In addition to his now bright blue appearance, Osterman gained the ability to alter his size at will, to teleport himself and others instantly from one location to another, and to be aware of the future, that is, to experience time-past, present, and future-simultaneously. That is to say, following the removal of his intrinsic field and subsequent rebirth as Dr. Manhattan, Jon Osterman appears to have gained independent control over his quantum mechanical wave function.
His quantum mechanical what? We have now reached the point in our amazing story where we consider what a wave function is, and why gaining control over it would be akin to possessing superpowers. Here's the short answer: The quantum mechanical aspect of any object is reflected in its wave function. By performing simple mathematical operations on the wave function, one can calculate the probability density (that is, the probability per unit volume) of finding the object, whether it is an electron, an atom, or a large blue, naked physicist, at any point in s.p.a.ce and time. If you could indeed alter your wave function at will, you would gain the ability to instantly appear at some distant location, without ever technically traveling between your initial and final points; you could change your size (from either very large to tiny); you could diffract into multiple versions of yourself; and you would be cognizant of your future evolution. And you'd likely give off a blue glow, though as we'll see later, that is more a consequence of leaking high-energy electrons-a side effect of rebuilding yourself at the atomic level.
While Jon Osterman is not a real person, nor is there a wave a.s.sociated with an "intrinsic field," nor any such thing as an "intrinsic field," for that matter, the rest of the preceding discussion about the fundamental forces of nature (that is, electromagnetism and strong and weak nuclear forces) was correct. The experiments described in Section 1 demonstrated that there is a wave a.s.sociated with the motion of electrons and atoms, and in fact with the motion of any and all matter. The concept of a wave function, introduced by Erwin Schrodinger in his "matter-wave equation," is the key to understanding all atomic and molecular physics. It might as well be called the "intrinsic field" for the central role it plays in the understanding of chemical bonding, by which all matter is held together.
At the start of the twentieth century, physicists debated whether the electrical charges in an atom were spread out throughout the atom, uniformly distributed in s.p.a.ce, or existed as concentrated pointlike negatively charged electrons and positively charged protons. A series of experiments by Ernest Rutherford, Ernest Marsden, and Hans Geiger convinced scientists by 1911 that atoms were comprised of ma.s.sive, positively charged protons in a physically very small nucleus, toward which the lighter, negatively charged electrons are electrically attracted. This is the familiar "solar system" picture of the atom that you might remember from grade school. But an electrically charged solar-system atom has a big problem in stability.
The Earth is pulled toward the sun by gravity, so what keeps our planet from falling into the sun? It turns out that this is a common misconception-the Earth is falling into the sun all the time! Don't panic-we're not on a death spiral to a fiery end. The Earth is moving at a high velocity at nearly a right angle to an imaginary line connecting us to the sun. The gravitational pull toward the sun deflects the Earth away from its straight-line path (an object in motion will remain in motion-unless acted upon by an outside force, such as gravity from the sun). The combination of the acceleration toward the sun, and the motion at a right angle away from the sun, results in a circular trajectory (the actual orbit is an ellipse-a distorted circle). The stable orbits of the planets in our solar system are possible only through the continual falling toward the sun. The Earth maintains its...o...b..tal motion, as there is nothing to slow it down (collisions with s.p.a.ce particles provide a very small frictional drag that we can neglect), while the conditions that led to the Earth's original velocity about the sun are, as they say, a subject of current research.
As the force of electrical attraction between the electrons and protons in an atom is mathematically similar to the attractive force of gravity, a completely a.n.a.logous argument suggests that the electrons should move in circular or elliptical orbits about the nucleus, not unlike the way the planets...o...b..t the sun in our solar system. The main difference is that planets do not emit energy as they orbit the sun, but orbiting electrons do lose energy-in fact, quantum mechanics was developed in part to explain why all atoms don't suffer a death spiral to oblivion.
It was known from the days of the American Civil War that whenever an electric charge changes its direction, as in an elliptical orbit about an atomic nucleus, it emits electromagnetic radiation-that is, light. Since light carries energy, the electrons should lose energy as they orbit, and the slower they move, the less they are able to resist the attractive pull of the positively charged protons in the nucleus. In a short time (actually, less than a trillionth of a second), they should spiral into the nucleus. However, atoms form chemical bonds with other atoms, by which materials such as table salt, sand, and DNA are possible. The chemical bonds holding molecules and solids together involve interactions of the orbiting electrons among neighboring atoms, which would not be possible if the electrons were sitting on the nuclei. Something in the picture had to be wrong. If accelerated electric charges did not emit electromagnetic radiation, then radio and TV would not be possible.21 An important step in reconciling this puzzle was Niels Bohr's suggestion in 1913 that the electrons in an atom can a.s.sume only particular trajectories about the nucleus. That is, only certain planetary-like orbits are allowed. Electrons can jump from one orbit to another, but they may not follow any arbitrary path around the nucleus. An a.n.a.logy: The city of Minneapolis in Minnesota contains a series of lakes that can be circ.u.mnavigated by paved paths. There are several paths encircling each lake, one intended for pedestrians, another for bicyclists, and a third for automobiles, and each pathway is separated from the others by a gra.s.sy median. Bohr's electronic orbits were a.n.a.logous to these pathways, where electrons were free to travel but were forbidden from walking on the gra.s.s, as shown in Figure 12. The closer the orbit was to the nucleus, the more tightly bound the electron would be, so that more energy would have to be supplied to remove an inner-orbit electron than would be required to remove one from an outer ring. The electron could jump from an outer pathway to an inner loop, with the emission of the appropriate amount of energy, say by emitting light. Alternatively, by absorbing just the right amount of energy, it could be promoted from an inner orbit to a higher-energy, outer orbit (provided that the path had an open, available s.p.a.ce for the electron). Bohr proposed that, for some reason that he could not explain, the electron would not emit light during its...o...b..t, despite the requirements of Maxwell's equation for a charge that is constantly changing direction, but rather give off light only when moving from one orbit to another.
