From b7fc23ae039d4064c564d2b3ab34b7774adaffe1 Mon Sep 17 00:00:00 2001 From: cyqm <2784841400@qq.com> Date: Wed, 11 Oct 2023 19:36:46 +0800 Subject: [PATCH] complete 2023/10/11 notes --- src/20231011.md | 191 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 191 insertions(+) create mode 100644 src/20231011.md diff --git a/src/20231011.md b/src/20231011.md new file mode 100644 index 0000000..70a4c2f --- /dev/null +++ b/src/20231011.md @@ -0,0 +1,191 @@ +## 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 ■ red square moves with the phase velocity, and the ● 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 × 10−1 |−1 | +| 1 | 1 × 100 |0 | +| 5 | 0.5 × 101 |1| +| 6 | 0.6 × 101 |1| +| 31 | 3.1 × 101 | 1| +| 32 | 0.32 × 102 | 2| +| 999 | 0.999 × 103| 3| +| 1000 | 1 × 103| 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". + +A container with two sides + +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 + +$$\rho_P = {m_P \over {l_P}^3} = {c^5 \over \hbar G^2}$$ +