Poisson Brackets

Consider a dynamical 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, $p_i$ [55]. For instance, a Cartesian coordinate has an associated linear momentum, an angular coordinate has an associated angular momentum, et cetera [50]. As is well known, the behavior of a classical system can be specified in terms of either Lagrangian or Hamiltonian dynamics [55]. For instance, in Hamiltonian dynamics,

$$\frac{d q_i}{d t} = ~\frac{\partial H}{\partial p_i},$$ (2.1)

$$\frac{d p_i}{dt} = - \frac{\partial H}{\partial q_i},$$ (2.2)

where the function $H(q_i, p_i, t)$ is the system energy 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 [55].

We are interested in finding some construct in classical dynamics that consists of products of dynamical variables. If such a construct exists then 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 that involves products of dynamical variables. The classical Poisson bracket of two dynamical variables, $u$ and $v$ , is defined [55]

$$[u, v]_{cl} = \sum_{i=1,N} \left(\frac{\partial u}{\partial q_i}\... ... p_i} - \frac{\partial u}{\partial p_i}\frac{\partial v}{\partial q_i} \right),$$ (2.3)

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

$$[q_i, q_j]_{cl} =0,$$ (2.4)

$$[p_i, p_j]_{cl} =0,$$ (2.5)

$$[q_i, p_j]_{cl} = \delta_{ij}.$$ (2.6)

(See Exercise 1.) The time evolution of a dynamical variable can also be written in terms of a Poisson bracket by noting that

$$\frac{du}{dt} = \sum_{i=1,N} \left(\frac{\partial u}{\partial q_i... ...c{\partial u}{\partial p_i}\frac{\partial H}{\partial q_i}\right) =[u, H]_{cl},$$ (2.7)

where use has been made of Hamilton's equations, Equations (2.1)-(2.2).

Let us attempt to construct a quantum mechanical Poisson bracket in which $u$ and $v$ are non-commuting operators instead of functions. Now, the main properties of the classical Poisson bracket are as follows:

$$[u, v]_{cl} = - [v, u]_{cl},$$ (2.8)

$$[u, c]_{cl} =0,$$ (2.9)

$$[u_1+ u_2, v]_{cl} = [u_1, v]_{cl} + [u_2, v]_{cl},$$ (2.10)

$$[u, v_1 + v_2]_{cl} = [u, v_1] _{cl}+ [u, v_2]_{cl},$$ (2.11)

$$[u_1\, u_2, v]_{cl} = [u_1, v]_{cl} \,u_2 + u_1\, [u_2, v]_{cl},$$ (2.12)

$$[u, v_1 \,v_2]_{cl} = [u, v_1]_{cl} \,v_2 + v_1 \,[u, v_2]_{cl},$$ (2.13)

$$[u, [v, w]_{cl} ]_{cl}+ [v, [w, u]_{cl} ]_{cl} + [w, [u, v]_{cl}]_{cl} = 0.$$ (2.14)

(See Exercise 2.) The final relation is known as the Jacobi identity. In the previous expressions, $u$ , $v$ , $w$ , et cetera, represent dynamical variables, and $c$ represents a complex number. We wish to find some combination of non-commuting operators, $u$ and $v$ , that satisfies Equation (2.8)-(2.14). We shall refer to such a combination as a quantum mechanical Poisson bracket.

Actually, we can evaluate the quantum mechanical Poisson bracket, $[u_1 \,u_2, v_1 \,v_2]_{qm}$ , in two different ways, because we can employ either of the formulae (2.12) or (2.13) first. Thus,

$$[u_1\, u_2, v_1\, v_2]_{qm} = [u_1, v_1 \,v_2]_{qm}\,u_2 + u_1\,[u_2, v_1\, v_2]_{qm}$$

$$=\left([u_1, v_1]_{qm}\,v_2 + v_1\,[u_1, v_2]_{qm}\right) u_2 +u_1\left([u_2, v_1]_{qm}\,v_2 + v_1\,[u_2, v_2]_{qm}\right)$$

