Master of Science in Chemistry: Poisson bracket

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 q_i and p_i 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

Master of Science in Chemistry

Pages

Wednesday, March 30, 2011

Poisson bracket

Poisson Brackets

No comments:

Post a Comment