Bohr's proposal that only certain discrete orbits were possible was an attempt to account for the spectrum of light emitted by different atoms. Why are neon signs red, while the light from sodium lamps has a yellow tint? Neon, sodium, and in fact all atoms have the color they do because the electrons in these atoms have a predominant absorption (and emission) at only specific colors of light. The chemical differences between neon, with ten electrons (and ten protons in its nucleus), and sodium, with eleven electrons (balanced by eleven nuclear protons), leads to the separation between the relevant orbits corresponding to light in the red and yellow portions of the electromagnetic spectrum, respectively.
One can test this quantum principle tonight at dinner. Sprinkle a little bit of salt into the table candle (not so much that you smother the flame) and you will see a distinct yellow tinge to the candle's light. The energetic atoms in the flame will excite electronic transitions in the sodium in the salt crystals, with electrons moving from one quantum orbit to another, and when the electrons return to their original orbits, they give off yellow light. In fact, independent of sodium, the light we see from a candle results from the electrons in the hot gas atoms near the wick being excited into high-energy orbits. When the electrons return to their lower energy states, they emit photons-which is the source of the light in a flame. We cannot have a complete understanding of fire-the oldest technology-without an understanding of quantum mechanics.
Figure 12: Sketch of Bohr's proposed discrete electron orbits about a positively charged nucleus. Only certain trajectories are allowed, and an electron has a different energy depending on which orbital path it is on. The electron emits or absorbs light only when moving from one orbit to another.
Each element has its own unique spectrum of absorption (or emission) lines, specific wavelengths of light that correspond to allowed transitions from one electron orbit to another. Just as in the case of fingerprints or snowflakes, no two elements have exactly the same spectrum of absorption lines. By measuring the different wavelengths of the light emitted by an atom, one can identify the element or molecule. In fact, this is how the element helium, the second most common element in the universe (after hydrogen), was discovered. When absorption lines from light from the sun were studied, the line spectra for hydrogen were observed, but there was another series of lines that did not correspond to any element known on Earth. This newly detected element was named helium after the Greek G.o.d of the sun, Helios.
If you want to see what a line spectrum for a previously un-known element would look like, consider the November 1955 issue of Strange Adventures, a science fiction anthology comic book published by National Allied Periodicals, home of Superman and Batman. As shown in Figure 13, scientist Ken Warren uncovers the existence of a "radioactive metallic element hitherto22 undiscovered in the entire solar system." In order to trace the source of this new metallic element, Dr. Warren uses a scintillometer. A caption box in the story informs the readers that this is a device capable of detecting "even the smallest amount of radioactivity." While nowadays a scintillometer refers to a device that detects small variations in the optical properties of the atmosphere, back in 1955 this was indeed the term used to measure the presence of ionizing radiation. Dr. Warren discovers that the radioactive element was brought to Earth by an alien s.p.a.cecraft. This in turn leads to his capture by two would-be invaders who intend to set the entire planet aflame. Apparently if Earth could be converted into a flaming sun, then the temperature on the moon would rise to a point that would accommodate the aliens' physiology. The fact that the sun is not actually a large planet that has been set on fire seems to have escaped these aliens, who have nevertheless managed to master interstellar flight, but it turns out to be a moot point, as Dr. Warren and his chemist colleague Hank Forrest are able to trick the aliens into abandoning their plan and leaving our solar system.
Figure 13: Accurate ill.u.s.tration of an absorption spectrum from a 1955 DC Comics science fiction comic book. The element in question only absorbs light at very particular wavelengths, providing a unique signature as to its elemental composition. This line spectrum is a hallmark of the quantum nature of atoms.
In a sense, Bohr's proposal that atomic line spectra arise from energy emission or absorption from electrons residing in discrete orbits managed only to reframe the mystery of atomic spectra. Now, instead of wondering why atoms absorbed or emitted light at only particular wavelengths, one could ask why only certain electronic orbits were possible about the atomic nucleus. Prince de Broglie's matter-waves provided an answer.
Schrodinger was one of the first physicists to recognize that de Broglie's matter-wave hypothesis could neatly solve this riddle of why the electrons did not collapse into the nucleus. For if there is indeed a wave a.s.sociated with the motion of the electron, as demonstrated by Thompson's observations reported in the pages of Science Wonder Stories (and the experiments shown in Figure 7), then this wave has to be a "standing wave."
A guitar string, clamped at both ends, as shown in Figure 14, cannot oscillate with any wavelength; only those waves are possible for which the amplitude is zero at the two fixed ends. This commonsense constraint leads to a very restricted set of possible waves that the string can support-which is why when plucked the guitar string vibrates only at certain frequencies. Wavelengths as ill.u.s.trated in Figure 14b are simply not possible. This is the whole point of a musical instrument, after all, as a string that vibrated at all frequencies would be pretty useless for constructing harmonies. The possible waves of a clamped string, constrained by the fixed ends and unable to travel down the length of the string, are called "standing waves."
Figure 14: Sketch of an allowed standing wave for a vibrating string clamped at both ends (a) and a wave that is not possible (b).
Say there is a wave a.s.sociated with us as we walk along the pedestrian pathway circling Lake Harriet in Minneapolis (and we do in fact have a "matter-wave" moving along with us, but of such a small wavelength that it is undetectable). Once we have returned to our starting point, our wave must join up smoothly and perfectly with the wave started when we left for our stroll. If the wave were at its highest point, a crest, when we began our walk, then as we walked around the lake, the wave would oscillate down to a valley, back up to a crest, and so on. Once we are back at our starting point, the wave must again be at a crest. This simple, commonsense aspect of waves (what would it look like if the cycle of the wave were such that our wave was at a valley when we returned to our starting point, where the original wave was a peak?) leads immediately to the consequence that only certain pathways-those that correspond to different wavelengths that start and end correctly-around the lake are possible.
Similarly, a wave a.s.sociated with an electron in a closed orbit, returning to the same point after a full rotation, can a.s.sume only certain wavelengths. Once an orbit has been completed, the wave must be at the same point as when it left. This "single-valued" constraint, that at any point the wave can have only one amplitude (that is, it can't be a peak and a valley at the same time), restricts the infinite range of possible wavelengths to a very small set of allowed orbits. Just as the guitar string has a lowest pitch-it is impossible to excite a wave on the clamped string lower than its fundamental oscillation-there is a lowest standing-wave electron orbit that can be constructed around the nucleus. Thus, electrons do not continuously lose energy and spiral into the nucleus with ever-decreasing radii, as there is no way for the apparently very real matter-wave to form standing waves with a lower wavelength than the lowest possible orbit. The de Broglie hypothesis of matter-waves saves the stability of atoms and also accounts in a natural way for the line spectra (as shown in Figure 12) of atoms.
