complete 2023/10/11 notes

src/20231011.md

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+## 2023/10/11
+
+### 1. Phase an group velocity 相速度和群速度
+
+Phase velocity (相速度) $v_p$ is the velocity of a certain phase that travels:
+
+$$v_p = {\omega \over k}$$
+
+Group velocity (群速度) $v_g$ is the velocity with which the envelope of the wave (波包) propagates through space. 
+
+$$v_g = {\Delta \omega \over \Delta k}$$
+
+Energy is transmitted through *group velocity*, not 
+
+You can refer to this below:
+
+> ![Demo of a wave group](../assets/Wave_group.gif)
+>
+> The <span style="color: #DD0000">■</span> red square moves with the phase velocity, and the <span style="color: #77AC30">●</span> green circles propagate with the group velocity. (Source: Wikipedia)
+>
+> For more reference, you can see: [https://www.zhihu.com/question/29444240/answer/1833520606](https://www.zhihu.com/question/29444240/answer/1833520606).
+
+### 2. Order of magnitude 数量级
+
+For a number $N$, we usually define its *order of magnitude* as follows:
+
+Write the number in the form $$N =a \times 10 ^ b,$$ in which $$\dfrac{1}{\sqrt{10}} \leq a \leq \sqrt{10}, \ b \in \Bbb{Q},$$ and $b$ is the *order of magnitude* of the number.
+
+Of course this definition is not absolute, and some people tend to use $0.5 \leq a \leq 5$ or other criteria.
+
+For example:
+
+| $N$ | Expression in $N =a \times 10^b$ | Order of magnitude $b$ |
+|-|-|-|
+| 0.2 | 2 × 10<sup>−1</sup> |−1 |
+| 1 | 1 × 10<sup>0</sup> |0 |
+| 5 | 0.5 × 10<sup>1</sup> |1|
+| 6 | 0.6 × 10<sup>1</sup> |1|
+| 31 | 3.1 × 10<sup>1</sup> | 1|
+| 32 | 0.32 × 10<sup>2</sup> | 2|
+| 999 | 0.999 × 10<sup>3</sup>| 3|
+| 1000 | 1 × 10<sup>3</sup>| 3 |
+
+Some constants in physics: 
+
+- Avogadro constant $N_A = 6.02 \times 10^{23} \ \mathrm{mol^{-1}}$
+- Reduced Planck constant $\hbar = 1.054 \times 10^{-34} \ \mathrm{J \cdot s}$
+- Speed of light $c = 2.99792 \times 10^{8} \ \mathrm{m/s}$
+- Boltzmann constant $k_\mathrm{B} = 1.38 \times 10^{-23} \ \mathrm{J/K}$
+- Fundamental charge $e = 1.602 \times 10^{-19} \ \mathrm{C}$
+- Universal gravitational constant $G = 6.672 \times 10^{-11} \ \mathrm{N \cdot m^2/kg^2}$
+
+
+### 3. A particular model 某个模型
+
+Below is a container with two sides connected to each other by a "small hole".
+
+<img alt="A container with two sides" src="../assets/Model_Two_sides.png" height=240>
+
+To get the two sides to balance, we need to have
+
+$$
+\left\{
+\begin{align*}
+& p_1 = p_2 \ \ \  &\text{pressure} &\\[1ex]
+& T_1 = T_2 \ \ \ &\text{temperature} &\\[1ex]
+& \mu_1 = \mu_2 \ \ \ &\text{chemical potential (化学势)} &
+\end{align*}
+\right.
+$$
+
+Chemical potential $\mu$ is defined as follows:
+
+In a chemical system where there are $n$ kinds of species (物种), define **Gibbs free energy** (吉布斯自由能) $$G=U-TS+pV$$ and the **chemical potential** (化学势) of species $i$ $$\mu_i = \left({\partial G \over \partial n_i}\right)_{T, p, n_j (j \neq i)}.$$
+
+### 4. Something about dimensional analysis and unit systems 一些关于量纲分析和单位制的东西
+
+Maxwell points out that:
+
+1. For a mechanical quantity (力学量), we only meed three dimensions $\mathrm{M, L, T}$ to form its unit and
+
+2. Sometimes the dimensions of combined quantities are more useful.
+
+There are two commonly-used unit systems in the world:
+- *Système International* (SI) 国际单位制
+- Centimetre–gram–second system of units (CGS) 厘米—克—秒制
+
+Let's look at a few examples.
