Browse & Filter

The motion of a particle undergoing constant acceleration \(a\) in one dimension is described by
\begin{eqnarray*}
x &=& x_0 + \frac{p_0t}{m} + \frac{1}{2}a t^2\\
p&=&p_0+m a t
\end{eqnarray*}Show that the transformation from the present ``old'' variables \((x,p)\) to initial ``new'' variables \((x_0,p_0)\) is a canonical transformation

(a) by Poisson bracket test
(b) by finding ( for \(t\ne0\) ) the type 1 generating function \(F_1(x,x_0,t)\)

Source:Calkin

Does there exist an invertible matrix \(S\) such that \[ S \gamma_\mu  S^{-1} = \gamma_\mu'\] where  \[\gamma_1'= \gamma_2\gamma_3, \quad \gamma_2'=\gamma_3\gamma_1, \quad \gamma_3'= \gamma_1\gamma_2, \gamma_4'=\gamma_5 \gamma_4?\] 

In the position representation a function of \(\vec{r}\), \(f(\vec{r})\), can be thought of as an operator \(\hat{f}\):  \[  \hat{f} \psi(\vec{r}) = f(\vec{r}) \psi(\vec{r}). \] Use spherical harmonics, \(Y_{\ell m}(\theta, \phi)\) to define operators \(\widehat{Y}_{\ell m}\) in the above sense. Use properties of spherical harmonics and known expressions for angular momentum operators in polar coordinates to show  \begin{eqnarray}   [L_z, \widehat{Y}_{\ell m}] &=& m\hbar\, \widehat{Y}_{\ell m}, \nonumber\\{}  [L_{\pm}, \widehat{Y}_{\ell m}] &=& \sqrt{\ell(\ell+1) - m(m\pm1)}\,\hbar\,\widehat{Y}_{\ell\pm 1 m}.\nonumber  \end{eqnarray}