8.6 Problems

Problem 8-1: Invariance of Poisson Brackets

Show that the Poisson Bracket of two functions, \(U\), \(V\), is invariant even under time-dependent canonical transformations, \(Q_i(q_i,p_i,t)\), \(P_i(q_i,p_i,t)\):

\begin{align*} \{U,V\}_{q,p}=\{U,V\}_{Q,P} \end{align*}

Problem 8-2: Canonical prescription and angular momentum

Use the canonical prescription for quantization to show that the angular momentum in the z-direction is quantized.

Problem 8-3: Canonical transformation types

Show that the transformation equations for type 3 and type 4 canonical transformations are given by the following relations, respectively:

\begin{align*} F&=F_3(p_i,Q_i,t)+\sum_ip_iq_i \\ q_i&=-\die{F_3}{p_i}\\ P_i&=-\die{F_3}{Q_i}\\ \end{align*}

and:

\begin{align*} F&=F_4(p_i,P_i,t)+\sum_ip_iq_i-\sum_iQ_iP_i\\ q_i&=-\die{F_4}{p_i}\\ Q_i&=\die{F_4}{P_i}\\ \end{align*}

Problem 8-4: Identifying canonical transformations

Determine and show which of the following transformations are canonical:
a):

\begin{align*} Q&=\frac{1}{2}(q^2+p^2)\\ P&=-\tan^{-1}(\frac{q}{p}) \end{align*}

b):

\begin{align*} Q&=\sqrt{2q}e^t \cos(p)\\ P&=\sqrt{2q}e^{-t}\sin(p) \end{align*}

c):

\begin{align*} Q&=\ln\frac{\sin(p)}{q}\\ P&=q\cot(p) \end{align*}

Problem 8-5: Canonical transformation of a Hamiltonian

a) Show that the following transformation is canonical:

\begin{align*} Q&=\frac{1}{2}(q^2+p^2)\\ P&=-\tan^{-1}(\frac{q}{p}) \end{align*}

b) If the Hamiltonian, \(H(q,p)\), is given by:

\begin{align*} H=\frac{1}{2}(q^2+p^2) \end{align*}

Write an expression of the new Hamiltonian, \(K(Q,P)\), and write the equations of motion for the new canonical variables (\(Q\), \(P\)).

c) Show that the transformed Hamiltonian, \(K(Q,P)\), is a constant of the motion
d) Write an expression for the Lagrangian of this system in terms of \(q\) and \(\dot q\)
e) Write out the equation of motion for \(\ddot q\), and give an example of a physical system that is described by this Lagrangian/Hamiltonian