Following a seminar presentation by Schrodinger of the matter-wave model of electrons in an atom, Pieter Debye, a senior physicist in the audience, pointed out that if electrons had waves, there should be a corresponding wave equation to describe them (just as Maxwell had found a wave equation for electric and magnetic fields that turned out to accurately describe the properties of electromagnetic waves, that is, light) and challenged Schrodinger to find it. Taking up this a.s.signment, Schrodinger accomplished the deed in just six months. A consequence of Schrodinger's work was the realization that the electrons could not really be considered as mini-planets in a nuclear solar system. The resulting equation, which now bears his name, would garner him a n.o.bel Prize, change the future, and lead to philosophical arguments over the nature of measurements of quantum systems that so disturbed Schrodinger that he would claim that he was sorry he brought the whole thing up.
The Equation That Made the Future!
The Schrodinger equation plays the same role in atomic physics that Newton's laws of motion play in the mechanics of everyday objects. Those with long memories may recall that back in the seventeenth century it was well-known that the application of external forces was required to change the motion of objects. What was lacking was a method by which one could calculate exactly how the motion of any given object would change as a consequence of the pushes and pulls of external forces. Newton found a remarkably elegant expression (the net force is equal to the ma.s.s multiplied by the acceleration, or F = m a) that despite its surface simplicity could account for a wide range of complex motions.
As any student in an introductory physics cla.s.s can tell you, it is not enough to identify the forces acting in a situation-one must be able to show how these forces will lead to a change in the object's motion. Armed with Newton's law, one can determine the trajectory of skiers and boaters, of runners and automobiles, of rockets fired at the moon, and of the moon itself (not to mention falling apples). We know that Newton's laws of motion are correct, for comparisons of the theoretically calculated motion agree exactly with what is experimentally observed.
Similarly, in the quantum realm, it is not enough to say that there is a wave a.s.sociated with the motion of all matter whose wavelength is inversely proportional to its momentum. One also needs a process-an equation-by which, if one knows the external forces acting on the object, the resulting behavior of its "matter-wave" can be determined.
Consider a simple hydrogen atom with one proton in its nucleus and one electron attracted to the proton electrostatically, the symbol that Dr. Manhattan etched into his forehead in the Watchmen comic and film. Given that we know the nature of the electrostatic attractive force between the negatively charged electron and the positively charged nucleus and that there is a wave a.s.sociated with the motion of the electron in an atom, what we would like is a "matter-wave equation" that enables us to calculate the properties of the electron in the atom. These calculated properties, such as the average diameter of the atom, or the electron's average momentum, could then be compared to experimental measurements, in order to test the correctness of the matter-wave approach. Any equation that does not yield testable results is useless from a physics point of view.
Schrodinger found just such an equation, though the procedure for calculating the wave function involves a formula much more complicated than Newton's F = m a. In fact, one often requires a computer in order to solve Schrodinger's equation, except in a handful of simple cases. But the fact that the calculations can be difficult does not invalidate the Schrodinger approach. Applying Newton's law of motion, F = m a, to the motion of the more than trillion trillion air molecules in a room exceeds the calculating capability of the largest supercomputer. Even though we can't do the sums, we know that in principle they could be done. The importance of Schrodinger's equation is that the process by which one calculates an object's wave function, given the relevant forces acting on it, is known. The math might be hard, but the path is clear.
We will not go into the process, involving mathematical trial and error, physical intuition, and a creative use of the principle of conservation of energy that enabled Schrodinger to develop his equation. It is, indeed, a fascinating story, filled with false starts, suspense, and plenty of s.e.x!23 We are interested in how quantum mechanics brought about our futuristic lifestyle of light-emitting diodes, laptop computers, cell phones, and remote controls. Consequently, for our purposes, it is not important how the Schrodinger equation was developed, nor will we try to solve it, even for the simplest case of a single electron moving in a straight line in empty s.p.a.ce (and it doesn't get more plain vanilla than that). Schrodinger's equation involves the rates of change of the wave function in both s.p.a.ce and time; consequently it can't be solved without calculus. In addition, it involves imaginary numbers (the term mathematicians use when referring to the square root of negative numbers), and thus calls upon considerable imagination to interpret. What we will do is discuss the significance of the equation and point out how various solutions lead, in some cases, to semiconductor transistors and, in other situations, to semiconductor lasers.
Enough with the tease: The Schrodinger equation, as it is now universally known, is most often expressed mathematically as follows: 2/2m 2/x2 + V(x, t) = i /t.
where represents our old friend, Planck's constant (the angled bar through the vertical line in the letter h is a mathematical shorthand that indicates that the value of Planck's constant here should be divided by 2, that is, = h/2); m is the ma.s.s of the object (typically this would be the electron's ma.s.s) whose wave function (p.r.o.nounced "sigh") we are interested in determining; V is a mathematical expression that reflects the external forces acting on the object; and i is the square root of -1.24 The rate of change of in time is represented by /t, while the rate of change of the rate of change of in s.p.a.ce is represented by 2/x2. As messy and complicated as this equation may seem, I have taken it easy and written this formula in its simpler, one-dimensional form, where the electron can move only along a straight line. The version of this equation capable of describing excursions through three dimensions has a few extra terms, but we won't be solving that equation either.
The Schrodinger equation requires us to know the forces that act on the atomic electrons in order to figure out where the electrons are likely to be and what their energies are. The term labeled V in the Schrodinger equation represents the work done on the electron by external forces, which can change the energy of the atomic electron. For reasons that are not very important right now, V is referred to as the "potential" acting on the electron.25 In the most general case, these forces can change with distance and time. The fact that these forces, and hence the potential V, can be time t- and s.p.a.ce x-dependent is reflected in the notation V(x,t) in this equation, which implies that V can take on different values at different points in s.p.a.ce x and at different times t. Sometimes the forces do not change with time, as in the electrical attraction between the negatively charged electron and the positively charged protons in the atomic nucleus. In this case we need only to know how the electrical force varies with the separation between the two charges, to determine the potential V at all points in s.p.a.ce.
