Poisson brackets

Consider a dynamic system whose state at a particular time $t$ is fully specified by $N$ independent classical coordinates $q_i$ (where $i$ runs from 1 to $N$). Associated with each generalized coordinate $q_i$ is a classical canonical momentum $p_i$. For instance, a Cartesian coordinate has an associated linear momentum, an angular coordinate has an associated angular momentum, etc. As is well-known, the behaviour of a classical system can be specified in terms of Lagrangian or Hamiltonian dynamics. For instance, in Hamiltonian dynamics,

$$\frac{d q_i}{d t} = \frac{\partial H}{\partial p_i},$$ (91)

$$\frac{d p_i}{dt} = - \frac{\partial H}{\partial q_i},$$ (92)

where the function $H(q_i, p_i, t)$ is the energy of the system at time $t$ expressed in terms of the classical coordinates and canonical momenta. This function is usually referred to as the Hamiltonian of the system.

We are interested in finding some construct of classical dynamics which consists of products of dynamical variables. If such a construct exists we hope to generalize it somehow to obtain a rule describing how dynamical variables commute with one another in quantum mechanics. There is, indeed, one well-known construct in classical dynamics which involves products of dynamical variables. The Poisson bracket of two dynamical variables $u$ and $v$ is defined

$$[u, v]= \sum_{i=1}^N \left(\frac{\partial u}{\partial q_i}\fr... ...rtial u}{\partial p_i}\frac{\partial v}{\partial q_i} \right),$$ (93)

where $u$ and $v$ are regarded as functions of the coordinates and momenta $q_i$ and $p_i$. It is easily demonstrated that

$$[q_i, q_j] = 0,$$ (94)

$$[p_i, p_j] = 0,$$ (95)

$$[q_i, p_j] = \delta_{ij}.$$ (96)

The time evolution of a dynamical variable can also be written in terms of a Poisson bracket by noting that

$$\frac{du}{dt} = \sum_{i=1}^N \left(\frac{\partial u}{\partial q_i}\frac{d q_i}{dt} + \frac{\partial u}{\partial p_i}\frac{dp_i}{dt}\right)$$

$$= \sum_{i=1}^N \left(\frac{\partial u}{\partial q_i}\frac{\partial ... ...ial p_i} -\frac{\partial u}{\partial p_i}\frac{\partial H}{\partial q_i}\right)$$

$$= [u, H],$$ (97)

where use has been made of Hamilton's equations.

Can we construct a quantum mechanical Poisson bracket in which $u$ and $v$ are noncommuting operators, instead of functions? Well, the main properties of the classical Poisson bracket are as follows:

$$[u, v] = - [v, u],$$ (98)

$$[u, c] = 0,$$ (99)

$$[u_1+ u_2, v] = [u_1, v] + [u_2, v],$$ (100)

$$[u, v_1 + v_2] = [u, v_1] + [u, v_2]$$ (101)

$$[u_1 u_2, v] = [u_1, v] u_2 + u_1 [u_2, v],$$ (102)

$$[u, v_1 v_2] = [u, v_1] v_2 + v_1 [u, v_2],$$ (103)

and

$$[u, [v, w]]+ [v, [w, u] ] + [w, [u, v]] = 0.$$ (104)

The last relation is known as the Jacobi identity. In the above, $u$, $v$, $w$, etc., represent dynamical variables, and $c$ represents a number. Can we find some combination of noncommuting operators $u$ and $v$, etc., which satisfies all of the above relations?