$$= [u_1, v_1]_{qm}\, v_2 \,u_2 + v_1\,[u_1, v_2]_{qm} \,u_2 + u_1\,[u_2, v_1]_{qm}\,v_2$$

$$\phantom{=}+ u_1 \,v_1\,[u_2, v_2]_{qm},$$ (2.15)

and

$$[u_1\, u_2, v_1\, v_2]_{qm} = [u_1 \,u_2, v_1 ]_{qm}\,v_2 + v_1\,[u_1\, u_2, v_2]_{qm}$$

$$= [u_1, v_1]_{qm} \,u_2 \,v_2 + u_1\,[u_2, v_1]_{qm} \,v_2 + v_1\,[u_1, v_2]_{qm}\,u_2$$

$$\phantom{=}+ v_1\, u_1\,[u_2, v_2]_{qm}.$$ (2.16)

Note that the order of the various factors has been preserved in the previous expressions, because these factors now represent non-commuting operators. Equating the previous two results yields

$$[u_1, v_1]_{qm}\, (u_2 \,v_2 - v_2 \,u_2) = (u_1 \,v_1-v_1\, u_1)\,[u_2, v_2]_{qm}.$$ (2.17)

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

$$u_1 \,v_1 - v_1\, u_1 = {\rm i}\,\hbar \,[u_1, v_1]_{qm},$$ (2.18)

$$u_2\, v_2 - v_2\, u_2 = {\rm i} \,\hbar \,[u_2, v_2]_{qm},$$ (2.19)

where $\hbar$ does not depend on $u_1$ , $v_1$ , $u_2$ , $v_2$ , and also commutes with $(u_1\, v_1- v_1 \,u_1)$ . Because $u_1$ , et cetera, are general operators, it follows that $\hbar$ is just a number. Now, we need the quantum mechanical Poisson bracket of two Hermitian operators to be itself an Hermitian operator, because 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

$$[u, v]_{qm} = \frac{u\,v - v\,u}{{\rm i} \,\hbar},$$ (2.20)

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

$$h = 6.6261 \times 10^{-34}~~{\rm J\,s}$$ (2.21)

is Planck's constant. (The quantity $\hbar$ is usually referred to as the reduced Planck constant.) The notation $[u, v]$ is conventionally reserved for the commutator, $u\,v-v\,u$ , in quantum mechanics. Thus,

$$[u, v]_{qm} = \frac{[u, v]}{{\rm i}\, \hbar}.$$ (2.22)

It is easily demonstrated that the quantum mechanical Poisson bracket, as defined in the previous equation, satisfies all of the relations (2.8)-(2.14). (See Exercise 2.)

The strong analogy we have found between the classical Poisson bracket, defined in Equation (2.3), and the quantum mechanical Poisson bracket, defined in Equation (2.22), leads us to assume 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 going to assume that Equations (2.4)-(2.6) hold for quantum mechanical, as well as classical, Poisson brackets. This argument yields the fundamental commutation relations

$$[q_i, q_j] =0,$$ (2.23)

$$[p_i, p_j] =0,$$ (2.24)

$$[q_i, p_j] = {\rm i} \,\hbar \,\delta_{ij}.$$ (2.25)

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 Equations (2.8)-(2.14) allows $[\xi, \eta]$ to be expressed in terms of the fundamental commutation relations (2.23)-(2.25).

Equations (2.23)-(2.25) provide the foundation for the analogy between quantum mechanics and classical mechanics. Note that the classical result that dynamical variables commute is obtained in the limit $\hbar\rightarrow 0$ . Thus, classical mechanics can be regarded as the limiting case of quantum mechanics as $\hbar$ goes to zero. In classical mechanics, each generalized coordinate and its conjugate momentum, $q_i$ and $p_i$ , respectively, correspond to a different classical degree of freedom of the system. It is clear from Equations (2.23)-(2.25) that, in quantum mechanics, the dynamical variables corresponding to different degrees of freedom commute. In fact, it is only those variables corresponding to the same degree of freedom that may fail to commute.