Frequently Used Formulas
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Quantum Mechanics
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Baker–Campbell–Hausdorff formula
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See `Wiki <http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula>`_ for more details.

.. math::
   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 
   :label: BCH

Some examples:

* Spin rotation

.. math::
   e^{-i \phi S_z} S_x e^{i \phi S_z} = \cos\phi S_x + \sin\phi S_y
   :label: BCH_SpinRot
   
* For Boson

.. math::
   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
   :label: BCH_Boson
   
Pauli matrices
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See `Wiki link <http://en.wikipedia.org/wiki/Spin_(physics)>`_ and `Pauli matrices <http://en.wikipedia.org/wiki/Pauli_matrices>`_ for more properties.

.. math::
   [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}

Mathematics
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Triangle
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