The Fall of the Trajectory to uncertainty principle

The Fall of the Trajectory to uncertainty principle

So much for the idea of using the classical model of a particle to describe a microscopic entity. But can we salvage the basic state descriptor of classical physics, the trajectory? The first step in determining the trajectory of a particle is measuring its initial conditions, x (t0) and Px(t0). To determine the accuracy of our results, we would perform such a measurement not on just one particle, but on a large number of identical particles, all in the same state. Each individual measurement yields a value for x and a value for Px (subject to experimental uncertainties). But the results of different measurements are not the same, even though the systems are identical. If graphed, these results are seen to fluctuate about a central peak, as illustrated in Fig. 1.2. (A similar spread of results characterizes measurement of the y- or z- components of position and momentum.) At first, the fact that the results of many identical experiments are not the same doesn't worry us; we think it's just a manifestation of the experimental error that bedevils

all measurements. According to classical physics, we can reduce the errors in x and Px to zero and thereby determine precisely the initial conditions. But we cannot unambiguously specify the values of these observables for a microscopic particle. This peculiarity of nature in the microscopic realm is mirrored in the Heisenberg Uncertainty Principle (HUP). In its simplest form, the HUP shows that any attempt to simultaneously measure x(to) and Px(to) necessarily introduces an imprecision in each observable. No matter how the experiment is designed, the results are inevitably uncertain, and the uncertainties. , which are measures of fluctuations like those in Fig. 1.2, cannot be reduced to zero. Instead, their product must satisfy the condition.

where h is Planck's Constant

Not a very big number, but not zero either. [Similar constraints apply to the pairs of uncertainties. .] Position and momentum are fundamentally incompatible observables, in the sense that knowing the precise value of one precludes knowing anything about the other. But there is a deeper dimension to the Heisenberg Uncertainty Principle. Quantum theory reveals that the limitation reflected by Eq. (1.4) on our ability to simultaneously measure x and Px is implicit in nature. It has nothing to do with a particular apparatus or with experimental technique. Quantum mechanics proves that a particle cannot simultaneously have a precise value of x and a precise value of Px.

Similar uncertainty principles constrain our ability to measure other pairs of incompatible observables. But uncertainty relations such as (1.4) are not veiled criticisms of experimental physicists. They are fundamental limitations on knowledge: the universe is inherently uncertain. Man cannot know all of existence. We might think of uncertainty relations as nature's way of restraining our ambitions.

The Heisenberg Uncertainty Principle strikes at the very heart of classical physics: the trajectory. Obviously, if we cannot know the position and momentum of a particle at t0 we cannot specify the initial conditions of the particle and hence cannot calculate its trajectory. Indeed, since the HUP applies at any time, it makes no sense to ascribe a trajectory to a microscopic particle. We are forced, however reluctantly, to conclude that microscopic particles do not have trajectories. But once we throw out trajectories, we must also jettison Newton's Laws. And there goes the ball game: stripped of its basic elements and fundamental laws, the whole structure of classical physics collapses.

The demise of classical physics occurred around 1930. Sitting amidst the rubble, physicists of that era realized that their only alternative (other than to change careers) was to rebuild-to construct from scratch a new physical theory, one based on elements other than trajectories and on laws other than those of Newton and Maxwell. Thus, began the quantum revolution, whose reverberations are still being felt.

The source:

Michael A. Morrison - Understanding Quantum Physics.

By. Fady Tarek