$h=6.63\cdot {10}^{-34}[J\cdot s]$

$c=3.0\cdot {10}^8[m\cdot s^{-1}]$

$E=nh\nu$

$\rho (\nu ,T)d\nu ={\displaystyle \frac{8\pi h{\nu }^3}{c^3}}{\displaystyle \frac{1}{e^{h\nu /k_{B}T}-1}}d\nu$

$E=h\nu$

$p={\displaystyle \frac{h}{\lambda }}$

$\tilde{\nu }={\displaystyle \frac{1}{\lambda }}=R_H({\displaystyle \frac{1}{n_1^2}}-{\displaystyle \frac{1}{n_2^2}})$

$E_n=-{\displaystyle \frac{m_e{e}^4}{8{\epsilon }_0^2{h}^2{n}^2}}$

${\displaystyle \frac{\partial \mathrm{\Psi }(x,t)}{\partial x^2}}={\displaystyle \frac{1}{v^2}}{\displaystyle \frac{\partial \mathrm{\Psi }(x,t)}{\partial t^2}}$

$\nu ={\displaystyle \frac{1}{T}}$

$v=\lambda \nu$

$\begin{array}{rl}\mathrm{\Psi }(x,t)& =A\sin (kx-\omega t+\varphi )\\ & =\mathrm{Re}(A{e}^{i(kx-\omega t+{\varphi }^{\prime })})\end{array}$

$-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{d^2\psi (x)}{d{x}^2}}+V(x)\psi (x)=E\psi (x)$

$\mathrm{\Psi }(x,t)=\psi (x)e^{-i(E/\hslash )t}$

${\int }_D{f}^{\ast }gdx=0$

$\hat{x}=x$

$\hat{\mathbf{x}}=\mathbf{x}$

$\hat{T}={\displaystyle \frac{{\hat{p}}_x^2}{2m}}=-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}$

$\hat{\mathbf{T}}=-{\displaystyle \frac{{\hslash }^2}{2m}}{\mathrm{\nabla }}^2$

$\hat{H}=\hat{T}+\hat{V}=-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}+V(x)$

$\hat{\mathbf{H}}=-{\displaystyle \frac{{\hslash }^2}{2m}}{\mathrm{\nabla }}^2+V(\mathbf{x})$

$\hat{H}\mathrm{\Psi }(x,t)=i\hslash {\displaystyle \frac{\partial \mathrm{\Psi }(x,t)}{\partial t}}$

$\mathrm{\Psi }(x,t)=\psi (x)T(t)$

$\mathrm{Prob}(x,x+dx)=|\mathrm{\Psi }(x,t){|}^{2}dx=|\psi (x){|}^{2}dx$

$c_n=⟨{\varphi }_n|\psi ⟩=\int {\varphi }_n^{\ast }\psi dx$

$[-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{d^2}{d{x}^2}}+V(x)]\psi (x)=E\psi (x)$

$E_n={\displaystyle \frac{h^2}{8m{L}^2}}{n}^2={\displaystyle \frac{{\hslash }^2{\pi }^2}{2m{L}^2}}{n}^2$

$[-{\displaystyle \frac{{\hslash }^2}{2m}}{\mathrm{\nabla }}^2+V(\mathbf{x})]\psi (\mathbf{x})=E\psi (\mathbf{x})$

$E_n={\displaystyle \frac{h^2}{8m}}({\displaystyle \frac{n_x^2}{L_x^2}}+{\displaystyle \frac{n_y^2}{L_y^2}}+{\displaystyle \frac{n_z^2}{L_z^2}})$

$V(x)=\{\begin{array}{ll}0& x\in [0,L]\\ V_0& \mathrm{elsewhere}\end{array}$

$T={\displaystyle \frac{4\sqrt{E(E-V_0)}}{(\sqrt{E}-\sqrt{E-V_0}{)}^2}}$

$[-{\displaystyle \frac{{\hslash }^2}{2\mu }}{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left(r^2{\displaystyle \frac{\partial }{\partial r}}\right)+V(r)]\psi (\mathbf{x})=E\psi (\mathbf{x})$

$V(r)\approx \frac{1}{2}k{r}^2$

$[-{\displaystyle \frac{{\hslash }^2}{2\mu }}{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left(r^2{\displaystyle \frac{\partial }{\partial r}}\right)+{\displaystyle \frac{1}{2}}k{r}^2]\psi (r)=E\psi (r)$

