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THE QUANTUM WORLD

        The story of the quantum world sometimes is bizarre, sometimes wierd, but always is exciting. We find that our normal expectations about how the world behaves don't apply anymore. It is something akin to Alice in Wonderland. We enter a strange place when we (eat the cookie or whatever) shrink down to the atomic size. No longer do Newton's descriptions, or any other familiar ways to describe the world, work. Our models (and let us not forget that is just what they are) must be changed to explain this Wonderland.

        The following discussions attempt to address the nature of matter from the quantum perspective. It is not assummed that the reader has a strong background in physics and necessary details will be presented as needed in the development of the quantum picture.

        One would first ask the question, what does "quantum" mean and why are we interested in it anyway. Quantum is a Latin word meaning "how much" or quantity. It is the purpose of our discussions here to make the idea relevant to our study of the world, and see how the quantum world in fact is important to our very understanding of the universe.

        We saw in the last chapter that we are but in one small place in the universe. Certain aspects of it are many orders of magnitude larger and others are many orders of magnitude smaller. We try to understand the universe as a whole from a very limited perspective in size, and even time in the history of the universe itself. The universe has changed. It is continuously changing and we are examining it during one second of its lifetime. From these perspectives we are really trying to describe it as a whole. We need the parts; we need to put the parts together and then try to describe the whole, and the relationships of the parts to other parts and to the whole itself.

        Looking at the world at this moment of time is like taking a picture at Canterbury Park Racetrack, when the horses are just coming out of the first turn. From this, can you make a prediction of the winner? You might say that you would pick the one in front, but you really don't know the horses' speeds. It may very well be that the second horse is gaining so fast on the leader that it will pass it up in a few seconds. Trying to tell all about our world, how it was a million, or a billion years ago, or twenty billions years ago from looking at it now is extremely difficult. Similarly, it is difficult to describe what will happen; i.e., what the ultimate destiny of the universe will be.
 
 

Basic Forces of Nature

        We start our voyage in understanding the universe by looking first at the four basic forces in nature. Simply stated, we can describe all phenomena observed since the beginning of the known universe through today, and can make predictions about the future, all based on four well defined forces. we do not pretend that there cannot be any others. We have not observed in nature any effects that would require a force different from those we are familiar with to explain. i.e. the four basic forces of nature are adequate to explain every known phenomena. It is possible that other forces, unkown  to us now, may in fact exist. Ephraim Fishbach  of Purdue University contends that he has discovered  a fifth force, "hype rcharge", which tends to counteract  gravity at extraordinarily short distances. Indeed if  there is any truth to the amazing characteristics  attributed to UFOs, there surely must be some other kind of force that can cause such space craft to fly with incredible accelerations and even the presence of them on our planet would attest to the fact that only some very advanced mechanisms could allow the intelligent beings to be "transported" to our   solar system from some distant star system or  even galaxy. It would also mean  that our present theories would have  to be drastically altered (and perhaps even simplified to explain such phenomena.) We look at the nature of the four basic forces that have been discovered. We should keep in mind that the forces are present because matter is present. We do not hold in space free gravity forces or electromagnetic ones. They act as a consequence of the presence of matter. The four basic forces include: Gravitation, Electromagnetic, Weak and Strong Nuclear Forces.

       Gravity is the weakest force we know of in nature. We shall see later how Einstein modified Newton's concept of gravitation. It was Newton who first gave us the understanding of gravitation as a force. Galileo, the father of modern science, died in 1642. Newton was considered God's Christmas gift to mankind since he was born on December 25th of that same year. The life of Newton is filled with interesting facts. That is not our purpose to discuss here, but the reader is encouraged to investigate the background of the individual and his many contributions to science that formed the foundation of scientific perspectives for centuries. From the times of the Greeks and up to Newton, it was felt that natural motion itself was circular. Objects were thought to naturally go in circles, modeling after the motion of the universe. Newton shook the world with an idea that challenged what the state of "natural" motion really was. He said that an object naturally stays at rest, or will continue in motion in a straight line, unless acted upon by an external force (which causes it to follow another path.) He went on to say that an applied force causes a change in the motion of the object and that this acceleration is proportional to the mass of an object. While Newton considered mass to be a fundamental characteristic of an object, describing the amount of matter in it, we can also see that mass also determines the degree of inertia of an object to continue at rest or in motion in a straight line. The more massive an object is, the greater the tendency to resist a change of motion, or the greater the inertia. He further pointed out that any two masses are attracted to each other-(gravitationally) by the relationship:

where G is a constant (Universal Gravitation Constant), m₁ and m₂ are the two masses respectively and r is the distance betwein them. So the force of gravity depends directly on the product of the masses. This means that bigger masses attract more strongly and smaller masses less so. The force also depends inversely on the square of the distance between the masses. This tells us that as the masses move further apart the force of gravity between them decreases dramatically. As the distance is doubled, the force decreases by a factor of four! If the distance is quadrupled, the force decreases sixteen fold.
 

