3.1. Classical Mechanics and Quantum Mechanics

Throughout this section, we use Einsten’s summation rule.

3.1.1. Poisson Brackets in Classical Mechanics

Definition of Poisson Bracket

Poisson bracket is defined as

\[\{A,B\}= \frac{\partial A}{\partial q_i} \frac{\partial B}{\partial p_i} - \frac{\partial A}{\partial p_i} \frac{\partial B}{\partial q_i}.\]

In classical mechanics, we can find conserved quantities without the help of symmetries, by using Hamilton’s form.

For an conserved quantity \(A(q_i,p_i)\), we have

\[\partial_t A(q_i,p_i) = 0.\]

By using Hamilton’s equations, we can find that

\[\partial_t A(q_i,p_i) = \{ A(q_i,p_i), H \}.\]

Thus the conservation condition is

\[\{ A(q_i,p_i), H \} = 0.\]

Derivation of the Relation between Conserved Quantities and Poisson Bracket

On the other hand,

(1)\[\partial_t A(q_i,p_i) = \frac{\partial A(q_i,p_i)}{\partial q_i} \partial_t q_i + \frac{\partial A(q_i,p_i)}{\partial p_i} \partial_t p_i.\]

Recall that the Hamilton’s equations are

\[\begin{split}\partial_t q_i = & \frac{\partial H}{\partial p_i}, \\ \partial_t p_i = & -\frac{\partial H}{\partial q_i}.\end{split}\]

Plug them back into Eq. (1), we have

\[\begin{split}\partial_t A(q_i,p_i) =& \frac{\partial A(q_i,p_i)}{\partial q_i} \frac{\partial H}{\partial p_i} + \frac{\partial A(q_i,p_i)}{\partial p_i} \left( -\frac{\partial H}{\partial q_i} \right) \\ =& \{A(q_i,p_i), H\}.\end{split}\]

Self-consistancy Check of Poisson Bracket and Conserved Quantities

We can easily find the Poisson brackets \(\{q_i,H\}\) and \(\{p_i,H\}\), which are in fact \(\partial q_i\) and \(\partial_t p_i\).

We would actually get the Hamilton’s equations.