Question 8.7: By means of the Lie series (8.139), express a finite rotatio......
By means of the Lie series (8.139), express a finite rotation about the z-axis as a canonical transformation.
\overline{u}=u+\kappa \mathcal{L}_{X}u+\frac{\kappa ^{2}}{2!}\mathcal{L}^{2}_{X}u + \frac{\kappa ^{3}}{3!}\mathcal{L}^{3}_{X}u +· · · . (8.139)
As shown in Section 8.6, the z-component of the angular momentum generates infinitesimal rotations about the z-axis. Therefore, for a clockwise finite rotation by angle θ, the Lie series (8.139) applies with κ = θ and X =L_{z} = xp_{y }− yp_{x}. Taking u = x we have
\mathcal{L}_{L_{z}}x = \left\{x, xp_{y }− yp_{x }\right\} = −y\left\{x, p_{x}\right\} = −y ; (8.141)
\mathcal{L}^{2}_{L_{z}}x = −\mathcal{L}_{L_{z}}y = −\left\{y, xp_{y} − yp_{x}\right\} = −x\left\{y, p_{y}\right\} = −x ; (8.142)
\mathcal{L}^{3}_{L_{z}}x = −\mathcal{L}_{L_{z}}x = y ; \mathcal{L}^{4}_{L_{z}}x = \mathcal{L}_{L_{z}}y = x . (8.143)
It is easy to conclude that, in general,
\mathcal{L}^{2n}_{L_{z}}x =(−1)^{n}x , \mathcal{L}^{2n+1}_{L_{z}}x =(−1)^{n+1}y. (8.144)
Thus, the Lie series (8.139) takes the form
\overline{x} =\left(\sum\limits_{n=0}^{\infty }{\left(-1\right)^{n} \frac{\theta^{2n} }{\left(2n\right)! } } \right) x-\left(\sum\limits_{n=0}^{\infty }{\left(-1\right)^{n} \frac{\theta^{2n+1} }{\left(2n+1\right)! } } \right) y=x\cos \theta -y\sin \theta . (8.145)
Similarly, one checks that \overline{y} = x sin θ + y cos θ. This establishes the validity of the Lie series (8.140) to express a finite rotation as a canonical transformation.
\overline{u}=u+\kappa \left\{u,X\right\}+\frac{\kappa ^{2}}{2!}\left\{\left\{u,X\right\},X\right\}+\frac{\kappa ^{3}}{3!}\left\{\left\{\left\{u,X\right\},X\right\},X\right\}+· · · . (8.140)