The Wizard of Id

by Parker and Hart
 
 
 
 
 

        This force (gravity) is the weakest of all the forces known to humanity, yet on the large scale, it accounts for the very structure of the entire universe. That is why stars and galaxies have evolved the way they did. Look at a star like our own sun. The theory of stellar evolution tells us that the sun formed from a vast cloud of dust and gas called the solar nebula, probably 100 Astronomical Units (AU, the mean distance from the Earth to the Sun) in diameter. The weak, but steady pull of gravity (in the absence of any other forces to keep them apart) of the particles on one another caused them to condense in roughly a spherical shape. The astronauts demonstrated this vividly during the Skylab missions where even a glass of water took on a spherical shape when the glass was suddenly removed. The gravitational collapse of the sun continued into a dense ball with increasing temperature and pressure until radiation pressure pushing outward due to nuclear energy generation in the interior, resulted in a state of equilibrium.

        The Electromagnetic Force is another one we are somewhat familiar with. While we frequently think of electricity and magnetism as being two different forces, we should recognize that one exists because the other does. Electricity does two things. It makes heat, and it makes magnetism. We do not intend here to explain all about electromagnetism. That is a year long course in itself. We should recognize, however, that Newton's theories of gravitation were inadequate to explain electromagnetic phenomena. Many contributed to our knowledge of electromagnetism, but it was James Clerk Maxwell, who in 1865, quantified all that we know about electromagnetic theory in four relations. They are- called Maxwell's Equations. They tell us simply that charge (which today we know is also due to matter itself) is the source of an electric field. In fact, it has a form very similar to the gravitation relation:

where k_e is a constant, q₁ and q₂ are the charges on the objects respectively, and  r  is the distance between the charged objects. While we can describe all the effects of charges themselves, we recognize today that matter itself, electrons and protons in atoms are the sources of these effects. We don't have such things as free charge by itself, but rather a characteristic of matter itself. Electrons, protons and other basic forms of matter have electric charge and manifest the corresponding effects.

         Like gravity, the strength of the charge is proportional to the product of the charges. Stronger charges have stronger attraction or repulsion (like charges repel and opposites attract) while the magnitude of the force also depends inversely on the square of the distance between the charges. This is just like gravity in many ways. It makes us realize that both charge and mass are characteristic of matter. By knowing something about the nature of interaction of masses or charged particles we, in effect, learn something about the nature of matter itself. It has been said that all there is in the universe is matter and energy and the interactions between them.

        Maxwell also explained in his equations of electromagnetism that we can't separate magnetic poles like we can electric charges. We can hold positive charges in one hand and negative charges in the other, but we have never been able to hold a north pole in one hand and a south pole in the other. Everytime we break a magnet in two we have two complete magnets. As we divide them further we get more magnets. We will see later that atoms themselves act as elementary magnets. Maxwell further quantified relations between moving charges and changing magnetic fields. Basically, moving charges create magnetic fields and changing magnetic fields create electric currents. You don't have one without the other. Using these relationships we generate electricity for our uses. Even nuclear power does this. Whether we burn coal or control nuclear fusion, we use the heat generated to make steam which in turn causes big magnets to rotate, generating electric currents in wires wrapped about them.

        Even though electromagnetic forces drop off in strength as the square of the distance between them, they are still about 10³⁷ times stronger than the gravitational forces. Even so, matter on the large scale is neutral (charge) so gravitational forces still determine the large scale structure of the universe.

        The above two forces are not adequate to explain all of nature. They do account many of the phenomena visible on the already defined macroscopic scale. In fact the basic structure of the atom and atomic bonding is primarily electromagnetic, balancing the positive charges in the nucleus with the negative charges on the electrons surrounding the nucleus. But it fails to explain why the sun generates energy at the rate it does; nor do these two forces explain radioactivity. We look at these later.

