Quantum Theory, the Uncertainty Principle, and Pauli's Exclusion Principle

In the early 1900's, physics enjoyed a series of major discoveries that forever changed our view of electromagnetic radiation, the atom, and science in general. It is important that all of the work described in these pages--discoveries of subatomic particles, explaining spectrum lines, and the following research--was occurring at the about same time, so it took several years for science to integrate all of the new knowledge into one consistent theory.

Quantum mechanics had some surprising implications for all of science. When we observe something--an atom, for example--we shoot electromagnetic radiation at it. The radiation then reflects off the atom and returns to our detecting device (our eyes, a microscope, or other apparatus). However, when the photons hit the particle being studied, it disturbs the position of the particle. The more accurately we try to observe the particle, using higher-energy radiation, the more we disturb the particle's position. Using this type of example as a guide, Werner Heisenberg formulated a law he called the uncertainty principle, which states that the uncertainty in the position of particle multiplied by the uncertainty in its velocity can never be smaller than a constant (specifically, Planck's constant over 4 pi). Therefore, the more precisely we know a particle's position, the more imprecisely its velocity is. The uncertainty principle seems to be a fundamental characteristic of our universe, and numerous experiments have confirmed its effects.

The uncertainty principle impacts all of science. Instead of precisely-defined mathematical values for position, velocity, and other important variables, quantum mechanics expresses values in terms of probability: for example, instead of saying an atom is at a given point and is moving at a certain speed, quantum mechanics forces us to say that the atom has a 90% probability of being within a certain region and a 75% probability of being in another region at a given time.

However, in a strange reversal, quantum mechanics has also been applied to traditional particles like electrons. All types of waves, whether water waves EM waves, interact with each other. For instance, if the crest of one wave coincides with the trough of another, the waves cancel out. If the crest of one wave matches the crest of another, the waves are "added," or amplified. An experiment with a light source confirmed that EM radiation can be thought of as a wave. A screen with two tiny slits was mounted in front of a light source. Another screen was mounted behind the first, so that light passing through the slits would show up and be seen. When the light waves passed through the slits, they interacted (or interfered) with each other, producing a series of dark and light bars as the waves sometimes cancel and sometimes amplify each other. When the experiment was repeated using electrons instead of light waves, the interference pattern remained! Even when one electron at a time was sent through the screen, the pattern still appeared: the electron must have passed through both slits and interfered with itself! The electron was behaving as a wave instead of a particle! Because particles such as electrons can now be though of as waves, the uncertainty principle now seems more reasonable, because the "position" of a wave cannot be determined precisely. This strange particle/wave duality is a hallmark of quantum mechanics.

In 1925, Louis Victor de Broglie proposed just this theory: that the electron could be considered as a wave as well as a particle. Using Einstein's equation, E = mc2 and the E = c/λ, de Broglie formulated an equation relating the speed and mass of a particle, usually an electron, to its wavelength:

λ = h/mv;

where m is the mass of the particle (an electron weighs 9.109 x 10-31 kg) and v is its velocity in meters per second. At normal speeds, the wavelengths of particles are so incredibly tiny that we don't notice them. However, at very high speeds such as those found in moving electrons, the wavelength becomes important. This discovery was key to the forthcoming revolution in atomic science.

The final great discovery in physics we will discuss in this page was Pauli's exclusion principle. It is obvious that no two atoms can have the same position; matter cannot overlap. However, under the uncertainty principle, position and velocity are both uncertain. Therefore, Wolfgang Pauli stated that two particles cannot have both the same position and velocity, within the limits dictated exclusion principle. Therefore, two given particles must stay a certain distance apart, in order to obey the exclusion principle. As with the other breakthroughs discussed above, a multitude of experimental data confirms Pauli's hypothesis.

Using these new theories, a new model of the atom and its electrons was invented by Erwin Schrödinger called the wave-mechanical theory. The main feature of the theory was the integration of quantum mechanics, the wave view of an electron, and the exclusion principle. The resulting theory requires complex equations, called wave equations, that predict a region of high probability in which an electron can be found. Wave equations are represented by the symbol ψ , and because the equations are so complex, we will not present the mathematical details. However, Schrödinger's theory makes the following points:

• The fixed orbit levels of an electron are regions where the electron waves amplify each other instead of canceling; in the particle view, they are the only permitted energy levels.
• Each wave function defines one allowed energy level.
• As stated above, wave equations result in probabilities, not exact values, due to their integration of the uncertainty principle.
• Four variables are necessary to solve the wave equation: n, l, ml, and ms. These values are called quantum numbers, and together they define one energy state; only certain values for each number are allowed. The first three specify the orbital's location in space, while the third specifies which electron the equation describes.

The next page will explore the allowed energy states, called orbitals, that result from particular sets of quantum numbers.

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