## Introduction

The uncertainty principle is one of the most fundamental characteristics that distinguishes quantum mechanics from classical physics. It is also one of the most difficult concepts in theoretical physics to understand even on an intellectual level, but its mastery will provide you with an intuitive core around which other theoretical concepts will crystallize. The uncertainty principle, discovered by Werner Heisenberg in 1927, originates from the dilemma of wanting to know a particle's position and velocity at the same time.

## The Problem

Return for a moment to the two-slit light experiment presented in The Electromagnetic Spectrum. If you have not read this section, a brief recap will be given here. The experiment consisted of a beam of light shining through two slits in a barrier between it and a photographic plate. When one slit was open, one one bar of light appeared on the plate. When two slits are open, an interference pattern is produced. However, modern physics holds that light is composed of particles, called photons. How can photons interfere with each other? Even when the rate of photons passing through the slits is reduced so drastically that only one photon is traveling toward the plate at a time, the pattern still appears.

Through a series of experiments, physicists determined that all particles exhibit both wavelike and particlelike tendencies - the principle of wave-particle duality. Further advances in theory led to the idea that a particle's wave function, or its probability of being found in a certain place, determined its location after it passed through the slit. Therefore, places where light is observed on the plate are places where the photons were most likely to be found. In an alternative (but equally accurate) description, called Feynman's sum-over-paths method, particles travel not one but every possible path from one point to another. Numbers assigned to these paths cancel out to the path that a macroscopic object actually takes, but not every path cancels out for particles, thus producing the interference pattern. However, many physicists objected to these ideas, claiming that the photons passing through each slit must somehow be detectable.

## Complications

Many physicists of the time - and probably most readers as well - wondered why the photons' paths, and thus which slit they entered, couldn't simply be measured. So imagine a photon detector placed on the slits. When physicists performed this experiment, they found that the interference pattern disappeared - two slits of light appeared on the plate! The act of measurement had interfered with the particles' trajectories, in effect forcing them to pass through one slit or the other. This is called reducing the wave-function because it takes what was a fundamentally uncertain, indeterminate quantity - which slit the particles passed through - and measured it. However, the act of measurement destroyed the experiment itself, so which slit the particle passed through not only cannot be known, but, according to wave functions or sum-over-paths, is not even a meaningful statement because the particle actually passed through both.

A convenient rephrasing of these findings utilizes a similar experiment using electrons, which behave the same way. If electrons can be detected with light, by bouncing photons off them, we should be able to know which slit an electron went through. However, since the smallest increment of light is one photon, there is a certain minimum disruption of the electron that destroys the experiment. The end summary of all these experiments is the following statement: we can only determine an object's position by bouncing waves, such as light, off it; when we use this method, we can only determine the object's position within a "margin of error" equal to the wavelength of the probe wave used.

Heisenberg analyzed this relationship mathematically, and his results lead to the "standard" way to sum up the uncertainty principle - the more precisely you measure an object's position, the less precisely you can measure its velocity, and vice versa. Therefore, you cannot possibly know both position and velocity with absolute precision. This does not, as was widely believed at first, represent incomplete knowledge - it is a fundamental limit of the universe's structure.

## A Step Further

While the position/velocity relationship has proven important in theoretical physics, there is another manifestation of the uncertainty principle that is even more influential. This is the relationship, also quantified by Heisenberg, between the precision of a measurement of energy and the amount of time it takes to perform the measurement. This implies that energy held by a particle can fluctuate provided that it does so within a brief enough time. The shorter the time of the fluctuation, the more drastic the fluctuation (the greater change in energy), and vice versa. Therefore, quantum mechanics allows a particle to temporarily "borrow" energy provided that it is relinquished within the time determined by Heisenberg's equations.

These results allow a number of events forbidden by classical physics to occur. For example, a classical particle fired against a concrete wall would have no chance of passing through, but a quantum particle has a small but calculable chance of "borrowing" enough energy to penetrate the wall and reemerge on the other side. To phrase this result in the language of particle motion instead of energy, it can be said that the particle's wave function included a small probability that it would end up on the other side of the wall. In one particular case, that small chance became reality and the particle ended in that position described by the wave function.

## Quantum Foam

It should be noted that Planck's constant, denoted , determines the ability of particles to "borrow" energy. Planck's constant, the proportionality factor between the frequency of a wave and its minimum energy, is extremely small - specifically, 1.05 × 10-27 g-cm2/sec. This tiny value causes "quantum weirdnesses" such as the ability to pass through concrete walls to be very limited in scale. These occurrences are generally accepted not to happen on a macroscopic scale, although the probabilities are calculable, because of their extreme unlikeliness. However, such events can and do happen on a microscopic scale; specifically, when lengths smaller than the Planck length (about 10-33 cm) are examined, a "quantum foam" is observed. This foam consists of numerous pairs of particles and antiparticles suddenly leaping into existence for an extremely short period of time and using "borrowed" energy, then annihilating each other through matter/antimatter interactions. This froth of particles does not normally affect anything, but its action has prevented physicists from developing a theory of gravity that is fully integrated into the quantum framework. Nevertheless, the development of string theory may solve not only the gravity problem but also permanently eliminate the "quantum foam" from calculations.

Created by Dan Corbett, Kate Stafford, and Patrick Wright for ThinkQuest.