Atomic Structure

Suppose that we repeat our diffraction experiment but only allow one photon at a time to approach the slitted surface. The applet below illustrates the results of this experiment, displaying the changing intensity as more and more photons hit the screen:

You need a Java-capable browser to be able to use the applets. If they do not work with your Windows system, download the Java VM (Virtual Machine) for your version of Windows at the download section at java.sun.com.

As you can see, the same interference pattern which was created by a beam of light is produced over time by the sequence of photons. Yet the existence of the interference pattern was explained by Huygens' Principle as due to each slit acting as a coherent source. This implies that each photon passed through both slits, and the resulting phase difference at any point on the screen correlates with the number of photons which eventually hit there: no photons hit where the interference is destructive, and many photons hit where the interference is constructive. Hence we conclude that traveling quanta of the electromagnetic field behave as traveling waves do.

In what is perhaps the most amazing experiment in modern physics, we find that electrons and other elementary particles do exactly the same thing! We will therefore think in terms of an electron field, made up of tiny oscillators at every point in space, just as with the electromagnetic field. This implies that a traveling electron has an associated wavelength, which we call the de Broglie wavelength:

l = h / m v.

This relationship gives us our first clue to the length scale by which particles are measured. Setting the scale using the speed of light for the velocity we find the Compton wavelength of an electron to be about 2.42 * 10^-12 m, while for a proton it is 1.32 * 10^-15 m. These represent minimum sizes, since velocities are always less than the speed of light.

Bound Electrons

If traveling particles share the wave characteristics of field quanta, what does this tell us about electrons bound to nuclei in an atom? Here, the appropriate model is the standing wave: it has no well-defined position or momentum, but it does have a well-defined wavelength and frequency. Consider the Hydrogen atom. Since it exists in three dimensions, we envision the electron as a sort of spherical standing wave surrounding the nucleus. Because the wave must be continuous as we travel around the sphere, we should think of it is having the same boundary conditions at both "ends". This implies that the circumference of the sphere must be equal to an integer times the wavelength:

2 p r = n h / m v.

Rearranging this equation so that both sides have units of angular momentum, we obtain

m v r = n h / 2 p:

the angular momentum of the electron relative to the nucleus is quantized. This is known as the Bohr quantization condition.

It turns out that an electron bound to an atom has four distinct quantum numbers:

n, a positive integer called the energy level;

l, which can take values from 0 to n - 1, and is called the angular momentum quantum number;

m, which can take any value from -l to l, and is called the magnetic quantum number, and

m_s, the spin quantum number.

The images below are plots of the amplitude of the electron wave function for the Hydrogen atom. Because the atom is three dimensional, we have shown the intersection of the electron with the x-z plane. Each plot is labeled by the quantum numbers n, l and m; the spin quantum number does not affect the appearance of the wave function.

The first three are spherically symmetric because l is zero:

Note the nodes (black rings) in the amplitude for n = 2 and n = 3. Just as in the case of the one dimensional standing wave with fixed boundary conditions at both ends, these waves have n - 1 nodes. These electrons are called 1s, 2s and 3s, respectively, by chemists.

For nonzero l, the wave functions are no longer spherically symmetric, but pick up lobes:

These are the 1p, 2p and 3p electrons. Larger values of l produce more, narrower lobes:

These are the 3d, 4d and 4f electrons. The magnetic quantum m labels the orientation of the wave functions:

These are all 2p electrons.

The following applet will show you a full three dimensional plot of the wave function for one or two electrons in a Hydrogen atom (for a single electron, set "n2" to zero). The first three scroll bars allow you to choose a point of view: "r scale" allows you to move farther from or closer to the atom; "theta" is the polar angle (theta=0 is above the positive z axis, theta=p/2 is on the x-y plane, and theta=p is below the -z axis); and "phi" is the azimuthal angle (rotating around the z axis from 0 to 2p). Push the "replot" button to redraw the plot. The other scroll bars allow you to change n, l and m for each electron. Electron 1's wave function is plotted in magenta, and electron 2's wave function is plotted in cyan. The two colors add to blue or white where the wave functions overlap.

Please be patient: each plot requires the evaluation of each wave function at over 3.5 million points! The "Replot" button will be labeled with ellipsis while the evaluation process is taking place.

The scales appropriate to bound electrons are determined by the nature of the Coulomb attraction as well as by their mass. The length scale is set by the Bohr radius:

a₀ = 5.29 * 10^-11 m,

which is about half an Angstrom. This value can only depend on the electrical attraction between the electron and the nucleus, the mass of the electron, and Planck's Constant, and is derived from dimensional considerations (see the problem below).

The average radius of a spherical electron wave function is

n² a₀ / Z,

where Z is the atomic number: the number of protons in the nucleus. The energy scale is set by the Rydberg energy:

E₀ = 2.18 * 10^-18 J.

This value can be derived from the sum of the kinetic energy (through the Bohr quantization condition) and the electrical potential energy of the electron.

