Show that the bosonic commutation relations of the creation and annihilation operators are preserved by the Bogoliubov transformation

acting on \(\hat{a}\) and \(\hat{a}^{\dagger}\)

 * 1) Substitute the Bogoliubov transformation \(\begin{pmatrix}\hat{\alpha} \\ \hat{\alpha}^{\dagger}\end{pmatrix} = \begin{pmatrix}\cosh\,{\theta}\, & \sinh\,{\theta}\, \\ \sinh\,{\theta}\, & \cosh\,{\theta}\,\end{pmatrix} \begin{pmatrix}\hat{a} \\ \hat{a}^{\dagger}\end{pmatrix}\) into the standard commutator, \([\hat{\alpha}, \hat{\alpha}^{\dagger}]\): \([\hat{\alpha}, \hat{\alpha}^{\dagger}] = [\cosh\,{\theta}\,\hat{a} + \sinh\,{\theta}\,\hat{a}^{\dagger}, \sinh\,{\theta}\,\hat{a} + \cosh\,{\theta}\,\hat{a}^{\dagger}]\)
 * 2) Expand the commutator: \([\hat{\alpha}, \hat{\alpha}^{\dagger}] = \cosh\,{\theta}\,\hat{a}\sinh\,{\theta}\,\hat{a} + \cosh\,{\theta}\,\hat{a}\cosh\,{\theta}\,\hat{a}^{\dagger} + \sinh\,{\theta}\,\hat{a}^{\dagger}\sinh\,{\theta}\,\hat{a} + \sinh\,{\theta}\,\hat{a}^{\dagger}\cosh\,{\theta}\,\hat{a}^{\dagger} - \sinh\,{\theta}\,\hat{a}\cosh\,{\theta}\,\hat{a} - \sinh\,{\theta}\,\hat{a}\sinh\,{\theta}\,\hat{a}^{\dagger} - \cosh\,{\theta}\,\hat{a}^{\dagger}\cosh\,{\theta}\,\hat{a} - \cosh\,{\theta}\,\hat{a}^{\dagger}\sinh\,{\theta}\,\hat{a}^{\dagger} \)
 * 3) Simplify into commutators: \([\hat{\alpha}, \hat{\alpha}^{\dagger}] = \cosh\,{\theta}\,\sinh\,{\theta}\,[\hat{a},\hat{a}] + \cosh\,{\theta}\,\cosh\,{\theta}\,[\hat{a},\hat{a}^{\dagger}] + \sinh\,{\theta}\,\sinh\,{\theta}\,[\hat{a}^{\dagger},\hat{a}] + \cosh\,{\theta}\,\sinh\,{\theta}\,[\hat{a}^{\dagger},\hat{a}^{\dagger}]\)
 * 4) Solve the commutators: \([\hat{\alpha}, \hat{\alpha}^{\dagger}] = \cosh^2\,{\theta}\, - \sinh^2\,{\theta}\,\)
 * 5) Simplify with the trigonometric identity \(\cosh^2\,{\theta}\, - \sinh^2\,{\theta}\, = 1\): \([\hat{\alpha}, \hat{\alpha}^{\dagger}] = 1\)

acting on \(\hat{a}\) and \(\hat{b}\)

 * 1) Substitute the Bogoliubov transformation \(\begin{pmatrix}\hat{\alpha} \\ \hat{\beta}\end{pmatrix} = \begin{pmatrix}\cosh\,{\theta}\, & \sinh\,{\theta}\, \\ \sinh\,{\theta}\, & \cosh\,{\theta}\,\end{pmatrix} \begin{pmatrix}\hat{a} \\ \hat{b} \end{pmatrix}\) into the standard commutator, \([\hat{\alpha}, \hat{\beta}]\): \([\hat{\alpha}, \hat{\beta}] = [\cosh\,{\theta}\,\hat{a} + \sinh\,{\theta}\,\hat{b}, \sinh\,{\theta}\,\hat{a} + \cosh\,{\theta}\,\hat{b}]\)
 * 2) Expand the commutator: \([\hat{\alpha}, \hat{\beta}] = \cosh\,{\theta}\,\hat{a}\sinh\,{\theta}\,\hat{a} + \cosh\,{\theta}\,\hat{a}\cosh\,{\theta}\,\hat{b} + \sinh\,{\theta}\,\hat{b}\sinh\,{\theta}\,\hat{a} + \sinh\,{\theta}\,\hat{b}\cosh\,{\theta}\,\hat{b} - \sinh\,{\theta}\,\hat{a}\cosh\,{\theta}\,\hat{a} - \sinh\,{\theta}\,\hat{a}\sinh\,{\theta}\,\hat{b} - \cosh\,{\theta}\,\hat{b}\cosh\,{\theta}\,\hat{a} - \cosh\,{\theta}\,\hat{b}\sinh\,{\theta}\,\hat{b} \)
 * 3) Simplify into commutators: \([\hat{\alpha}, \hat{\beta}] = \cosh\,{\theta}\,\sinh\,{\theta}\,[\hat{a},\hat{a}] + \cosh\,{\theta}\,\cosh\,{\theta}\,[\hat{a},\hat{b}] + \sinh\,{\theta}\,\sinh\,{\theta}\,[\hat{b},\hat{a}] + \cosh\,{\theta}\,\sinh\,{\theta}\,[\hat{b},\hat{b}]\)
 * 4) Solve the commutators: \([\hat{\alpha}, \hat{\beta}] = \cosh^2\,{\theta}\,[\hat{a},\hat{b}] - \sinh^2\,{\theta}\,[\hat{a},\hat{b}]\)
 * 5) Simplify with the trigonometric identity \(\cosh^2\,{\theta}\, - \sinh^2\,{\theta}\, = 1\): \([\hat{\alpha}, \hat{\beta}] = [\hat{a},\hat{b}]\)