        The two remaining forces both involve the nucleus. The Weak Nuclear Force is much weaker than the electromagnetic, but yet stronger than gravity. It accounts for the nature of radioactivity and the ejection of particles from within the nucleus itself. The Strong Nuclear Force is the strongest force discovered. It accounts for the basic structure of the nucleus. As particles are brought together to form the nucleus, we see neutrons (no charge) and protons combining. Neutrons have no electromagnetic force to overcome, but without the strong nuclear force, all the protons would tend to make the nucleus "fly apart." It holds the nucleus together as the distance decreases and the force of repulsion gets larger. Both the Strong and Weak Nuclear forces are not apparent beyond the bounds of the nucleus itself and act only at that tremendously short range. We look more at their nature when we study the nucleus in the next chapter.
 

        If we stop for a moment and look back through the history of man's search for understanding of the physical world, we find that it was the Greeks who first showed us that the universe can be understood by man. Through many trials and tribulations, mankind developed some rather sophisticated theories of the worldly phenomena by the end of the 19th century. These were largely discovered during the preceeding two hundred years. Newton gave us a marvelous theory of mechanics. It worked so well that we were able to fly to the moon (Appollo Missions) and land exactly where we wanted to, so well that we went to Mars (unmanned) and landed there (Viking Missions, 1976) , so well that the Voyagers have gone to Jupiter, Saturn as well as Uranus, and reached Neptune (August 24,1989). Maxwell described all that man had learned about electromagnetism, even to the point of predicting electromagnetic waves and demonstrating that light itself was a special form of an electromagnetic wave. He contributed to Boltzmann's work of the kinetic theory of matter, linking the macroscopic world with the microscopic. This theory helped us understand how the gross features of gases could be explained with an atomic picture of matter. For example, the pressure, temperature and related properties could be explained on the macroscopic  level by treating gases as fluids, but this atomic picture gave the same results! Optics, too, had been developed to such a sophisticated level that it was not
surprising that scientists in general felt good about the "modern" picture of the world at the end of the 19th
century. Their models explained the physical phenomena quite, well and we were able to make predictions that convinced us that we were obviously right. (Remember, one characteristic of science that Kuhn pointed out is the consensus of the practitioners. When the models worked so good, scientists knew they had some basis to be the foundation of correct knowledge.)

         But there were a few little things that weren't quite explained. Lord Kelvin called these little clouds on the horizon. Yet, because the other theories were so well established that these few unanswered questions were frequently overlooked. Explanations for these phenomena led to an whole new and profound vision of the microscopic world, called the quantum theory of matter.

Black Body Radiation

        One of these unexplained phenomena was called the Black Body. A Black Body is defined as a perfect absorber of radiation and when heated to incandescence, a perfect radiator of energy. If one explored the relationship between the wavelength of radiation and the amount of energy per unit area per second the following curve is obtained:

         It was noticed that the curves seemed to be quite independent of the material and in fact depended only on temperature. This meant that wood and steel at the same temperature gave the same effects or same curves! Weird! The higher the temperature the more the peak of the curve moved to the left, or towards shorter wavelengths. In addition, it also was more intense or higher overall, i.e.; it gave off more energy per unit area per second for the higher temperature. If one added up the energy emitted at all wavelengths, it was also found that the total energy per unit area per second is proportional to the fourth power of the temperature. Weird! It is called the Stefan-Boltzmann Law.

Note that as the temperature is doubled, the energy increases by a factor of 2 to the fourth power, or sixteen
times! Weird again!

        Classical physics had been so strongly ingrained in the scientists of the world that this phenomena could not be  explained. (We cover the history of the development of the qunatum world to see how this revolution took place, ala Kuhn.) Radiation, or the emitted energy was of course classically treated as a wave in nature. In the low wavelength limit or the  long wavelength limit there were good approximations, but could predict the curves of the observed phenomena exactly. Max Karl Ernst Ludwig Planck sought out such a theory. Using some of Boltzmann's statistics from the kinetic theory of gases he found that he could in fact reproduce the observed
relationships if- he made the assumption that the atoms act like tiny electromangnetic oscillators, each emitting  electromagnetic wave. But he went further and required that the oscillators have only discrete values given by:

where f is the oscillator frequency, h is a constant (called Planck's constant) and n is a number (called  a quantum number) that can have only discrete positive integer values. The value of h is very small:

His assumptions were radical, yet they worked.  He failed to accept them as an accurate description of matter, or a helping to describe the real nature of matter on the atomic level. He was a classical physicist, remained that way, but his contributions to our present picture of the world at the atomic level were earth shaking. It effectively undermined the very foundations of physics. He described his work to the Berlin Physical Society on Friday, December 14, 1900. This started a new era in science.

