# Poisson Brackets

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## Contents

## Introduction

Let's imagine that we have a quantity *f* that is a function of the coordinates, momentaand possibly time as well. Let's take its total time derivative to get

There is lots of different notation for the Poisson brackets. To wit, it is often written with braces { } rather than brackets [ ], probably to avoid confusion with the quantum mechanical commutator to which it is intimately related. Since we call them Poisson brackets it seems natural to use brackets rather than braces.

If the function *f* is an integral of the motion we have
In particular if *f* doesn't explicitly depend on time, we find that that

## Definition of the Poisson Brackets

We can define the Poisson brackets for any two quantities

## Properties of the Poisson Brackets

The brackets have the following properties

We can also look at the time derivative of the brackets

If one of the functions *f* and *g* is a momentum or a coordinate we have

Furthermore we can look at the Poisson brackets among the momenta and coordinates themselves

We also have the important relation

known as *Jacobi's identity*.

## Integrals of the Motion

Let's say the functions *f* and *g* are integrals of the motion, then we can prove that [*f*, *g*] is also an integral of the motion.

so

where I have used Jacobi's identity to replace the final term in the first equation. Rearranging yields

The right-hand side vanishes because *f* and *g* are integrals of the motion. This result is known as Poisson's theorem.

### An Example with Angular Momentum

The angular momentum about the *x*-axis and *y*-axis are given by

Let's calculate

where we only used the properties of the brackets themselves and the results for the brackets among the momenta and coordinates.

## Quantum Mechanics

The Poisson bracket defined here is connected to the commutator of quantum mechnanics. Specifically, the commutator is

Classically the commutator would vanish because real numbers commute. Quantum mechanically think of *A* and *B* as "operators" that act on vectors or functions (you can imagine these as matrices that don't necessarily commute). We take

and scale all of the results from the algebra of the Poisson brackets. For example

One way to make sense of these relations is to think of the operators acting on a particular function of position and time (the wave function), so we have

which tells us that the momentum operator is

Classically the Hamiltonian of a system often looks like

so we have in quantum mechanics

Classically, the Hamiltonian tells quantities how to evolve forward in time. It is similar in quantum mechanics
we have

so if the Hamiltonian commutes with a particular quantity and that quantity is not an explicit function of time, the quantity is conserved by the motion even in quantum mechanics.

Again let's assume that these operators act on a function ψ and that *f* is an integral of the motion

As with the momentum we have that

so

This result is known as the Schrödinger equation.