I've belabored the fact that V can take on different values depending on which point in s.p.a.ce x we are, and at what time t we consider, because what is true of V is also true of the wave function . Mathematically, an expression that does not have one single value, but can take on many possible values depending on where you measure it or when you look, is called a "function." Anyone who has read a topographic map is familiar with the notion of functions, where different regions of the map, sometimes denoted by different colors, represent different heights above sea level. This is why is called a "wave function." It is the mathematical expression that tells me the value of the matter-wave depending on where the electron is (its location in s.p.a.ce x) and when I measure it (at a time t). Though as we'll see in the next chapter with Heisenberg, where and when get a little fuzzy with quantum objects.
For an electron near a proton, such as a hydrogen atom, the only force on the electron is the electrostatic attraction. Since the nature of the electrical attraction does not change with time, the potential V will depend only on how far apart in s.p.a.ce the two charges are from each other. One can then solve the Schrodinger equation to see what wave function . will be consistent with this particular V. The wave function will also be a mathematical function that will take on different values depending on the point in s.p.a.ce. Now, here's the weird thing (among a large list of "weird things" in quantum physics). When Schrodinger first solved this equation for the hydrogen atom, he incorrectly interpreted what represented-in his own equation!
Schrodinger knew that itself could not be any physical quant.i.ty related to the electron inside the atom. This was because the mathematical function he obtained for involved the imaginary number i. Any measurement of a real, physical quant.i.ty must involve real numbers, and not the square root of -1. But there are mathematical procedures that enable one to get rid of the imaginary numbers in a mathematical function. Once we know the wave function , then if we square it-that is, multiply it by itself so* = 2 (p.r.o.nounced "sigh-squared")-we obtain a new mathematical function, termed, imaginatively enough, 2.26 Why would we want to do that? What physical interpretation should we give to the mathematical function 2?
Schrodinger noticed that 2 for the full three-dimensional version of the matter-wave equation had the physical units of a number divided by a volume. The wave function itself has units of 1/(square root (volume)). This is what motivated the consideration of 2 rather than -for while there are physically meaningful quant.i.ties that have the units of 1/volume, there is nothing that can be measured that has units of 1/(square root (volume)). He argued that if 2 were multiplied by the charge of the electron, then the result would indicate the charge per volume, also known as the charge density of the electron. Reasonable-but wrong. 2 does indeed have the form of a number density, but Schrodinger himself incorrectly identified the physical interpretation of solutions to his own equation.
Within the year of Schrodinger publis.h.i.+ng his development of a matter-wave equation, Max Born argued that in fact 2 represented the "probability density" for the electron in the atom. That is, the function 2 tells us the probability per volume of finding the electron at any given point within the atom. Schrodinger thought this was nuts, but in fact Born's interpretation is accepted by all physicists as being correct.
What Schrodinger discovered was the quantum a.n.a.log of Newton's force = (ma.s.s) (acceleration). Newton said, in essence, you tell me the net forces acting on an object, and I can tell you where it will be (how the motion will change) at some other point in s.p.a.ce and at some later time. Schrodinger's equation a.s.serts that if you know the potential V (which can be found by identifying the forces acting on the electron), then I can tell you the probability per volume of finding the electron at some point in s.p.a.ce and time, now and in the past and future.
Once I know how likely it is to find an electron throughout s.p.a.ce, I can calculate its average distance from the proton in the nucleus, which we would call the size of the atom. The energy within the atom, its average momentum, and in fact anything I care to measure can be calculated using the Schrodinger equation. We know that the Schrodinger equation is a correct approach for determining the wave function for electrons in an atom, for it enables us to calculate properties of the atom that are in excellent agreement with experimental observations. This is the only true test of the theory and the only reason why we take seriously notions of matter-waves and wave functions.
Schrodinger first applied his equation to the simplest atom, hydrogen, with a nucleus of a single proton and only one orbiting electron. For the force acting on the electron he used the familiar law of electrostatic attraction, extensively confirmed as the correct description of the pull two opposite charges exert on each other. With no other a.s.sumptions or ad hoc guesses, his equation then yielded a series of possible solutions, that is, a set of different functions that corresponded to the electron having different probability densities. This is not unlike the set of different possible wavelengths that a plucked guitar string can a.s.sume. For each probability density there was a different average radius, and in turn a different energy. Obviously the solitary electron in the hydrogen atom could have only one energy value and only one average radius at any given time. What Schrodinger found was that there was a collection of possible energies that the electron could have, not unlike an arrangement of seats in a large lecture hall (shown schematically in Figure 15). There are some seats close to the front blackboard, some in the next row a little bit farther away, and so on all the way to seats so far from the front of the room that a student sitting here could easily sneak out of the cla.s.s with very little effort.
An atom with only one electron is akin to a lecture hall with only one student attending. If there is a tilt to the hall, so that the front of the room is at a lower level than the rear (as in many audi-toriums), then the student would lower his or her energy by sitting in the front seat in the front row. This would be the lowest energy configuration for the student in the lecture hall, and similarly Schrodinger found that a single electron would have a probability density that corresponded to it having the lowest possible energy value. This configuration for an atom is termed the "ground state." If the electron received additional energy, say, by absorbing light or through a collision with another atom, it could move from its seat closest to the front of the room to one farther away. With enough energy it could be promoted out of the auditorium entirely, and in this case it would be a free electron, leaving behind a positively charged ionized atom (which for the simple case of hydrogen would be just a single proton).
Figure 15: Cartoon sketch of the possible quantum states, represented as seats in a lecture hall that an electron can occupy, as determined by the Schrodinger equation for a single electron atom. In this a.n.a.logy the front of the lecture hall, at the bottom of the figure, is where the positively charged nucleus resides. Upon absorbing or releasing energy, the electron can move from one row to another.
Schrodinger's equation at once explained why atoms could absorb light only at specific wavelengths. There are only certain energies that correspond to valid solutions to the Schrodinger equation for an electron in an atom, pulled toward the positively charged nucleus by electrostatic attraction. Electrons ordinarily sit in the lowest energy level-the ground-state configuration. Upon absorbing energy from outside the atom, they can move from the row closest to the front of the hall to another row at a higher level, closer to the exit, provided the seats they move to are empty. Those who recall their high school chemistry may know that each "seat" can actually accommodate two electrons-this is discussed in some detail in Section 4. There is a precise energy difference between the row where the electron is sitting and the empty seat it will be promoted to. It must move from seat to seat and cannot reside between rows. Different atoms with different numbers of protons in their nucleus will have slightly different functions and different s.p.a.cing between the rows of possible seats, similar to each string on a guitar having a different fundamental frequency as well as different overtones.
