Frequently Used Formulas¶
Quantum Mechanics¶
Baker–Campbell–Hausdorff formula¶
See Wiki for more details.
(1)\[\begin{split}e^{X} Y e^{-X} &=& e^{\text{ad}_X}Y \equiv Y + [X,Y] + \frac{1}{2!}[X,[X,Y]] + \frac{1}{3!}[X,[X,[X,Y]]]+\cdots\\
e^{t(X+Y)} &=& e^{tX} e^{tY} e^{-\frac{t^2}{2}[X,Y]} e^{\frac{t^3}{6}(2[Y,[X,Y]+[X,[X,Y]]])}\cdots\end{split}\]
Some examples:
- Spin rotation
(2)\[e^{-i \phi S_z} S_x e^{i \phi S_z} = \cos\phi S_x + \sin\phi S_y\]
- For Boson
(3)\[\begin{split}U_0^{\dagger}(t) a U_0(t) &=& e^{i \omega t a^{\dagger}a} a e^{-i \omega t a^{\dagger}a} = a e^{-i \omega t}\\
D^{\dagger}(\alpha)a D(\alpha) &=& e^{-\alpha a^{\dagger} + \alpha^{*}a} a e^{\alpha a^{\dagger} - \alpha^{*}a} = a + \alpha\end{split}\]
Pauli matrices¶
See Wiki link and Pauli matrices for more properties.
\[\begin{split}[s_{\alpha}, s_{\beta}] &=& i \hbar \epsilon_{\alpha\beta\gamma} s_{\gamma}\\
[\sigma_{\alpha}, \sigma_{\beta}] &=& 2i \hbar \epsilon_{\alpha\beta\gamma} \sigma_{\gamma}\\
\{\sigma_{\alpha}, \sigma_{\beta}\} &=& 2\delta_{\alpha\beta}\\
(\mathbf{a}\cdot\boldsymbol{\sigma})(\mathbf{b}\cdot\boldsymbol{\sigma})&=& \mathbf{a}\cdot\mathbf{b} + i (\mathbf{a}\times\mathbf{b})\cdot \boldsymbol{\sigma}\end{split}\]