         This magnificent idea was exciting. The world was convinced that light and the entire spectrum of electromagnetic radiation, from radio waves at the long end, through microwaves, the visible, to the ultraviolet, in fact were really wave-like in nature. All of these were special cases of wave radiation, just different wavelengths and frequencies. (This will be explained in detail later, in Chapter 7.) For now, let us recognize that a wave is a mathematical construct that allows energy to be transmitted from one place to another without matter also being transferred. Even in the presence of matter, such as air or water, the wave is a disturbance in the medium itself, and it is energy that moves. Note water waves the next time you are at the beach. The entire lake or ocean does not "pile" up on the beach, but rather the wave travels through the medium. Light waves travel from the sun to us, even through the vacuum of space, and we feel its warmth. Light behaves like a wave and fits all of the characteristics like refraction, reflection, diffraction, etc. This all applies of course, on the macroscopic level of our normal experiences. It works so well that we can explain all optical phenomena, television communication, etc. Planck's Hypothesis (yes, a hypothesis) said that all of this was wrong.

        Planck's idea said that light (at the atomic level) acted like it was a particle carrying its own energy (hf) and momentum. As unacceptable as this hypothesis seemed, it was soon demonstrated that this was the only way to explain Black Body Radiation. We have been witness to the birth of the idea of the dual nature of light. On the macroscopic level it acts like a particle. We'll see more ramifications of this later.

The Photoelectric Effect

        Another phenomena that gave us further insight into the quantum world was called the photoelectric effect. Still, it was generally accepted that radiant energy was really in the form of electromagnetic waves. In fact Heinrich Hertz demonstrated "conclusively" how this happened. But one day the sparks from his apparatus were stronger than normal when illuminated with an unltraviolet light. He ignored this, as he was concerned primarily with electromagnetic radiation and not the reverse effect. Others, however, did study this phenomena, which became known as the Photoelectric Effect.

        Basically, this phenomena involves certain metals being illuminated with various kinds of light. Upon this stimulation, beams of electrons were emitted. In fact, some built tubes that utilized "beams of these emitted electrons and applied reverse voltages to "stop" the flow of electrons or the flow of electric current. These kinds of devices are still~used when we enter a store that has a buzzer at the door which "rings" or "dings" when we enter. As we enter the doorway we pass in front of the light beam, thereby disrupting the flow of electrical current which normally keeps a switch in the open position. When the current is stopped, the switch closes and a buzzer or other alerting device is activated.

        Another one of Einstein's three significant papers of 1905 explained the PhotoElectric Effect which basically renforced Planck's ideas.of a quantum picture of matter. Using all the power of "classical" physics, the effect cannot be explained. Suppose we liken the electrons in the metal to boats beached upon the shore of an ocean. Classically, if a very large wave came upon the shore, the boats would be removed violently. Yet, the most intense light could not yield one electron. It was observed that a very weak light of the right frequency would free electrons. Classically, it would be predicted that a low intensity beam of long duration would eventually free electrons. i.e., if the electrons continue to absorb the energy, they would eventually gain enough to free themselves from the metal. Observation tells us that if the light has the riqht threshold frequency, the time to be released is of the order of 10^-8 seconds! The effect violates all the rules of classical physics. Weird!
 
 

        Einstein said (and he received the Nobel Prize in 1921 for this triumph) that Planck was right: Light is quantized. And one electron absorbs one photon, provided the photon has enough energy. Einstein is the one who coined the word "photon" to describe this quantum or bundle of light energy, hf. (What we mean here is h, the Plank constant multiplied by the frequency, f.) Photos is the Greek word meaning light. There is a threshold energy value, different for differing metals, that must be reached for the electron to be freed. If it absorbs a photon of at least that frequency, and hence that minimum energy value, the elctron will be released. The excess energy will be in the form2of kinetic energy of the electron. (Energy of motion, 1/2 my .)

Many other exciting phenomena were observed during this time period. It is difficult to describe them sequentially. Instead, let us try to look at then as they affected new ideas about our world and their significance in developing new models of matter.

