Particles and Their Trajectories in Classical Physics

Particles and Their Trajectories in Classical Physics

Physicists, classical or otherwise, study the universe by studying isolated fragments of it. These little chunks of existence are called systems: classical physics pertains to macroscopic systems, quantum physics to microscopic systems. A system, then is just a collection of particles that interact among themselves via internal forces and that may interact with the world outside via external fields.

To a classical physicist, a particle is an indivisible mass point possessing a variety of physical properties that can be measured. In physical theory, measurable quantities are called observables. By listing the values at any time of the observables of a particle, we can specify its state. (The trajectory is an equivalent, more convenient way to specify a particle 's state.) The state of the system is just the collection of the states of the particles comprising it.

Extrinsic versus Intrinsic Properties.

Classical physicists often characterize properties of a particle as intrinsic or extrinsic. Intrinsic properties don't depend on the particle's location, don't evolve with time, and aren't influenced by its physical environment; rest mass and charge are intrinsic properties. Extrinsic properties, on the other hand, evolve with time in response to the forces on the particle; position and momentum are extrinsic properties.

According to classical physics, all properties, intrinsic and extrinsic, of a particle could be known to infinite precision. For example, we could measure the precise values of the position and momentum of a particle at the same time. Of course, precise knowledge of everything is a chimera-in the real world, neither measuring apparatus nor experimental technique is perfect, and experimental errors bedevil physicists as they do all scientists. But in principle both are perfectible: that is, our knowledge of the physical universe is limited only by ourselves, not by nature.

The trajectory.

How does a classical theorist predict the outcome of a measurement? He uses trajectories. The trajectory of a single particle consists of the values of its position and momentum at all times after some (arbitrary) initial time t0:

where the (linear) momentum is, by definition,

with m the mass of the particle.

Trajectories are the "state descriptors" of Newtonian physics. To study the evolution of the state represented by the trajectory (1.1), we use Newton's Second Law,

where V(r, t) is the potential energy of the particle." To obtain the trajectory for t > to, we need only know V (r,t ) and the initial conditions, the values of rand p at the initial time to- With the trajectory in hand, we can study various properties of the particle in the state that the trajectory describes, e.g., its energy or orbital angular momentum.

Notice that classical physics tacitly assumes that we can measure the initial conditions without altering the motion of the particle-e.g., that in measuring the momentum of the particle, we transfer to it only a negligible amount of momentum. Thus, the scheme of classical physics-which is illustrated in Fig. 1.1 is based on precise specification of the position and momentum of a particle.

I do not mean to imply by this cursory description that classical physics is easy. Solving the equations of motion for a system with lots of particles interacting among themselves or exposed to complicated external fields is a formidable task. Contemporary applications of Newtonian mechanics to problems like celestial mechanics (the motion of the planets in our solar system) require massive computers and budgets to match.

At worst, one may be unable, using present technology, to solve these equations. The important point is that in principle, given sufficient time and scratch paper, we could predict from the initial positions and momenta of all particles the precise future behavior of any system-indeed, of the entire universe. That's the classical view.

The source:

Michael A. Morrison - Understanding Quantum Physics.

By. Fady Tarek