One intriguing consequence of the Schrodinger equation is that it explained that electrons in an atom could have only certain energy values, and that all other electronic energies were forbidden. Schrodinger's solution has nothing to do with circular or elliptical orbits, but rather with the different possible probability densities the electron can have. Another way of stating this is that the probability of an electron having a "forbidden energy" is zero and thus will never be observed to occur.
Only if the energy supplied to the atom, for example, in the form of light, is exactly equal to this difference will the electron be able to move to this empty seat. Too much energy or too little would promote the electron between rows. Since the electron cannot be at these energy values (the probability of it occurring is zero), the electron cannot absorb the light of these energies. There will thus be a series of wavelengths of light that any given atom can absorb, or emit, when the electron moves from one row to another; all other light is ignored by the atom.
Now for the cool part. When Schrodinger used the electrostatic attraction between a proton and an electron for a simple hydrogen atom, he found a set of possible wave functions that corresponded to different probability densities. When he made sure that was "normalized"-which is a fancy way of saying that if you add up the probability density of the electron over all s.p.a.ce, the total has to be 100 percent-then his equation yielded a set of possible energy levels for the electron that exactly corresponded to the energy transitions experimentally observed in hydrogen. No a.s.sumptions about orbits, no ignoring the fact that electromagnetism requires the electron to radiate energy in a circular trajectory. All you have to tell the Schrodinger equation is the nature of the interaction between the negatively charged electron and the positively charged nucleus in the atom, and it automatically tells you the possible line spectra of light for emission or absorption.
Schrodinger's equation destroyed the notion of a well-defined elliptical trajectory for the electron and replaced it with a smoothly varying probability of finding the electron at some point in s.p.a.ce. In essence, physics had come full circle. At the start of the twentieth century, scientists thought that the atom consisted of a jelly of positive charges, in which electrons sat like marshmallows in a Jell-O dessert. In this model the atom emitted or absorbed light at very particular wavelengths as these wavelengths corresponded to the fundamental frequencies of oscillation of the trapped electrons. Rutherford demonstrated that the atom actually had a small positively charged nucleus, around which the electrons...o...b..ted. But this model could not explain what kept the electron from spiraling into the nucleus, nor did it account for the absorption line spectra. With Schrodinger the small positively charged nucleus remained, but the negatively charged electrons were like jelly smeared over the atom, in the form of a "probability density." The source of the "uncertainty" in the electron's location was accounted for by Werner Heisenberg, independently of Schrodinger, who was mixing business with pleasure in the Swiss Alps.
The Uncertainty Principle Made Easy.
Just prior to the introduction of the Schrodinger equation, Werner Heisenberg was developing an alternative approach to atomic physics. The wavelike nature of matter is also at the heart of the famous Heisenberg uncertainty principle. Heisenberg took a completely different tack to the question of how de Broglie's matter-waves could account for an atom's optical absorption line spectra. As we've discussed, the conventional view that the electron was executing cla.s.sical orbits around the nucleus could not be reconciled with the absence of light that should be continuously emitted by such an electron. Heisenberg's struggle to envision what the electron was doing and where it was located within the atom finally convinced him to give up and not bother trying to figure out where the electron was. This turned out to be a winning strategy.
Physics is an experimental science, and the development of quantum mechanics was driven by a need to account for atomic measurements that were in conflict with what was expected from electromagnetic theory and thermodynamics. Heisenberg realized that any theory of the atom needed only to agree with measurements, rather than make predictions that could never be tested! Who cares where the electron "really was" inside the atom? Could you ever definitely measure its location to confirm or disprove this prediction? If not, then forget about it. What could be measured? For one, the wavelengths of the light emitted from an atom in a line spectrum. Well then, let's construct a theory that describes the energy difference when an atom makes a transition from one state to another. Heisenberg's model for the atom consisted of a large array of numbers that characterized the different states the electron could be in, and rules that governed when the electron could go from one state to another. When compared to the observed spectra of light absorbed or emitted from a hydrogen atom, Heisenberg's approach agreed exactly with measurements.
In 1925 Heisenberg exiled himself to the German island of Helgoland in the southeast corner of the North Sea as he worked out the details of this approach. He had known for years that, in contrast to Schrodinger, he did his best work removed from any distractions. While the isolation suited his requirements for extended contemplation, his motivation for decamping to the island was a bit more mundane. Heisenberg suffered from severe allergies, and it was to escape the pollens of Gottingen that he traveled to the treeless Helgoland. The stark island's only recreational option consisted of mountains, of which the hiking enthusiast Heisenberg availed himself as he struggled with his theory.
On this island Heisenberg constructed arrays of values for the different states in which electrons in an atom could be found, and the rules for how they would make the transition between states. When he returned to Berlin and showed his preliminary efforts to Max Born, the older professor was initially confused. As he read and reread Heisenberg's work, trying to understand these transition rules, he had a strong sense of familiarity. Eventually Born and Pascual Jordan, another physicist at Gottingen, recognized that Heisenberg had independently developed a mathematical notation known as a "matrix" to describe his theory-a notation that had been invented a hundred years earlier by mathematicians interested in solving series of equations that had many unknown variables. This branch of mathematics is called linear algebra, or matrix algebra, and Heisenberg had unknowingly reinvented a wheel that had been rounded years earlier.