The Spectral Lines of Hydrogen

        One interesting observation, first made by chemists, but important in explaining matter, was that every element known to mankind had its own signature, or set of spectral lines. This is not unfamiliar to us. For example, throw some table salt into a fire and a burst of yellow flame is observed. Chemists have long used this flame test to identify the element Sodium (table salt is sodium chloride.) When heated, elements give off light. If that light is allowed to pass through a prism or diffraction grating, a set of lines is seen.

One particularly interesting set of lines is that from hydrogen. Consider the following arrangement:

         The resulting emission spectrum for hydrogen in the visible and near ultra-violet has characteristic lines at 410 nm (violet), 434 nm (blue), 486 nm (blue-green) and 656 nm (red). It had been measured quite accurately in the mid 1800s, but could not be represented by a model until 1885 when J.J. Balmer devised an empirical formula that could be used to predict this series of spectral lines:

where R is a constant. Recall also that wavelength and frequency are related by the speed of light ( c = l f) so
that the above formula also to within a constant, can be used to give the frequency. Note, however, that an integer, n, appears strangely in the relation. No one could give any reason for the dependence of wavelength and frequency of the spectral lines on an integer. Recognize that any acceptable model for the hydrogen atom must explain this phenomena too.

Modeling the Atom

        Once Boltzmann, Maxwell and others demonstrated that the world could indeed be made of tiny particles (which we call atoms) instead of being continuous in nature. The search for atoms and their structure began. Finding the right model is like trying to determine what is in a wrapped present under the Christmas tree. You can pick up the box, estimate its weight, shake it, roll it, perhaps even move a magnet over its exterior to determine further properties, but all this may not be such an easy task unless you unwrap
the package. Unwrapping atoms is not so easy. We can't see atoms, even with electron microscopes. Thus, we have to determine their properties indirectly, mostly by their effects on the rest of the world.

        Developing a valid model of the atom depended on the known information and the kinds of predictions one should be able to make (and experimentally verify) using the model. It is not our purpose here to cover the details of all models conceived in the early days of atomic physics. For such historical background the reader is referred to tradtional physics texts covering the material.

        Actually, long before atomic models were sought, John Dalton (about 1808) suggested that chemical elements existed as tiny spheres. The first atomic model!!! About 1870 it was observed that in a low pressure tube, electrically shocked gas would "glow." This was interpreted as a kind of electromagnetic radiation or rays. Since the "rays" were emitted from the cathode (negative terminal of the tube) they were called cathode rays. The device was called a cathode ray tube. (Cathode Ray Tubes, or CRTs, are used in. television and computer monitors today.) By showing that these cathode rays were deflected in a magnetic field, J.J. Thomson convinced the world that these were not electromagnetic waves, but in fact were negatively charged particles, which we know as electrons. Since the atoma were electrically neutral, Thomson suggested that atoms might be more like a "sea" of positive charge. in which negatively charged electrons floated. This was called the "plum pudding" model.

        Electrically heated cathode causes electrons to "boil" off the negative terminal called the cathode. They are attracted to the positively charged anode. A magnetic field causes the beam of electrons to be deflected. Oscilloscopes, TVs, etc. use these principles. Thomson did not have enough information to definitely determine the charge and mass of the electron, but he did have enough to estimate their ratio, q/m (charge to mass ratio, in Alombs per kilogram.) This was experimentally performed at the turn of the century amid a flurry of activity in developing atomic models.

        The difficulties that Thomson experienced were resolved by Millikan. He suggested that droplets of a fluid sprayed between two charged plates with an atomizer could be suspended by balancing the forces of gravity with an electric field. Note how basic ideas help in solving the riddles of the universe.

        The weight of oil droplets are suspended between plates by an applied electric field. The electrical forces on the droplets balance the gravitational weight of them. At this point, Thomson's plum pudding model was the only one to show structure to the atom. We now knew the charge and mass of the electron, one of the major component's of the atom. The idea that atoms had a nucleus was first proposed by Lord Rutherford. He performed some scattering experiments with positively charged alpha particles. (These alpha particles we know now as the nucleus of helium atoms, i.e., a nucleus containing two protons, two neutrons, but no electrons at all. We discuss this in detail in the chapter on radiation.) He "fired" these positively charged alpha particles into a thin gold foil. According to Thomson's model, he expected them to be slowed up by the repulsive force of the sea of positive charge. Yet, instead of slowing up, some of them actually were scattered through large angles. He likened this to firing a cannon at tissue paper and seeing the shell come back at you: What this meant was  that the charge must be concentrated in a small volume which we today call the nucleus. He attempted to construct a planetary type model with a negatively charged electron orbiting the positively charged nucleus. But the well understood concepts of classical physics indicated that this could not be. It is known that when a charged particle is accelerated (even in a circular orbit) it radiates energy. If it loses energy, the orbit would decay and the electron would "spiral" into the nucleus. Since this obviously doesn't happen, the model failed. How can an atom have a centrally located positive nucleus with separate electrons, and be stable?

