r}}$ number $3 N$ as for $N$ independent particles. The set symbol ${\boldsymbol{r}}$ indicates that the coordinate includes the spin variable. Associated with each $N$ particle is a reduced mass. For simplicity we will consider that these particles are all electrons or possibly nuclei with a given, possibly time-dependent, coordinates and that the spin degrees of freedom in $H$ all refer to electron coordinates. Since $E$ is fixed the solutions are eigenstates of the energy operator $H$. To articulate the general theory as simply as possible it is assumed that $H$ describes a oneelectron species, which could be an atom or an $\mathrm{H}^{+}$-like molecular ion. In this case the set of coordinates ${\boldsymbol{r}}$ becomes just one spatial coordinate $r$. Where needed, generalizations to more than one electron will be indicated with a minimum of mathematical detail.

## 物理代写|原子物理代写Atomic and Molecular Physics代考|TIME IN QUANTUM MECHANICS

$$\left(-\frac{\hbar^{2}}{2 M} \frac{\partial^{2}}{\partial \varsigma^{2}}+H\right) \Psi=E \Psi$$

$$\left(-\frac{\hbar^{2}}{2 M} \frac{\partial^{2}}{\partial \zeta^{2}}-i \hbar^{2} \frac{K}{M} \frac{\partial}{\partial \zeta}+H\right) \psi=0$$

$$H \psi=i \hbar \frac{\partial \psi}{\partial t}$$

## 物理代写|原子物理代写Atomic and Molecular Physics代考|BASIS SET METHODS

$$\psi(\Gamma, \mathrm{t})=\phi(\mathrm{r}) \exp [-\mathrm{i} E \mathrm{t} / h]$$

$$\mathrm{H} \phi(\mathrm{r})=\mathrm{E} \phi(\mathrm{r}) .$$

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## MATLAB代写

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