+
+#### (1) Coulomb's law 库仑定律
+
+$$F = \left\{
+\begin {align*}
+& \dfrac{q_1q_2}{4 \pi \varepsilon_0 r^2} & \text{(SI)} \\[3ex]
+& \dfrac{q_1q_2}{r^2} & \text{(CGS)}
+\end {align*}
+\right.
+$$
+
+
+
+#### (2) Fine structure constant 精细结构常数
+
+$$\alpha = \left\{
+\begin {align*}
+& \dfrac{e^2}{4 \pi \varepsilon_0 \hbar c} & \text{(SI)} \\[3ex]
+& \dfrac{e^2}{\hbar c} & \text{(CGS)}
+\end {align*}
+\right.
+\approx {1 \over 137}
+$$
+
+*W. Pauli died in Room No. 137 in hospital. (地狱笑话了属于是)*
+
+#### (3) Bohr radius 玻尔半径
+
+$$r_B = \left\{
+\begin {align*}
+& \dfrac{4 \pi \varepsilon_0 \hbar^2}{m_ee^2} & \text{(SI)} \\[3ex]
+& \dfrac{\hbar^2}{m_ee^2} & \text{(CGS)}
+\end {align*}
+\right.
+$$
+
+### 5. Planck units 普朗克单位
+
+Planck units are a set of units that, by definition, are expressed using these universal constants below, which, have the numeric value $1$ when expressed:
+
+- the speed of light in vacuum $c$
+- the gravitational constant $G$
+- the reduced Planck constant $\hbar$
+- the Boltzmann constant $k_\mathrm{B}$
+
+Typically we would use dimensional analysis to derive these units.
+
+$$[c] = \mathrm{LT^{-1}}$$
+
+$$[G] = {[F][r^2] \over [m_1m_2]} = \mathrm{\dfrac{ML}{T^2} \cdot L^2 \over M^2} = \mathrm{M^{-1}L^3T^{-2}}$$
+
+$$[\hbar] = [E][t] = [mc^2][t] = \mathrm{ML^2T^{-1}}$$
+
+$$[k_\mathrm{B}] = {[E] \over [T]} = {[mc^2] \over [T]} = \mathrm{ML^2T^{-2}\Theta^{-1}}$$
+
+#### (1) Planck length 普朗克长度
+
+$$\left[{G \hbar \over c^3}\right] = \mathrm{M^{-1}L^3T^{-2} \cdot ML^2T^{-1} \over (LT^{-1})^3} = \mathrm{L^2}$$
+
+$$l_P = \sqrt{G \hbar \over c^3}$$
+
+#### (2) Planck time 普朗克时间
+
+$$t_P = {l_P \over c} = \sqrt{G \hbar \over c^5}$$
+
+#### (3) Planck mass 普朗克质量
+
+$$\left[{\hbar c \over G}\right]  = \mathrm{ML^2T^{-1} \cdot LT^{-1} \over M^{-1}L^3T^{-2}} = \mathrm{M^2}$$
+
+$$m_P = \sqrt{\hbar c\over G}$$
+
+#### (4) Planck temperature 普朗克温度
+
+$$\left[{\hbar c^5 \over Gk_\mathrm{B}^2}\right] = \mathrm{ML^2T^{-1} \cdot (LT^{-1})^5 \over M^{-1}L^3T^{-2} \cdot (ML^2T^{-2}\Theta^{-1})^2} = \mathrm{\Theta^2}$$
+
+$$T_P = \sqrt{\hbar c^5 \over Gk_\mathrm{B}^2}$$
+
+#### (5) Planck energy 普朗克能量
+
+$$E_P = m_Pc^2 = \sqrt{\hbar c^5\over G}$$
+
+#### (6) Planck momentum 普朗克动量
+
+$$p_P = m_Pc = \sqrt{\hbar c^3\over G}$$
+
+#### (7) Planck acceleration 普朗克加速度
+
+$$a_P = {c \over t_P} = \sqrt{c^7 \over \hbar G}$$
+
+#### (8) Planck force 普朗克力
+
+$$F_P = m_P a_P = {c^4 \over G}$$
+
+Note that the Planck force can be *hidden* in the Einstein gravitational field equations:
+
+$$R_{\mu\nu} - {1 \over 2}g_{\mu\nu}R = 8 \pi \red{G \over c^4}T_{\mu\nu} = {8 \pi \over \red{F_P}} T_{\mu\nu}$$
+
+*隐藏在引力方程中的力，被称为引力很合理吧？这恒河里！doge*
+
+#### (9) Planck density 普朗克密度
+
+$$\rho_P = {m_P \over {l_P}^3} = {c^5 \over \hbar G^2}$$
+