量子物理复习笔记

热辐射 黑体辐射

斯特藩一玻耳兹曼定律

\[M(T) = \int_{0}^{\infty}M_\lambda(T)d\lambda = \sigma T^4\]

维恩位移定律

\[\lambda_mT = b\]

普朗克经验公式

\[M_0(\nu, T) = \frac{2 \pi \nu^2}{c^2} \frac{h \nu}{e^{h \nu /k_B T} - 1}\] \[M_0(\lambda, T) = \frac{2 \pi h c^2}{\lambda^5}\frac{1}{e^{hc/\lambda k T} - 1}\]

康普顿散射

康普顿散射

\[\Delta \lambda = \lambda - \lambda_0 = \frac{h}{m_0 c} (1 - cos \theta)\]

其中 \(\frac{h}{m_0 c} = \lambda_c\)为康普顿波长，\(m_0\)为电子静止质量

氢原子光谱、波尔理论

里德伯公式

\[\tilde{\nu} = R_\infty (\frac{1}{2^2} - \frac{1}{n^2}), \space R_\infty = \frac{me^4}{8\varepsilon_0^2h^3c}\]

角动量量子化

\[m \nu_n r_n = n \hbar\]

波尔半径

\[r_1 = \frac{\varepsilon_0 h^2}{\pi m e^2} = 0.053nm\]

电子能量

\[E_n = - \frac{m e^4}{8 \varepsilon_0^{2} h^2 n^2}\]

物质波

德布罗意公式

\[\lambda = \frac{h}{p} = \frac{h}{m_0 v} \sqrt{1 - \frac{v^2}{c^2}}\] \[\lambda \approx \frac{h}{m_0 v} \approx \frac{h}{\sqrt{2m_0 E_k}}, v \ll c\]

波函数

\[\Psi(x, t) = \psi_0 \cos(\frac{2\pi}{\lambda}x - \omega t) = \psi_0 e^{i(kx - \omega t)} = \psi_0 e^{-\frac{i}{\hbar}(Et-px)}\]

不确定关系

\[\Delta x \cdot \Delta p_x \ge \frac{\hbar}{2}\] \[\Delta E \cdot \Delta t \ge \frac{\hbar}{2}\] \[(\Delta x)^2 =\overline{x^2} - (\overline{x})^2\]

薛定谔方程

\[i\hbar \frac{\partial \Psi(x, t)}{\partial t} = \left(- \frac{\hbar^2}{2m} \vec{\nabla}^2 + U \right )\Psi(x,t)\]

定态薛定谔方程

\[\Psi_E(\vec{r}, t) = \Phi_E(\vec{r}) e^{-\frac{i}{\hbar}Et}\] \[\left (- \frac{\hbar^2}{2m} \nabla^2 + U(\vec{r}) \right) \Phi(\vec{r}) = E \Phi(\vec{r})\]

概率密度

\[\rho = \Psi^\ast \Psi\] \[\vec{J} = \frac{\hbar}{2mi}\left [\Psi^\ast(\vec{r}, t)\nabla \Psi(\vec{r}, t) - \Psi(\vec{r}, t)\nabla \Psi^\ast(\vec{r}, t) \right]\] \[\frac{\partial \rho}{\partial t} + \nabla \cdot \vec{J} = 0\]

一维无限深势阱

\[\left\{ \begin{matrix} \begin{aligned} &\Phi(x) = \sqrt{\frac{2}{a}}\sin(\frac{n \pi}{a} x) \space (n = 1, 2, 3... \space 0 < x < a)\\ &\Phi(x) = 0 \space (x \le 0, \space x \ge a) \end{aligned} \end{matrix} \right.\] \[k^2 = \frac{2mE_n}{\hbar^2}\] \[ka = n\pi\] \[E_n = \frac{\pi^2 \hbar^2}{2ma^2}n^2, n=1,2,3...\]

力学量的算符表示

位置表象中算符的平均值为

\[\left \langle \hat{O} \right \rangle = \iiint_{- \infty}^{\infty} \Psi^\ast(\vec{r}, t)\left[\hat{O} \Psi(\vec{r}, t)\right]dxdydz\]

动量算符

\[\left \langle \vec{p} \right \rangle = \iiint_{-\infty}^{\infty} |C(\vec{p}, t)|^2 \vec{p}dp_xdp_ydp_z\] \[\hat{\vec{p}} = -i\hbar \nabla\]

动量空间的波函数

\[\Psi(x, t) = \int_{-\infty}^{\infty} C(p, t) e^{\frac{ipx}{\hbar}} \frac{dp}{\sqrt{2\pi \hbar}}\] \[C(p, t) = \int_{-\infty}^{\infty} \Psi(x, t) e^{-\frac{ipx}{\hbar}} \frac{dx}{\sqrt{2 \pi \hbar}}\]

