15.2: Poisson bracket Representation of Hamiltonian Mechanics
Hamilton’s canonical equations of motion→ time evolution of the canonical variables$(q,p)$ in phase space
Poisson bracket formalism → Hamiltonian mechanics ⇒ a direct link between classical mechanics and quantum mechanics
Poisson bracket of any two continuous functions $F(p,q), \ G(p,q):$
$$ \{F,G\}_{qp}\equiv\sum_i\left(\frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}\right) $$
leads to the following
identity$\{F,F\}=0$
antisymmetry $\{F,G\}=-\{G,F\}$
linearity$\{G,F+Y\}=\{G,F\}+\{G,Y\}$
Leibniz rules$\begin{aligned}&\{G,FY\}=\{G,F\}Y+F\{G,Y\}.\end{aligned}$
Jacobi identity$0=\{F,\{G,Y\}\}+\{G,\{Y,F\}\}+\{Y,\{F,G\}\}$
the poisson brackets of the canonical variables themselves.
$$ \begin{gathered}\{q_k,q_l\}{qp}=\sum_i\left(\frac{\partial q_k}{\partial q_i}\frac{\partial q_l}{\partial p_i}-\frac{\partial q_k}{\partial p_i}\frac{\partial q_l}{\partial q_i}\right)=\sum_i(\delta{ki}\cdot0-0\cdot\delta_{li})=0 \\\{p_k,p_l\}{qp}=\sum_i\left(\frac{\partial p_k}{\partial q_i}\frac{\partial p_l}{\partial p_i}-\frac{\partial p_k}{\partial p_i}\frac{\partial p_l}{\partial q_i}\right)=\sum_i(0\cdot\delta{li}-\delta_{ki}\cdot0)=0 \\\{q_k,p_l\}{qp}=\sum_i\left(\frac{\partial q_k}{\partial q_i}\frac{\partial p_l}{\partial p_i}-\frac{\partial q_k}{\partial p_i}\frac{\partial p_l}{\partial q_i}\right)=\sum_i(\delta{ki}\cdot\delta_{li}-0\cdot0)=\delta_{kl} \end{gathered} $$
the only non-zero fundamental Poisson bracket is for conjugate variabls where $k=l$.
$$ \{q_k,p_k\}_{pq}=1 $$
canonical variables↔$[q,p]\ne0$ in quantum theory
Canonical transformation正则变换|form invariance形式不变性
invariant under a canonical transformation$(q_k,p_k)\to (Q_k,P_k)$