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Poisson Brackets

For functions that are defined on the phase space we can define the following operation. Let F = F(q, p, t) and G = G(q, p, t). Then a Poisson bracket of these two functions is defined by:

 \begin{displaymath}
\left\{
F, G
\right\} =
\sum_{i=1}^n
\left(
\frac{\pa...
...al G}{\partial q_i}
\frac{\partial F}{\partial p_i}
\right)
\end{displaymath} (4.10)

This operation has the following neat properties:
$\displaystyle \left\{ F, G \right\}$ = $\displaystyle - \left\{ G, F \right\}$ (4.11)
$\displaystyle \left\{ F, F \right\}$ = 0 (4.12)
$\displaystyle \left\{ F_1 + F_2, G \right\}$ = $\displaystyle \left\{ F_1, G \right\} + \left\{ F_2, G \right\}$ (4.13)
$\displaystyle \left\{ F_1 F_2, G \right\}$ = $\displaystyle F_1 \left\{ F_2, G \right\} + F_2 \left\{ F_1, G \right\}$ (4.14)
$\displaystyle \left\{ F, q_i \right\}$ = $\displaystyle - \frac{\partial F}{\partial p_i}$ (4.15)
$\displaystyle \left\{ F, q_i \right\}$ = $\displaystyle \frac{\partial F}{\partial q_i}$ (4.16)
$\displaystyle \left\{ q_i, q_j \right\}$ = 0 (4.17)
$\displaystyle \left\{ p_i, p_j \right\}$ = 0 (4.18)
$\displaystyle \left\{ q_i, p_j \right\}$ = $\displaystyle \delta_{ij}$ (4.19)
0 = $\displaystyle \left\{ F_1, \left\{ F_2, F_3 \right\} \right\} +
\left\{ F_2, \left\{ F_3, F_1 \right\} \right\} +
\left\{ F_3, \left\{ F_1, F_2 \right\} \right\}$ (4.20)
$\displaystyle \frac{\partial}{\partial t}
\left\{ F, G \right\}$ = $\displaystyle \left\{ \frac{\partial F}{\partial t}, G \right\}
+ \left\{ F, \frac{\partial G}{\partial t} \right\}$ (4.21)

Poisson brackets can be used to express time derivatives of phase space functions:
$\displaystyle \frac{\textrm{d} F}{\textrm{d} t}$ = $\displaystyle \sum_{i=1}^n\left(\frac{\partial F}{\partial q_i}
\frac{\textrm{d...
...p_i}
\frac{\textrm{d} p_i}{\textrm{d} t}\right)
+ \frac{\partial F}{\partial t}$  
  = $\displaystyle \sum_{i=1}^n\left(
\frac{\partial F}{\partial q_i}
\frac{\partial...
...al p_i}
\frac{\partial H}{\partial q_i}
\right) + \frac{\partial F}{\partial t}$  
  = $\displaystyle \left\{F, H\right\} + \frac{\partial F}{\partial t}$ (4.22)

This equation can then be applied to qi and pi itself to re-express the Hamilton equations in the following form:
$\displaystyle \frac{\textrm{d} q_i}{\textrm{d} t}$ = $\displaystyle \left\{ q_i, H \right\}$ (4.23)
$\displaystyle \frac{\textrm{d} p_i}{\textrm{d} t}$ = $\displaystyle \left\{ p_i, H \right\}$ (4.24)

In turn, substituting H in place of F yields:

\begin{displaymath}\frac{\textrm{d} H}{\textrm{d} t}
= \left\{H, H\right\} + \frac{\partial H}{\partial t}
= \frac{\partial H}{\partial t}
\end{displaymath} (4.25)

Expressions such as $\left\{q_i, p_j\right\} = \delta_{ij}$ ought to tug at the heart of everyone acquainted with Quantum Mechanics, where one of the expressions of the Heisenberg Uncertainty Principle is

\begin{displaymath}\left[\hat{q}_i, \hat{p}_j\right] = i\hbar\delta_{ij},
\end{displaymath}

where

\begin{displaymath}\left[\hat{q}_i, \hat{p}_j\right] =
\hat{q}_i \hat{p}_j - \hat{p}_j \hat{q}_i
\end{displaymath}

is a commutator of operators that represent position and momentum. Similarly time evolution of any Quantum Mechanical operator that does not depend on time explicitly is given by

\begin{displaymath}\left[\hat{\Psi}, \hat{H}\right]
= i\hbar \frac{\textrm{d} \hat{\Psi}}{\textrm{d} t}
\end{displaymath}

This is not entirely an accident. Poisson brackets lead directly to the so called canonical quantization. Canonical quantization is a procedure which converts a classical field theory or a classical mechanical theory into the corresponding Quantum theory. One of its rules is:

\begin{displaymath}\left\{\Psi, \Phi\right\}
\rightarrow
\frac{1}{i\hbar}
\left[\hat{\Psi}, \hat{\Phi}\right]
\end{displaymath}

But the truth about canonical quantization carried out like that is that it has to be interfered with frequently in order to deliver a meaningful Quantum theory, and the reason for that is that Quantum theories cannot be derived formally from classical theories. The opposite is the case, i.e., Quantum theories are a lot richer than classical theories, and it is the latter that are derivable from the former in thermodynamic limit. But canonical quantization was useful in its day in providing a bridge between XIXth century classical physics and XXth century quantum physics.


next up previous index
Next: Hamilton-Jacobi Equation Up: Kinematics and Dynamics of Previous: Lagrange and Hamilton Equations
Zdzislaw Meglicki
2001-02-26