Bohr's Model of Hydrogen

       Neils Bohr resolved this dilemma with a seemingly bizarre idea that worked. (Bohr is credited with remarking at a scientific meeting that the only concern about a crazy idea is only if it is crazy enough to be correct.) Bohr's model turned out not be correct, but the fundamental idea that resolved the difficulty of electrons in atoms gave the world the first glimpse at the quantum atom. He combined known facts with an entirely new concept, quantum numbers and quantum levels.

        Remember that all the exciting work on atoms was happening concurrently, not necessarily sequentially. Finding the nature of the world brought together many ideas-that were found separately and needed to be integrated into a comprehensive model. The word quantum itself comes from the Latin word meaning quantity or amount. In science we use it to describe discrete steps in discrete quantities, for example energy or momentum.

Bohr's model consisted of three basic postulates:
 

1. Electrons are in orbit about the nucleus, but only in certain allowable orbits in which they do not radiate energy.

2. As electrons transition up or down to other allowable orbits, they either absorb or radiate one photon or quantum of light energy, E = hf.

3. The specific stable orbits are prescribed by angular momentum (an orbital characteristic), L = nh.
 

        Still, as a model, it certainly had some utility. You may even recognize this 1921 model as the basic atomic structure children learn about in elementary school, namely that electrons orbit a positively charged nucleus. For electrically neutral atoms, the number of electrons equals the number of protons. If one or more electrons are "stripped" off of the atoms, we call it an ion.

        Bohr knew it was not a complete picture of the atom, but it was useful in making certain predictions about the atomic world. It had some degree of correctness.
 
 

        The electron transitions to a higher state when it absorbs a photon of the proper frequency (according to Planck, this amount of increased energy is E = hf) and as it "steps" downward it emits a photon of a specific frequency. The amount of energy it gains or loses is just Planck's constant multiplied by the frequency of light or photon.
 
 

DeBroglie Wavelength

        Hardly had Bohr's model been in place when deBroglie (1923), in his PhD thesis, questioned the basic nature of the world. He asked that if light has a dual nature such that on the classical, macroscopic level, it acts like waves, but on the microscopic level it acts like a particle, why shouldn't matter act like a particle on the classical level (which it obviously does) and perhaps exhibit the properties of a wave on the atomic scale. This would make nature more symmetric in a sense. He suggested that matter be described by a wavelength, l , called the deBroglie wavelength.

            His model prescribed a resonant wave be imposed on lengths equal to the paths of electrons in Bohr's model of the atom. Resonant waves are like the waves that appear to stand still on a jump rope if you "whip" it just right. They are reenforced; for example, if n=3, the path would contain exactly three wavelengths. These wavelengths were-related to the momentum by Planck's constant:
 
 

        This model blended the best of Bohr's ideas, but the meaning of waves left a lot to be desired. The wave idea described the stability of orbits quite well, but was conceptually quite abstract. Did this model mean to imply that matter consisted of waves? How would this be reconciled?

Probability to the Rescue

       Erwin Schrodinger helped to answer these questions. Classically, the mathematical construct of waves describes, with a rather complex looking second order differential equation, (beyond the mathematical level we wish to consider here) how waves propogate outward in time and space, maintaining their shape and speed:

        The above wave equation relates 3-dimensional propogation in space and time. Schrodinger forced this classical equation onto atomic models. The  equation describes particles on the atomic level. Rather than suggesting that waves make up matter, it must be interpreted in terms of probability. Indeed, much discussion and debate led to the meaning of this equation and exactly what the wave functions represent. Solutions to the equation are not trivial, but they give us more quantum numbers than we had before. Recall, that according to Bohr we had n, the principle quantum number n describing the energy level of the electron in the hydrogen atom. In Bohr's model the angular momentum was described by its orbital position, but here the quantum number l determines the angular momentum of the electron in the atom. the quantum number m gives the z-component of th angular momentum, and an additional quantum number, s, called the spin quantum number, tells us that the electrton acts as if it were spinning with its axis pointing up or down. The solutions come from boundary conditions applied to the equation which describe initial and final conditions of the atoms in the real world. The quantum numbers are integers, such that multiples of them also describe certain properties, such as energy, angular momentum, etc. The important observation to be made here is that the values of all of these characteristics are discrete in nature, i.e., they vary in discrete steps, not in smooth, continuous values. Experiment agrees with this conclusion. The solutions and quantum numbers really tell us the probability of finding the elctron in a certain place in the atom, with certain characteristics. The periodicity of solutions to the equation forms a logical requirement for the stability of the orbits.
 

