Poisson Brackets – A Tutorial

The Poisson bracket, named for Siméon-Denis Poisson [simeõ
dəni
pwasõ] (1781 Jun 21 – 1840 Apr 25), a French mathematician, geometer, and
physicist, gives us a differential spatio-temporal operator that has no more
physical meaning than does the curl operator, which it faintly resembles. It
merely gives us more a convenient notation for a complex mathematical operation
that we carry out in phase space, where we use generalized coordinates q_{i}
and p_{i}. If we have two functions f(q_{i},p_{i},t) and
g(q_{i},p_{i},t), then we define the Poisson bracket as

(Eq’n 1)

Given an additional function φ(q_{i},p_{i},t)
and a constant k, we can lay out the basic algebraic properties of Poisson
brackets:

(Eq’ns 2)

Poisson brackets appear most notably in the Hamiltonian
version of dynamics. If we can determine the Hamiltonian function, H=T(p_{i})+U(q_{i}),
of a system of bodies and/or particles and then determine some other function f(**q**,**p**,t)
pertaining to that system, then we can calculate the rate at which that function
changes with the elapse of time as

(Eq’n 3)

Because the system obeys the laws of physics, it must conform to the Hamilton-Jacobi equations of motion,

(Eq’ns 4)

Substituting from those equalities into Equation 3 gives us

(Eq’n 5)

A dynamical system may have fixed integrals, which
correspond to constants of the motion, associated with it. The system’s total
energy or total linear momentum offer obvious examples. Equation 5 tells us that
the function f(**q**,**p**,t) represents a constant of the motion if the
function commutes with the Hamiltonian within the Poisson bracket. We have

(Eq’n 6)

This equation is known as the Liouville equation. The content of Liouville's theorem is that the time evolution of a measure (or "distribution function" on the phase space) is given by the above. If the function represents a constant of the motion, then certainly it cannot have an explicit dependence upon elapsed time, so we have ∂f/∂t=0 necessarily. Thus we must have {f,H}=0. But f does not represent a numerical constant, which necessitates that, in accordance with the second of Equations 2, {H,f}=-{f,H}, which can only stand true to mathematics if the Poisson bracket inherently equals zero. And that means that {f,H}={H,f}.

Distinct from Newtonian dynamics, Hamiltonian dynamics gives us the extra value of providing the mathematical means for working out and representing the quantum theory. In Werner Heisenberg’s matrix mechanics version of the quantum theory we have a postulate that states,

"The operators of quantum mechanics are such that their commutators are proportional to the corresponding classical Poisson brackets according to the prescription

where {q,r} is the classical Poisson bracket for the observables q and r."

Using that formulation we thus express Bohr’s correspondence principle as

(Eq’n 7)

In order to work out the algebraic properties of the commutators we represent the appropriate measurable variables with the corresponding operators and apply the commutator to some arbitrary function f(x) of the coordinates. For distance and linear momentum, for example, we get

(Eq’n 8)

Because we apply the operators to a purely arbitrary function, we can leave that function out of the equation, leaving it implicit, thereby giving us an operator-only equation,

(Eq’n 9)

We can apply that form of the equation directly in many cases to get, for example,

(Eq’n 10)

By applying that technique more generally we can work out the fundamental algebraic properties of the commutator brackets. If we have operators A, B and C and a numerical constant k, then we have the algebraic properties of the commutators as

(Eq’ns 11)

A comparison between those equations and Equations 2 tells us that commutator brackets obey the same algebraic rules that the Poisson brackets obey.

In addition, the quantum theory gives us an equation
analogous to Equation 5. We have a matrix M_{H} that describes the state
of a quantum system in the Heisenberg version, so we have the equivalent
equations of motion as

(Eq’n 12)

In that version the state functions themselves do not change with the elapse of time, but the dynamical variables change in accordance with Equation 12. In contrast, in Schrödinger’s wave mechanics version of the quantum theory the state functions change with the elapse of time and, unless they have an explicit dependence upon the elapse of time, the dynamical variables remain constant.

Finally, note that Equation 9 gives us an expression that looks very much like Heisenberg’s indeterminacy principle,

(Eq’n 13)

If we interpret an operator acting on a state function as the mathematical
equivalent of measuring the dynamical variable that the operator represents,
then we must interpret the commutator as representing the difference between the
products of two measurements taken in different order when applied to the same
state function. If we assume that one pair of measurements differs from the
other by the indeterminate amounts
Δx and
Δp_{x},
then the commutator gives us xΔp_{x}+p_{x}Δx+ΔxΔp_{x}.
Locating the measured system in a reference frame in which x=0 and p_{x}=0
gives us the product of the indeterminacies as a hard minimum, reflected in the
equality of Equation 13 (the inequality reflects the fact that the
indeterminacies can have larger values that make their product larger than one
Planck unit). We know that action exists as a relativistic invariant, so the
action formulation of Equation 13 remains valid in all inertial frames.

Thus we see how an entity devised for notational convenience in Hamiltonian dynamics yielded a quantum mechanical analogue that gives us a small piece of new knowledge.

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