算符的运算规则

对易恒等式

\[[\hat{A}, \hat{B}] = - [\hat{B}, \hat{A}] \\ [\hat{A}, \hat{B} + \hat{C}] = [\hat{A}, \hat{B}] + [\hat{A}, \hat{C}]\\ [\hat{A}, \hat{B}\hat{C}] = \hat{B}[\hat{A}, \hat{C}] + [\hat{A}, \hat{B}]\hat{C}\\ [\hat{A}\hat{B}, \hat{C}] = \hat{A}[\hat{B}, \hat{C}] + [\hat{A}, \hat{C}]\hat{B}\] \[[\hat{x}, \hat{p_x}] = i\hbar\]

本征方程

\[\hat{A}\phi = \lambda \phi\]

算符的转置

\[\int d^3r \varphi^\ast (\tilde{\hat{A}} \phi) = \int d^3r \phi(\hat{A} \varphi^\ast )\]

厄米算符

\[\int d^3r \varphi^\ast(\hat{A^+} \phi) = \int d^3r (\hat{A} \varphi)^\ast \phi\]

• 厄米算符的本征值必为实数；
• 本征值为实数的算符必为厄米算符；
• 厄米算符的平均值必为实数；
• 平均值为实数的算符必为厄米算符

角动量算符

\[L_x = y \hat{p}_z - z \hat{p}_y\\ L_y = z \hat{p}_x - x \hat{p}_z\\ L_z = y \hat{p}_z - z \hat{p}_x\]

算符的矩阵表示

\(L_x\)的本征方程为

\[\frac{\hbar}{\sqrt2} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 &0 \end{pmatrix} \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \lambda \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}\]

轨道角动量算符

角动量算符

\[\hat{L_j} = \varepsilon_{jkl} \hat{x}_k \hat{p}_l\]

角动量算符之间不对易

\[[\hat{L}_x, \hat{L}_y] = i\hbar\hat{L}_z\]

角动量平方算符与其各分量之间对易

\[[\hat{L}^2, \hat{L}_x] = 0\]

其他对易关系

\[[\hat{L}_x, \hat{x}] = 0\\ [\hat{L}_x, \hat{y}] = i\hbar \hat{z}\\ [\hat{L}_x, \hat{z}] = -i\hbar \hat{y}\\ [\hat{L}_x, \hat{p}_x] =0\\ [\hat{L}_x, \hat{p}_y] = i\hbar \hat{p}_z\\ [\hat{L}_x, \hat{p}_z] = -i\hbar \hat{p}_y\]

本征值与本征函数

\[\left \{ \begin{matrix} \begin{aligned} &\hat{L}^2 Y_{lm} = l(l+1) \hbar^2 Y_{lm} ,\space (l = 0, 1, 2,...)\\ &\hat{L}_z Y_{lm} = m \hbar Y_{lm} ,\space(m = 0, \pm1, \pm2,... \pm l) \end{aligned} \end{matrix} \right.\]

自旋算符和自旋波函数

自旋

\[\vec{M}_S = -\frac{e}{m_e} \vec{S}\] \[S_z = \pm \frac{\hbar}{2}\]

泡利算符

\[\hat{\vec{S}} = \frac{\hbar}{2} \hat{\vec{\sigma}}\] \[\hat{\vec{\sigma}} \times \hat{\vec{\sigma}} = 2i\hat{\vec{\sigma}}\]

反对易关系

\[\hat{\vec{\sigma}_x} \hat{\vec{\sigma}_y} = -\hat{\vec{\sigma}_y}\hat{\vec{\sigma}_x}\] \[\sigma_x = \left ( \begin{matrix} 0 & 1\\ 1 &0 \end{matrix} \right) ,\space \sigma_y = \left ( \begin{matrix} 0 & -i\\ i &0 \end{matrix} \right) ,\space \sigma_z = \left ( \begin{matrix} 1 & 0\\ 0 &-1 \end{matrix} \right)\]

一维有限深势阱

\[\Psi = \left \{ \begin{matrix} \begin{aligned} & A \sin kx , \space |x| < a\\ & Be^{-k'x}, \space x > a\\ & Ce^{k'x},\space x < -a \end{aligned} \end{matrix} \right.\]

奇宇称：\(B = -C\)，偶宇称：\(B = C\)

一维散射问题

\[J_R = -\frac{k \hbar}{m}|A'|^2\\ J_T = \frac{k \hbar}{m}|C|^2\]

一维谐振子

\[\hat{H} = -\frac{\hbar^2}{2m} \frac{d^2}{d^2 x} + \frac{1}{2} m \omega^2 x^2\] \[E_n = (n + \frac{1}{2}) \hbar \omega, \space n=0,1,2...\]

一个谐振子能级只有一个本征函数，即一个状态，所以能级是非简并的

氢原子

\[U(r) = - \frac{Ze^2}{4 \pi \varepsilon_0 r}\]

氨分子

\[|\Psi(t)\rangle = C_1(t) |1\rangle + C_2(t) |2\rangle = e^{-\frac{iE_0t}{\hbar}} \cos\frac{At}{\hbar}|1\rangle - e^{-\frac{iE_0t}{\hbar}} i\sin\frac{At}{\hbar}|2\rangle\] \[P_1 = |C_1(t)|^2 = \cos^2 \frac{At}{\hbar}\\ P_2 = |C_2(t)|^2 = \sin^2 \frac{At}{\hbar}\]