         Thus far, the quantum description of the atom seems to follow an almost straight forward path culminating in a very different, but yet very pleasing kind of result. There seems to be a beauty in how the world corresponds so well:

MATTER and  LIGHT Duality

        The essence of our search has been an explanation of ideas. What is the true nature of light? Is it a wave? Is it a particle? What is the nature of matter? It would be very nice to have a world that in the classical regime gave us the correct answer while at the same time giving very different results at the atomic level. This is not the case. The triumphantly successful Newtonian Mechanics works terrifically at the level of our normal experiences, but fails miserably when trying to describe and predict proper results at the very small level. Similarly, quantum mechanics describes phenomena on the atomic levelb ut fails for large scale events. This idea has been called the Complementarity Principle. The two pictures complement each other.

Hang on - we are yet done with quantum weirdness. It gets even more interesting.

Heisenberg Uncertainty Principle

        We've noticed that there are some fundamental constants that are intimately connected to the way that the universe works. The speed of light ( c ) is constant everywhere, c = 2.99 X 10⁸ m/s  , the Universal Gravitational constant (G): G = 6.67 X 10¹¹ and now, we see another value, Planck's constant ( h ),  h = 6.6 X 10 ³⁴ erg-sec
 that is virtually at the foundation of the quantum world. Note that Black Body Radiation, the Photoelectric Effect, Bohr's and deBroglie's models of the atom, and Schrodinger’s Equation, all depend on h. The value of h is small indeed, but if it were actually zero, (it is 34 decimal places small) there would not be any quantum effects. This small effect gives us clues not only to the nature of our world on the very small level, but perhaps to the very source of the world itself, and maybe even its ultimate destiny. We leave these topics for later discussions. If h were zero, Black Bodies would not radiate as they do, electrons could not absorb or emit photons of light energy, etc. So the existence of h, as well as its very special value are quite significant:

         An additional effect is very much related to the nature of the quantum world. On  the classical level we are used to being able to measure an object's position and momentum or speed, energy, etc., with a high degree of certainty. At the microscopic level we find an effect that is awesome. Ask yourself how do we observe the world? You might say that we look at it. To do so requires that a photon be involved, which in effect, could change what we see. By actually
observing the atom we are in effect "forcing" it into a specific state. But from the nature of the probability distribution as we, for example know more about position, we know less about the particle's momentum, and vice versa. Heisenberg expressed this in the uncertainty relation:
 

        It describes for us another weird effect of the quantum atom - as the uncertainty in position, D x, becomes small, the uncertainty in momentum, Dp, increases. The opposite is also true. We cannot measure position and momentum exactly!

         Today scientists use these very ideas to describe atoms and their interactions. Although the mathematical details can be extremely difficult and abstract, the use of computers makes their studies not as difficult as it was a few years ago. The reader should appreciate the ideas and thought processes as well as the creativity that resulted in these models. The reader should also remember that the basic structure of the atom is dependent primarily on the elect ro-magnetic force alone. The force of gravity is too weak by many many orders of magnitude to affect the structure of atoms while the two nuclear forces act at ranges far smaller than the size of atomic electron orbits. The loaction and energies of electrons are determined by mere electrical interactions with other electrons and the positively charged nucleus.

        The simplest atom is hydrogen, consisting of one electron and one proton to make the nucleus. Solutions to our mathematical models tell us that it  is about one Angstrom in diameter. (1 Angstrom = 10^-10 metres.) The nucleus is much, much smaller still, it is about 10,000 times smaller, about 10 metres or 10 Angstroms in size. In discussing the nucleus (next chapter) it is convenient to express distances in terms of the "fermi" or "femtometre."
 

The fermi is used because the nucleus is characteristically of the order of a few fermis. This not only is useful in describing size, but also to describe the range of the strong nuclear force itself.