(This is always happening, by the way. More often than not we find in physics that the necessary mathematics to solve a particularly challenging problem has already been developed, frequently no less than a century earlier. The mathematicians are just doing what mathematicians do and are not trying to antic.i.p.ate or solve any physics problems. There are two notable exceptions: Newton had to invent calculus in order to test his predictions of celestial motion against observations, and modern string theorists are inventing the necessary mathematics simultaneously with the physics.) Heisenberg's approach using matrices is an alternative explanation for the behavior of electrons in atoms, for which he was awarded the n.o.bel Prize in Physics in 1932. Heisenberg's theory was published in 1925. Less than a year later Schrodinger introduced his matter-wave equation to account for the interactions of electrons inside the atom. The two descriptions of the quantum world do not appear to have anything in common, aside from the fact that they both accurately predict the observed optical line spectra for atoms. In 1926 Schrodinger was able to mathematically translate one approach into the other, demonstrating that the two descriptions are in fact equivalent. Both theories rely on de Broglie's matter-wave hypothesis, though neither takes the suggestion of elliptical electronic orbits literally. Both approaches make use of what we now would characterize as the electron's wave function and have prescriptions for how one can mathematically calculate the average momentum, the average position, and other properties of an electron in an atom. For our purposes, we do not have to go too deeply into either theory, as our goal is to understand how the concepts of quantum mechanics underlie such wonders of the modern age as the laser and magnetic resonance imaging.
There is one aspect of Heisenberg's theory that has generated way too much blather and misinformation to be ignored-that is, the famed uncertainty principle. If you'll bear with me, I'd like to take a brief detour to set the record straight about what the uncertainty principle does and does not mean. It's actually not too complicated, that is, once one accepts that there is a wavelike nature to matter.
The uncertainty principle posits a relations.h.i.+p between the uncertainty of the location of a particle and the uncertainty of its momentum. Heisenberg saw that the product of these two uncertainties must be bigger than a constant that turns out to be (wait for it) Planck's constant, h, divided by 4 times . Why Planck's constant is divided by 4 has to do with technical aspects of waves that need not concern us here. The fact that h turns up again and again when describing the atomic regime indicates that Planck's initial guess about the graininess of energy was on target, and by introducing h he discovered a new fundamental constant of nature, as important for understanding how the universe is put together as the value of the speed of light or the charge of an electron.
Right off the bat let me emphasize that the uncertainty principle does not restrict the precision with which I can measure the position of a particle, nor that of its momentum. Neither does it state that one can never measure the position and momentum of a particle simultaneously, but it does get to the heart of what such a measurement entails.
Suppose that I can experimentally determine the position of an electron and at the same time the momentum of this electron (I'll describe how in a moment). I may find that the electron is at a location x = 2.34528976543901765438 cm as reckoned from a given point, and the momentum has a value of p = 14.254489765539989021 kg-meter/sec. There are certainly a large number of decimal places in these measurements, but in order to convince myself that this is indeed the electron's position and momentum, I repeat the experiment, under exactly the same conditions. If those are in fact the true values of position and momentum, I should be able to reproduce all of those decimal places. Performing the experiment a second time, I now find that the electron's position is x = 2.3452891120987693790 cm and the momentum is p = 14.2544100876495832002 kg-meter/ sec. On the one hand, the first few digits in both the x and p measurements agree exactly with what was found before, but on the other hand after a few decimal places the overlap with the first observation disappears.
What I find, after repeating the experiment many more times, is that the average position of the electron is 2.32428 cm and the average momentum is 14.254 kg-meter/sec, as ill.u.s.trated in Figure 16. That is, the only measurements that one really can trust are of the average position and momentum. I can make a plot of the num-ber of times a particular value of the position is observed against the value of the position. Such a plot would look, after a large number of trials, like the familiar bell-shaped curve, known and feared by students throughout recorded history. The peak in the curve would represent the value of the position that was observed most frequently and would also indicate the average location of the electron. At the highest point, the curve is very narrow. Halfway down the curve, there is a width that is referred to as the "standard deviation." The standard deviation is an indication of how much we can trust the average value resulting from this bell-shaped curve. The bigger the standard deviation, the greater the possible uncertainty in the average value. It is these standard deviations, also referred to in mathematics as uncertainties, that are constrained by the Heisenberg uncertainty principle.
Figure 16: Plot of the histogram of measured positions (a) and momenta (b). The vertical dashed lines represent the average values of position and momentum, and the small arrows indicate the standard deviation for each measurement.
As an ill.u.s.tration of standard deviations, consider a cla.s.s of 100 students taking a final exam. If all 100 students receive exactly the same score of 50 points out of 100, then the average grade will be 50 and the standard deviation will be 0. That is, if you are told that the average grade is 50 with a standard deviation of 0, then there is no ambiguity or uncertainty in what any given student scored on the exam. Now, if 90 students receive a score of 50 points, while 5 students score a 55 and the remaining 5 receive a grade of 45, the average grade would remain 50, but now there is a small width to the distribution of grades. While the average value is still 50, some students have a score that differs from the average value. There is now some small but non-zero question as to whether a student chosen at random will have a grade equal to the average score of 50 or not. In an extreme case, all 100 students in the cla.s.s could receive different grades, from as low as 1 or 2, through 48, 49, 50, and 51, up to 98, 99, and 100. The average grade would still be 50,27 but now the distribution would range from 1 through 100. Only 1 student out of 100 would have an actual grade on the final exam that is exactly equal to the average score of the cla.s.s, and the large size of the standard deviation would indicate that the average grade was not a particularly meaningful or insightful indicator of any given student's performance. When dealing with large numbers, the two questions one must ask are: What is the average? and What is the standard deviation?
Returning to my simultaneous measurements of an electron's position and momentum, I found that after repeated trials there was a bell-shaped curve for the position, which gave its average location, and another bell-shaped curve for the momentum. It is the standard deviations of these two bell-shaped curves that the Heisenberg uncertainty principle addresses. Heisenberg's theory tells us that the standard deviation of the electron's position is connected to the standard deviation of the electron's momentum, so that changes in one affect the other. The widths of the two distributions in Figure 16 are not independent, and efforts to narrow one will necessarily broaden the other. Heisenberg calculated that the product of the two standard deviations cannot be smaller than h/4. Any experiment that manages to decrease the standard deviation of the momentum (for example) will necessarily broaden the standard deviation of the position.
Why would the standard deviation of an electron's position be related to the standard deviation of its momentum? Because of the de Broglie wave a.s.sociated with the motion of the electron. The wavelength of this matter-wave is in a sense a measure of how precisely we can say where the electron is, and this wavelength is connected to the electron's momentum by the relation (momentum) (wavelength) = h.