Well, we can evaluate the Poisson bracket $[u_1 u_2, v_1 v_2]$ in two different ways, since we can use either of the formulae (102) or (103) first. Thus,

$$[u_1 u_2, v_1 v_2] = [u_1, v_1 v_2]u_2 + u_1[u_2, v_1 v_2]$$ (105)

$$= \left\{ [u_1, v_1]v_2 + v_1[u_1, v_2]\right\} u_2 +u_1\left\{[u_2, v_1]v_2 + v_1[u_2, v_2]\right\}$$

$$= [u_1, v_1] v_2 u_2 + v_1[u_1, v_2] u_2 + u_1[u_2, v_1]v_2 + u_1 v_1[u_2, v_2],$$

and

$$[u_1 u_2, v_1 v_2] = [u_1 u_2, v_1 ]v_2 + v_1[u_1 u_2, v_2]$$ (106)

$$= [u_1, v_1] u_2 v_2 + u_1[u_2, v_1] v_2 + v_1[u_1, v_2]u_2 + v_1 u_1[u_2, v_2].$$

Note that the order of the various factors has been preserved, since they now represent noncommuting operators. Equating the above two results yields

$$[u_1, v_1](u_2 v_2 - v_2 u_2) = (u_1 v_1-v_1 u_1)[u_2, v_2].$$ (107)

Since this relation must hold for $u_1$ and $v_1$ quite independent of $u_2$ and $v_2$, it follows that

$$u_1 v_1 - v_1 u_1 = {\rm i} \hbar [u_1, v_1],$$ (108)

$$u_2 v_2 - v_2 u_2 = {\rm i} \hbar [u_2, v_2],$$ (109)

where $\hbar$ does not depend on $u_1$, $v_1$, $u_2$, $v_2$, and also commutes with $(u_1 v_1- v_1 u_1)$. Since $u_1$, etc., are quite general operators, it follows that $\hbar$ is just a number. We want the quantum mechanical Poisson bracket of two Hermitian operators to be an Hermitian operator itself, since the classical Poisson bracket of two real dynamical variables is real. This requirement is satisfied if $\hbar$ is a real number. Thus, the quantum mechanical Poisson bracket of two dynamical variables $u$ and $v$ is given by

$$[u, v]= \frac{u v - v u}{{\rm i} \hbar},$$ (110)

where $\hbar$ is a new universal constant of nature. Quantum mechanics agrees with experiments provided that $\hbar$ takes the value $h/2\pi$, where

$$h = 6.6261 \times 10^{-34}  {\rm J s}$$ (111)

is Planck's constant. Somewhat confusingly, the notation $[u, v]$ is conventionally reserved for the commutator $u v-v u$ in quantum mechanics. We will use $[u, v]_{\rm quantum}$ to denote the quantum Poisson bracket. Thus,

$$[u, v]_{\rm quantum} = \frac{[u, v]}{{\rm i} \hbar}.$$ (112)

It is easily demonstrated that the quantum mechanical Poisson bracket, as defined above, satisfies all of the relations (98)-(104).

The strong analogy we have found between the classical Poisson bracket, defined in Eq. (93), and the quantum mechanical Poisson bracket, defined in Eq. (112), leads us to make the assumption that the quantum mechanical bracket has the same value as the corresponding classical bracket, at least for the simplest cases. In other words, we are assuming that Eqs. (94)-(96) hold for quantum mechanical as well as classical Poisson brackets. This argument yields the fundamental commutation relations

$$[q_i, q_j] = 0,$$ (113)

$$[p_i, p_j] = 0,$$ (114)

$$[q_i, p_j] = {\rm i} \hbar \delta_{ij}.$$ (115)

These results provide us with the basis for calculating commutation relations between general dynamical variables. For instance, if two dynamical variables, $\xi$ and $\eta$, can both be written as a power series in the $q_i$ and $p_i$, then repeated application of Eqs. (98)-(103) allows $[\xi, \eta]$ to be expressed in terms of the fundamental commutation relations (113)-(115).

Equations (113)-(115) provide the foundation for the analogy between quantum mechanics and classical mechanics. Note that the classical result (that everything commutes) is obtained in the limit $\hbar\rightarrow 0$. Thus, classical mechanics can be regarded as the limiting case of quantum mechanics when $\hbar$ goes to zero. In classical mechanics, each pair of generalized coordinate and its conjugate momentum, $q_i$ and $p_i$, correspond to a different classical degree of freedom of the system. It is clear from Eqs. (113)-(115) that in quantum mechanics the dynamical variables corresponding to different degrees of freedom all commute. It is only those variables corresponding to the same degree of freedom which may fail to commute.