$H_n(r)=(-{)}^n{e}^{x^2}\left({\displaystyle \frac{d^n}{d{x}^n}}\right)e^{-x^2}$

$E_n=(n+\frac{1}{2})\hslash {\omega }_0$

$\mathrm{\Delta }n=\pm 1$

$[-{\displaystyle \frac{\hslash }{2m_e}}{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial r}}\left(r^2{\displaystyle \frac{\partial }{\partial r}}\right)+{\displaystyle \frac{\overrightarrow{L}}{2m_e{r}^2}}-{\displaystyle \frac{e^2}{r}}]\psi (x)=E\psi (x)$

${\psi }_{nlm}(x)=R_{nl}(r)Y_l^m(\theta ,\varphi )$

$R_H=2.179\times {10}^{-18}\mathrm{J}=13.6\mathrm{eV}$

$P_{nl}(r)dr=r^2{R}_{nl}^2(r)dr$

$[\underbrace{-{\displaystyle \frac{{\hslash }^2}{2m}}{\mathrm{\nabla }}_1^2}\mathrm{\_}\text{KE of }{e}_1^{-}\overbrace{-{\displaystyle \frac{{\hslash }^2}{2m}}{\mathrm{\nabla }}_2^2}\text{^}\text{KE of }{e}_2^{-}\underbrace{-{\displaystyle \frac{2e^2}{|{\mathbf{x}}_1|}}}\mathrm{\_}{e}_1^{-}\text{-N attraction}\overbrace{-{\displaystyle \frac{2e^2}{|{\mathbf{x}}_2|}}}\text{^}{e}_2^{-}\text{-N attraction}+\underbrace{{\displaystyle \frac{e^2}{|{\mathbf{x}}_1-{\mathbf{x}}_2|}}}\mathrm{\_}{e}_1^{-}\text{-}{e}_2^{-}\text{repulsion}]\psi ({\mathbf{x}}_1,{\mathbf{x}}_2)=E\psi ({\mathbf{x}}_1,{\mathbf{x}}_2)$

$[-{\displaystyle \frac{{\hslash }^2{\mathrm{\nabla }}_i^2}{2m}}-{\displaystyle \frac{Z{e}^2}{|\mathbf{x}|}}+\sum _{j=1,j\ne i}^N{\displaystyle \int {\displaystyle \frac{e^2{\varphi }_j^{\ast }({\mathbf{x}}^{\prime }){\varphi }_j({\mathbf{x}}^{\prime })}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}}{d}^3{\mathbf{x}}^{\prime }}]{\varphi }_i(\mathbf{x})={\epsilon }_{ii}{\varphi }_i(\mathbf{x})$

$[{\hat{S}}_i,{\hat{S}}_j]=i\hslash {\hat{S}}_k$

${\hat{\mathbf{S}}}^2\leftrightarrow {\hslash }^{2}s(s+1)$

$P_{ij}\psi ({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_i,\dots ,{\mathbf{r}}_j,\dots ,{\mathbf{r}}_N)=\psi ({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_j,\dots ,{\mathbf{r}}_i,\dots ,{\mathbf{r}}_N)$

$\lambda =1$

$\frac{1}{2}$-integer spin, anti-symmetric

$\psi ({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_i,\dots ,{\mathbf{r}}_i,\dots ,{\mathbf{r}}_N)=-\psi ({\mathbf{r}}_1,{\mathbf{r}}_2,\dots ,{\mathbf{r}}_i,\dots ,{\mathbf{r}}_i,\dots ,{\mathbf{r}}_N)=0$

$[-{\displaystyle \frac{{\hslash }^2{\mathrm{\nabla }}^2}{2m}}-{\displaystyle \frac{Z{e}^2}{|\mathbf{x}|}}]{\varphi }_i(\mathbf{r})+{\displaystyle \sum _{j=1}^N[{\varphi }_i(\mathbf{r})\int {\displaystyle \frac{e^2{\varphi }_j^{\ast }(\mathbf{r}){\varphi }_j(\mathbf{r})}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}}{d}^3{r}^{\prime }-{\varphi }_j(\mathbf{r})\int {\displaystyle \frac{e^2{\varphi }_j^{\ast }(\mathbf{r}){\varphi }_i(\mathbf{r})}{|\mathbf{x}-{\mathbf{x}}^{\prime }|}}{d}^3{r}^{\prime }]={\epsilon }_i{\varphi }_i(\mathbf{r})}$

$E={\displaystyle \frac{⟨\psi |H|\psi ⟩}{⟨\psi |\psi ⟩}}$