# 物理代写|量子力学代写quantum mechanics代考|PHYS3101

## 物理代写|量子力学代写quantum mechanics代考|Lorentz covariants

By means of their transformation properties the following combinations of spinors and Dirac matrices can be classified as tensors of various ranks,
$\begin{array}{ll}\bar{\psi} \psi & \text { scalar } \ \bar{\psi} \gamma_5 \psi & \text { pseudoscalar } \ \bar{\psi} \gamma^\mu \psi & \text { vector } \ \bar{\psi} \gamma^\mu \gamma_5 \psi & \text { pseudovector, axialvector } \ \bar{\psi} \sigma^{\mu \nu} \psi & \text { 2nd rank tensor, } \quad \sigma^{\mu \nu}=\frac{i}{2}\left[\gamma^\mu, \gamma^\nu\right] .\end{array}$
The transformation properties can be immediately derived from $\psi \rightarrow S \psi$ and the relation (2.131) for the representation matrices $S$, together with
$$S^{-1}=\gamma^0 S^{\dagger} \gamma^0 .$$
Two examples are made explicit, the others are left as exercises.

• Scalar
$$\overline{\psi^{\prime}} \psi^{\prime}=(S \psi)^{\dagger} \gamma^0 S \psi=\psi^{\dagger} S^{\dagger} \gamma^0 S \psi=\bar{\psi}\left(\gamma^0 S^{\dagger} \gamma^0\right) S \psi=\bar{\psi} S^{-1} S \psi=\bar{\psi} \psi$$
• Vector
\begin{aligned} \overline{\psi^{\prime}} \gamma^\mu \psi^{\prime}=(S \psi)^{\dagger} \gamma^0 \gamma^\mu S \psi=\psi^{\dagger} S^{\dagger} \gamma^0 \gamma^\mu S \psi &=\bar{\psi}\left(\gamma^0 S^{\dagger} \gamma^0\right) \gamma^\mu S \psi \ &=\bar{\psi} S^{-1} \gamma^\mu S \psi=\Lambda_\nu^\mu \bar{\psi} \gamma^\nu \psi \end{aligned}
For covariants involving $\gamma_5$ one has to distinguish between $S=\gamma^0$ for inversions and $S(\Lambda)$ for Lorentz transformations $\Lambda$ without an inversion,
$$S^{-1} \gamma_5 S= \begin{cases}+\gamma_5 & \text { for } S=S(\Lambda) \ -\gamma_5 & \text { for } S=S(\mathbf{P})\end{cases}$$
yielding different signs under space inversions for pseudoscalar and axialvector in comparison to scalar and vector.

## 物理代写|量子力学代写quantum mechanics代考|Dirac Particle in an External Electromagnetic Field

Although relativistic one-particle quantum mechanics is not a consistent physics concept, it may be a useful tool for an approximative description of certain limiting cases. A typical example is a Dirac particle (electron or muon) at a low momentum scale in an external electromagnetic field which is treated as a classical field. The general case of an interaction with the electromagnetic quantum field corresponds to quantum electrodynamics and will be the content of the subsequent chapter.

Consider an electron (or muon) with mass $m$ and charge $e$ in a given electromagnetic field described by its classical 4-potential $\left(A^\mu\right)=\left(A^0, \vec{A}\right)$. The minimal substitution 3
$$P_\mu \rightarrow P_\mu-e A_\mu, \quad i \partial_\mu \rightarrow i \partial_\mu-e A_\mu$$
in the free Dirac equation (2.50) introduces the electromagnetic interaction between $A_\mu$ and the fermion described by the Dirac spinor $\psi$,
$$\gamma^\mu\left(i \partial_\mu-e A_\mu\right) \psi-m \psi=0$$
or equivalently expressed in terms of the $\vec{\alpha}$ and $\beta$ matrices,
$$i \partial_0 \psi=\left[\vec{\alpha} \cdot(-i \nabla-e \vec{A})+e A_0+\beta m\right] \psi \text {. }$$
For non-relativistic momenta this can be handled as a single-particle equation where $\psi$ plays the role of a 1-particle wave function. Two important special cases are:

• electron in a static Coulomb field,
• electron in a static magnetic field.
Historically, the derived phenomena discussed below had a significant impact on the success of the Dirac equation.

## 物理代写|量子力学代写quantum mechanics代考|Lorentz covariants

$\bar{\psi} \psi \quad$ scalar $\bar{\psi} \gamma_5 \psi \quad$ pseudoscalar $\bar{\psi} \gamma^\mu \psi$ vector $\bar{\psi} \gamma^\mu \gamma_5 \psi$ pseudovector, axialvector $\bar{\psi} \sigma^{\mu \nu} \psi$ 转换属性可以立即从 $\psi \rightarrow S \psi$ 以及表示矩阵的关系 (2.131) $S$ ， 和…一起
$$S^{-1}=\gamma^0 S^{\dagger} \gamma^0 .$$

• 标量
$$\overline{\psi^{\prime}} \psi^{\prime}=(S \psi)^{\dagger} \gamma^0 S \psi=\psi^{\dagger} S^{\dagger} \gamma^0 S \psi=\bar{\psi}\left(\gamma^0 S^{\dagger} \gamma^0\right) S \psi=\bar{\psi} S^{-1} S \psi=\bar{\psi} \psi$$
• 向量
$$\overline{\psi^{\prime}} \gamma^\mu \psi^{\prime}=(S \psi)^{\dagger} \gamma^0 \gamma^\mu S \psi=\psi^{\dagger} S^{\dagger} \gamma^0 \gamma^\mu S \psi=\bar{\psi}\left(\gamma^0 S^{\dagger} \gamma^0\right) \gamma^\mu S \psi \quad=\bar{\psi} S^{-1} \gamma^\mu S \psi=\Lambda_\nu^\mu \bar{\psi} \gamma^\nu \psi$$
对于涉及的协变量 $\gamma_5$ 必须区分 $S=\gamma^0$ 对于倒置和 $S(\Lambda)$ 对于洛伦兹变换 $\Lambda$ 没有反转，
$$S^{-1} \gamma_5 S=\left{+\gamma_5 \quad \text { for } S=S(\Lambda)-\gamma_5 \quad \text { for } S=S(\mathbf{P})\right.$$
与标量和向量相比，在伪标量和轴向向量的空间反转下产生不同的符号。

## 物理代写|量子力学代写quantum mechanics代考|Dirac Particle in an External Electromagnetic Field

$$P_\mu \rightarrow P_\mu-e A_\mu, \quad i \partial_\mu \rightarrow i \partial_\mu-e A_\mu$$

$$\gamma^\mu\left(i \partial_\mu-e A_\mu\right) \psi-m \psi=0$$

$$i \partial_0 \psi=\left[\vec{\alpha} \cdot(-i \nabla-e \vec{A})+e A_0+\beta m\right] \psi$$

• 静态库仑场中的电子,
• 静磁场中的电子。
从历史上看，下面讨论的衍生现象对狄拉克方程的成功产生了重大影响。

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