Commutations and Formulae

Commutations

\begin{align*} [x_i,p_j] = i \hbar \delta_{ij} \end{align*}

\[a = \sqrt{\frac{m\omega}{2\hbar}} \left( x + \frac{i p}{m\omega} \right), \qquad a^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \left( x - \frac{i p}{m\omega} \right),\] \[[a, a^\dagger] = \left( \frac{1}{2\hbar} \right) \left( -i[x,p] + i[p,x] \right) = 1.\]

Pauli Matrices

\[\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\qquad \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.\] \[\sigma_{\pm} = (\sigma_x \pm i\sigma_y)/2\] \[\{\sigma_i,\sigma_j\} = 2\delta_{ij}\] \[[\sigma_i,\sigma_j] = 2i \epsilon_{ijk}\sigma_k\] \[(\boldsymbol{\sigma}\cdot\mathbf{a})(\boldsymbol{\sigma}\cdot\mathbf{b}) = \mathbf{a}\cdot\mathbf{b} + i\boldsymbol{\sigma}\cdot(\mathbf{a}\times \mathbf{b})\]

Formulae

Baker–Hausdorff lemma \(\begin{aligned} \exp(iG\lambda)\, A\, \exp(-iG\lambda) =& A + i\lambda[G,A] + \frac{i^2\lambda^2}{2!}[G,[G,A]] + \\ &\cdots + \frac{i^n \lambda^n}{n!}[G,[G,[G,\dots[G,A]]]\dots] + \cdots \end{aligned}\)

Special case where $[G,A] = \pm A$, $ \exp(iG\lambda)\, A\, \exp(-iG\lambda) = A \exp(\pm i\lambda) $.

Let $ x \in \mathbb{R} $ and $ \hat{A}^2 = \mathbb{I} $: \begin{equation} \exp(i \hat{A} x) = \cos(x) \mathbb{I} + i\, \sin(x) \hat{A} \label{expiax} \end{equation}

One useful substitution of eq.(\ref{expiax}):

\[\exp(-\frac{i\vec{\sigma}\cdot\hat{\bf{n}}\phi}{2}) =\mathbb{I} \cos(\frac{\phi}{2}) - i\vec{\sigma}\cdot \hat{\bf{n}} \sin(\frac{\phi}{2})\]