If the nucleus were scaled up to about an inch in size, the entire atom itself (basically area the electron occupies) would be about the size of the MetroDome Stadium in Minneapolis. That means there is a lot of empty space in it. Perhaps on this scale one might be able to visualize how difficult it would be to pinpoint the exact location and energy of the electron in the atom. As atoms become more complex, the forces between electrons and the nucleus and electrons themselves become increasingly more complicated.
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SUMMARY

Models of the Atom and Progress in Developing Them:

• John Dalton - 1st real model - atoms were spheres, each kind of element was a different kind of sphere - about 1808
• Boltzmann and Maxwell - about 1865 - showed the kinetic theory of matter where atoms could explain the gross features of nature and matter. They linked the microscopic and macroscopic world.
• Electrical interactions not well understood until the middle of the 19th century- matter is the source of all charges -electrons give us - charge; protons give us + charge. The idea of electrons and protons did not come until later.
• Crookes Tube - Cathode Ray Tube - about 1870 - two opposing views - ":rays" were particles or waves
• J.J. Thomson - 1897 - measured charge to mass ratio of electrons. identified the "cathode rays" as electrons. Check lab # 5.
• J.J. Thomson - "plum pudding" model sea of + charge with electrons floating in it.
• Millikan - Oil Drop experiment - refer to lab # 6 measured the charge on electron and from the ratio q/m measured by Thomson, also determined the mass.
• Rutherford - 1st Nuclear model - scattering Experiments with alpha partciles- shot them at gold foil showed + charge is concentrated.
• Rutherford could not explain electron orbits- they should decay as they go in orbit- ie not stable.

Atomic Number - # of protons in nucleus

Atomic Mass # of protons and Neutron in nucleus

1920 Rutherford named proton after Greek "protos" meaning first.

Isotopes - element with differing # of neutrons

 Dual Nature of Light - Macroscopically acts like a wave One atomic level it acts like a particle

Photons - bundle of light energy - quanta of light

Bohr Atom -

•  Electrons stable in certain discrete orbits
•  radiate or absorb energy as they move up or down in orbits - E = hf
•  Orbits described by angular momentum = hf

deBroglie - Dual Nature of Matter:

1) Macroscopically acts like particles 2) On atomic level acts like waves

Wavelength = h/p

 Heisenberg Uncertainy Principle - Uncertainty in position x
uncertainty in momentum = h
 

Complementarity - the two pictures complement each other. On the atomic level, the classical picture of the world is unsatisfactory and the quantum mechanics must be used. On the classical level, the quantum picture is not acceptable. On the atomic or quantum level, light acts like particles and matter behaves as waves. On the classical level, light behaves as waves and matter as particles.

Erwin Schrodinger - Wave Equation - used wave equation as in classical wave theory to describe particles on the atomic level - must be interpreted in terms of probability - does God play dice with the universe? This came from the forcing of the classical wave equation that describes mathematically how waves propogate outward in time, maintaining their shape and speed. Solutions give us more quantum numbers than does
the Bohr model (only one quantum number). We then have three quantum numbers, n - the principle quantum number giving us the energy dependence; 1, used to determine the angular momentum of the electron in the atom, and m which gives us the z-component of the angular momentum.' An additional quantum number, m , the spin quantum number tells us if the electron acts as sif it were spinning up or down. The wave model does not mean matter is made of waves, but instead the probabilistic interpretation of a probability wave that tells us the probability of finding the lectron in a certain place in the atom. It also, due to the smearing out effect in space, better explains how a quantum of light is emitted as the electron transitions from one level to another.

Questions

1.

References

Asimov, Isaac, Understanding Ph sics: The Electron, Proton and Neutron, Signet science Library, The New American L rary, New York, 1969.
 

Hoffmann, Banesh, The Strange Story of the Quantum, Dover Publications, New York.
 

Bueche, Frederick J., Introduction to Physics for Scientists
and Engineers, McGraw-Hill Book ok Company, New York, 1986.

A physicist, an engineer, and a computer scientist were discussing the nature of God.  Surely a Physicist, said the physicist, because early in the Creation, God made Light; and you know, Maxwell's equations, the dual nature of electro-magnetic waves, the relativist consequences... An Engineer!, said the engineer, because before making Light, God split the Chaos into and and Water; it takes a hell of an engineer to handle that big
amount of mud, and orderly separation of solids from liquids... The computer scientist shouted: And the Chaos,
where do you think it was coming from, hmm?

---Anonymous