Figure 17: Sketch of two possible de Broglie matter waves for an electron. In the top curve, the electron is a.s.sociated with a single wave. As one needs only one wavelength to describe the wave, the momentum is perfectly known, but at the cost of an infinite uncertainty in the location of the electron. In the bottom curve many different waves, each with different wavelengths, have been added to yield a "wavepacket." The uncertainty in the spatial location of the electron is reduced, but there is a corresponding increase in uncertainty in the electron's momentum.
a.s.sume that the electron's momentum is precisely known, with absolutely no ambiguity. Then the average momentum is the momentum, just as in our cla.s.sroom example when all 100 students received the same grade on the final of 50. There is one value for the wavelength of the matter-wave for this perfectly known momentum, as shown in the top cartoon in Figure 17. A pure wave by definition extends forever, from one end of the universe to the other. Where would we say the location of the electron would be for such a perfect wave? Its average value might be well defined, but its standard deviations would be infinitely large.
The Heisenberg connection between the standard deviations of the position and momentum results from the fact that the location of the electron is connected to the wavelength of the matter-wave, which in turn is related to the object's momentum. In order to shrink the standard deviation of the electron's position, the electron's matter-wave should be zero except for a small region near the average position. But in order to construct a wave packet such as this, ill.u.s.trated by the lower sketch in Figure 17, one needs to add many waves together, all of slightly different wavelengths, so that they would destructively cancel out beyond a narrow region around the average position. Since the wavelength is connected to the momentum through (momentum) (wavelength) = h, adding many different wavelengths together is the same as saying that there is a broad range of momentum values for the localized electron. The tighter the limits with which we wish to specify the electron's location, that is, the smaller the position's standard deviation, the more different wavelengths we have to combine and the greater the corresponding standard deviation of the momentum. The two standard deviations are thus joined together, through the matter-wave relation (momentum) (wavelength) = h.
Of course, we can measure an electron's position to any arbitrary precision we wish-as long as we do not care about its momentum, and vice versa. Whether the electron has a well-defined momentum (with no standard deviation) or a well-defined position (also with no standard deviation) depends on what measurement I perform. There can be no answer until the question is posed, and how I ask the question determines what answer I'll obtain.
The bundle of waves described above, and shown in the lower sketch in Figure 17, is technically referred to as a "wave packet." It makes no sense to ask where the electron is located, on a scale smaller than the extent of the wave packet. How do I measure the momentum of an electron? One way would be to record its position at two different times. Knowing how far it moved, and how long it took to travel this distance, I can determine its velocity, and multiplying by the ma.s.s (a.s.suming I can avoid relativistic corrections) yields the momentum. How can I tell where the electron is at the two different times? It's easy to tell the velocity of an automobile-you just look at where it is at two different times. How do I look at an electron? The same way I look at a car-by s.h.i.+ning light on it and having the light be reflected to my eye (or some other detector). Cars are much larger than the wavelength of visible light, but to observe the electron I need light with a wavelength smaller than the extent of the electron's wave packet. The connection between wavelength and frequency for light is given by the simple relations.h.i.+p (frequency) (wavelength) = speed of light. Since the speed of light is a constant, the smaller the wavelength, the larger the frequency of the light, and from energy = h (frequency), the larger the energy of the light photon. Thus, to measure the momentum of an electron with a very small wave packet (small uncertainty in position), I must strike it with a very high-energy photon. When the photon reflects from the electron, the electron's recoil changes its momentum. When a careful mathematical a.n.a.lysis is performed, one finds that you cannot do better than the Heisenberg uncertainty principle. You may think there might be a clever scheme to get around this limitation of the wavelength of light to determine the electron's position and momentum-but many have tried and all have failed.
We might have been spared countless inane p.r.o.nouncements that "quantum mechanics has proven that everything is uncertain" if Heisenberg had simply named his principle something a little less catchy, such as "the principle of complementary standard deviations."
Armed now with an appreciation for the physical content of the famous uncertainty principle, we now consider the following cla.s.sic example of nerd humor: Werner Heisenberg is pulled over for speeding by a highway patrolman. The police officer walks over to Werner's car, leans over, and asks Heisenberg, "Do you know how fast you were going?"
Heisenberg replies, "No, but I know where I am!"
An understanding of average values and standard deviations of bell-shaped curves is also relevant to issues of climate change and explains why scientists are concerned about an increase in the average global temperature of a few degrees, which admittedly does not sound that menacing.
Consider a histogram plot of the temperature of North America, where the horizontal axis lists the average daily temperature, and the vertical axis charts the number of days per year for which that particular average temperature is recorded. We expect to find, and we do, a roughly bell-shaped curve, just as we have been discussing for grades in a large cla.s.s or measurements of an electron's position or momentum. There will be a large peak in the temperature histogram around the average daily temperature over the course of a year, and small tails at lower and higher temperatures.
What is the effect of the yearly average temperature being raised by a few degrees? The answer is easiest to see if we briefly return to the cla.s.sroom a.n.a.logy. Suppose we add 5 extra-credit points to every student's final exam paper. Thus, if the average had been a grade of 50 before, now it will be 55, and the lowest score will s.h.i.+ft from 0 to 5, while the highest possible grade will move from 100 to 105. A s.h.i.+ft in the average grade from 50 to 55 does not seem like much, and for most students there will be no significant effect. I am just adding a uniform 5 points to everyone's grade, so the shape of the curve does not change. In addition, I do not adjust the cutoff line for what grade merits an A, what score deserves a B, and so on. In this case I would find that by s.h.i.+fting the grades upward by only 5 points, the number of students that are at or above the A-cutoff threshold has increased dramatically. A minor s.h.i.+ft in the average, which does not have a large influence on the grade of the majority of the students, has a big impact on those students that were near but just below the threshold for a letter grade of A.