Physicists often express atomic energies in a more convenient unit called the electron Volt, or eV:

1 eV = 1.6 * 10^-19 J,

so that

E₀ = 13.6 eV.

The energy of an electron is approximately

E_n = - E₀ Z² / n²,

which neglects the effects of angular momentum and interactions between electrons. It essentially corresponds to an atom with an arbitrary number of protons but only one electron. The energies of electrons at various levels are negative, because they are bound. This reflects our choice of zero potential energy as corresponding to a free electron with zero velocity. As n becomes very large, the difference between two succeeding energy levels becomes vanishingly small:

In the limit of n approaching infinity, the energy spectrum (set of levels) becomes continuous. This is called the classical limit, since classical energies are not quantized but can be any real number. A basic tenet of the quantum theory is that as n approaches infinity, the equations (ie., for energy) should become equivalent to their classical (macroscopic) counterparts. This idea is called the correspondence principle.

Is there a smallest length or a shortest time? In fact, one can construct length, mass, energy and time scales using only three fundamental constants: Newton's Constant, Planck's Constant and the speed of light. These are called the Planck length, mass, energy and time:

l_Planck = 4.05 * 10^-35 m,

m_Planck = 5.46 * 10^-8 kg,

E_Planck = 4.91 * 10⁹ J and

t_Planck = 1.35 * 10^-43 s.

The Planck mass and energy are not minimums; rather, they are the mass and energy of a black hole whose size is of the order of the Planck length. A black hole is an object which has collapsed to such a massive density that even light is not fast enough to escape its gravitational attraction. Because these values involve Newton's Constant, they are relevant to Quantum Gravity: the still sought-after theory which unites quantum theory with General Relativity.

Now you have an opportunity to construct expressions for the various scales we have discussed in this section. Of particular use in some of these problems are the ratios G/c² and e²/e. Work out their units to figure out why they are so useful!

Notice the relationships between the Planck mass and energy, and between the Planck energy and time. These are useful in constructing energy and time scales for any particle.

So can you never get out of a black hole no matter what you do? To answer this question, we need to understand two new ideas: light cones and event horizons.

Every particle, at each point in its trajectory, has a future light cone and a past light cone:

The future light cone is the collection of all the places that particle might ever be at any given time in its future. The surface of the cone is where it could be if it travels at the speed of light, and the interior of the cone is where it could be if it travels more slowly. Therefore photons are destined to stay on the surface of their light cones and massive particles are destined to stay in the interior of their light cones.

The past light cone is the collection of all the places that could have influenced that particle at any given time in the past.

So your future light cone contains all of your possible future paths, and your past light cone contains all that you could know from the past.

A black hole is created by an object which has undergone gravitational collapse to an infinitesimal size (which means its density approaches infinity!). The laws of General Relativity cannot predict the final state of the collapsed object; it simply does not produce meaningful answers at those length scales. But it does predict the existence of a boundary around the collapsed object, the event horizon, which is an imaginary surface that defines the black hole.

As anything (even light) approaches the event horizon, the gravitational attraction toward the collapsed object becomes so great that its future light cone is tipped further and further toward the object. As it crosses the event horizon, one direction on its future light cone is parallel to the horizon, and the rest is pointing into the black hole. So their is NO escape once you cross the event horizon!

Let's assume for a moment that you are falling into a black hole feet first. What you experience until you cross the horizon depends on how much mass the collapsed object has. Larger objects have horizons which are so far away from the object that the gravitational force is essentially constant over the length of your body. Smaller objects have horizons which are much nearer, so that the gravitational pull on your feet is much larger than the pull on your head, resulting in significant (and perhaps fatal!) stretching.

But even if you survive to reach the horizon, regardless of the mass of the collapsed object, it's all over. The future light cone of your feet is pointing into the center of the black hole, so no nerve impulses can reach your head from there. When your heart crosses, no blood can flow from it to your head. And when your head crosses the horizon, the parts of your brain cannot communicate with each other and your consciousness ceases.

You may have heard that black holes can evaporate. While this phenomenon has not been experimentally observed, quantum mechanics tells us that black holes can emit Hawking Radiation (named after physicist Stephen Hawking, who first came up with this idea). Hawking Radiation is a thermal emission which causes the collapsed object to lose mass. Eventually, the object and everything it has swallowed disappears. But that doesn't help you if you've crossed the horizon: all of your mass has been converted into energy and radiated off, but that thermal radiation contains none of the structure that defined you.

There is another more important consequence of the one-way nature of the event horizon: since nothing (except perhaps Hawking Radiation) can leave the interior of the black hole, no information about it can ever be communicated to the outside. The interiors of black holes are not subject to experiment! Therefore, in a very real sense, the interiors of black hole event horizons, including the ultimate fates of the collapsed object and anything else which falls inside, lie outside the realm of science.

In the next section we see how the transitions of electrons between energy levels is the basis for all that we see.

Table of Contents

Arithmetic without a Calculator

Symbols and Abbreviations

Index:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

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