A small s.h.i.+ft in the average annual global temperature is akin to giving everyone in the cla.s.s 5 extra-credit points. There will thus be more days with higher-than-normal average temperatures (corresponding to those students whose exam performance warranted an A), and those days are what drive extreme weather situations. A s.h.i.+ft of the average upward by a few degrees is not a big deal on an average day and is even welcome in the winter in states such as Minnesota, where we would not have such extreme cold snaps. But other parts of the nation in the summer would see a greater number of days where it is hotter than normal. It takes a lot of energy to warm up a large ma.s.s of water such as the Gulf of Mexico, and it also takes a long time to cool it back down. The energy of these hotter-than-normal days can be viewed as "stored" in the ocean, and warmer water temperatures can provide energy for hurricanes, tropical storms, and other extreme events. Moreover, the more days with higher temperatures, the more ice will melt in northern regions. Fresh snow reflects 80 to 90 percent of all sunlight s.h.i.+ning on it, while liquid water absorbs (and stores) 70 percent of the sunlight. There is thus a positive feedback mechanism by which higher temperatures lead to additional warming. Just as in the case of the Heisenberg uncertainty principle, it's not the averages that matter so much as the width of the standard deviations. Those long tails will get us in the end.28
Why So Blue, Dr. Manhattan?
In Chapter 5 I stated that at least some of the amazing superpowers displayed by Dr. Manhattan in the graphic novel and motion picture Watchmen are a consequence of his having control over his quantum mechanical wave function. Now that we know a little bit more about wave functions, let's see how that might work.
While it is certainly true that all objects, from electrons, atoms, and molecules to baseb.a.l.l.s and research scientists, have a quantum mechanical wave function, one can safely ignore the existence of a matter-wave for anything larger than an atom. This is because the larger the ma.s.s, the larger the momentum-and the bigger the momentum, the smaller the spatial extent of the wave function. Anything bigger than an atom or a small molecule has such a large ma.s.s that its corresponding de Broglie wavelength is too small to ever be detected. So, right off the bat we must grant Dr. Manhattan a miracle exception from the laws of nature such that he can control his wave function's spatial extent independently of his momentum. While the de Broglie wavelength for an adult male is typically less than a trillionth trillionth of the width of an atom, Dr. Manhattan must be able to vary his wave function so that it extends a great distance from his body-even as far as the distance between the Earth and Mars!
The quantum mechanical wave function contains all the information about an object. If we want to know the object's average position, its average speed, its energy, its angular momentum for rotation about a given axis, and how these quant.i.ties will change with time, we perform various mathematical operations on the wave function, which yield calculated values for any measured characteristic of the object.
The "wave function" is so named because it is a mathematical function that has the properties of an actual wave. To review: In mathematics a "function" describes any situation where providing one input value leads to the calculation of a related number. The simple equation relating distance to time spent driving at a constant speed-that is, distance = speed time-is a mathematical function. If your speed is 60 miles per hour, and if you tell me the time you spent driving-1/2 hour, 1 hour, 3 hours-then this simple function enables me to calculate the distance you have covered (30 miles, 60 miles, or 180 miles, respectively, in this example). Most mathematical functions are more complicated than this, and sometimes they get as involved as the Schrodinger wave function, but they all relate some input parameter or parameters to an output value. For the quantum mechanical wave function of an electron in an atom, if you tell me its location in three-dimensional s.p.a.ce relative to the nucleus, then the solution to the Schrodinger equation returns the amplitude of the electron's wave function at that point in s.p.a.ce and time.
What does it mean to say that the wave function has the properties of a wave, such as a vibrating string or the series of concentric circles created on the surface of a pond when a rock is tossed into the water? Waves are distinguished by having amplitudes that vary periodically in s.p.a.ce and time. Consider the ripples created when a rock is tossed into a pond. At some points of the wave there are crests, where the height of the water wave is large and positive (that is, the surface of the water is higher than normal); at some points there are troughs, where the height of the water is lower than normal; and in other regions the amplitude of the wave is zero-the height of the water's surface is the same as it would be without the rock's disturbance.
The amplitude of the peaks and valleys typically becomes smaller with distance from the source of the waves. This is why on the California sh.o.r.e we don't notice if a rock is dropped into the center of the Pacific Ocean. Certain large disturbances can create tsunamis that maintain large amplitudes even when traveling great distances. Dr. Manhattan, presumably, is able to change the amplitude of his quantum mechanical wave function so that it can have an appreciable amplitude at some large distance away from him. This would be how he teleports, though in quantum mechanics we would say that he is "tunneling."
Schrodinger's equation enables us to calculate the wave function of an object as a function of the forces acting on it. If there are no net forces, the electron, for example, can have uniform straight-line motion, with a well-defined de Broglie wavelength determined by its momentum. If this electron strikes a barrier and it lacks sufficient energy to go over the obstacle, then the electron will be reflected, bouncing off the barrier and returning from where it came.
We are familiar with such wave phenomena whenever we use a mirror. Light waves move in straight lines, pa.s.sing through the gla.s.s covering of the mirror, until they reach the silvered backing. Unable to penetrate the metal, the waves are reflected back in another straight-line trajectory, along a path that makes the same angle with a line perpendicular to the mirror's surface as the incoming beam.
In fact, one does not need the metal backing to see this reflection effect. We all know that a single pane of gla.s.s can act like a mirror, when we look out the window from a well-lit room at night. In this case just the difference in optical media, gla.s.s and air, can cause light reflection, particularly when we look at the window at an angle. The reflection is more noticeable if the direction we are looking, relative to the gla.s.s surface, is larger than a particular angle that depends on the optical properties of gla.s.s and air. When we place our face against the gla.s.s, this reflection effect goes away, for then most of the light rays that we see from outside travel perpendicular to the surface. Light travels slower in gla.s.s than in air (more on this in a moment), and this difference in light velocities (characterized by the material's index of refraction, for technical reasons) accounts for the reflection effect. This can occur during daytime as well but is less noticeable when more light comes into the room from the outside than goes out from the interior.
Suppose that we are looking at the window at night from the interior of a strongly lit room. The gla.s.s reflects our image as if it were a conventional mirror. Now imagine a second sheet of gla.s.s placed behind the first, as in a double-paned window, only the separation between the two sheets isn't a quarter of an inch, but more like a millionth of a centimeter. In this case, even though the light would have been completely reflected without the second sheet of gla.s.s, the presence of the second pane enables some of the light to pa.s.s through both sheets of gla.s.s, even though they are not touching each other. This phenomenon is a hallmark of the wavelike properties of light (so for the sake of argument we will ignore for the moment that light is actually comprised of discrete photons). It turns out that the light wave is not completely reflected at the first gla.s.s-air interface, but instead a small amount of the oscillating electric and magnetic fields leak out into the air. The small leakage is limited