Quantum Man: Richard Feynman's Life in Science

by Lawrence M. Krauss

  CHAPTER 2

  The Quantum Universe

  I was always worried about the physics. If the idea looked lousy, I said it looked lousy. If it looked good, I said it looked good.

  —RICHARD FEYNMAN

  Feynman was fortunate to have stumbled upon Ted Welton in his sophomore year at MIT, while both were attending, as the only two sophomores, an advanced graduate course in theoretical physics. Kindred spirits, each had been checking advanced mathematics texts out of the library, and after a brief period of trying to outdo each other, they decided to collaborate “in the struggle against a crew of aggressive-looking seniors and graduate students” in the class.

  Together they pushed each other to new heights, passing back and forth a notebook in which each would contribute solutions and questions on topics ranging from general relativity to quantum mechanics, each of which they apparently had taught themselves. Not only did this encourage Feynman’s seemingly relentless quest to derive all of physics on his own terms, but also it provided some object lessons that would stay with him for the rest of his life. One in particular is worth noting. Feynman and Welton tried to determine the energy levels of electrons in a hydrogen atom by generalizing the standard equation of quantum mechanics, called the Schrödinger equation, to incorporate the results of Einstein’s special relativity. In so doing they rediscovered what was actually a well-known equation, the Klein-Gordon equation. Unfortunately, after Welton urged Feynman to apply this equation to understand the hydrogen atom, the attempt produced results that completely disagreed with experimental results. This is not surprising because the Klein-Gordon equation was known to be the wrong equation to use to describe relativistic electrons, as the brilliant theoretical physicist Paul Dirac had demonstrated only a decade earlier, in the process of earning the Nobel Prize for deriving the right equation.

  Feynman described his experience as a “terrible” but very important lesson that he never forgot. He learned not to rely on the beauty of a mathematical theory or its “marvelous formality,” but rather to recognize that the test of a good theory was whether one could “bring it down against the real thing”—namely, experimental data.

  Feynman and Welton were not learning all of physics completely on their own. They also attended classes. During the second semester of their sophomore year they had sufficiently impressed the professor of their theoretical physics course, Philip Morse, that he invited the two of them, along with another student, to study quantum mechanics with him in a private tutorial one afternoon a week during their junior year. Later he invited them to start a “real research” program in which they calculated properties of atoms more complicated than hydrogen, and in the process they also learned how to work the first generation of so-called calculating machines, another skill that would later serve Feynman well.

  By the time of his final year as an undergraduate, Feynman had essentially mastered most of the undergraduate and graduate physics curricula, and he had already become excited enough by the prospect of a research career that he made the decision to proceed on to graduate school. In fact, his progress had been so impressive that during his junior year the physics department recommended that he be granted a bachelor’s degree after three years instead of four. The university denied the recommendation, so instead, during his senior year, he continued his research and wrote a paper on the quantum mechanics of molecules that was published in the prestigious Physical Review, as was a paper on cosmic rays. He also took some time to reinforce his fundamental interest in the applications of physics, and enrolled in metallurgy and laboratory courses—courses that would later serve him well in Los Alamos—and even built an ingenious mechanism to measure the speeds of different rotating shafts.

  Not everyone was convinced that Feynman should take the next major step in his education. Neither of his parents had completed a college education, and the rationale for their son completing yet another three or four years of study beyond an undergraduate degree was unclear. Richard’s father, Melville Feynman, visited MIT in the fall of 1938 to speak to Professor Morse and ask if it was worth it, if his son was good enough. Morse answered that Feynman was the brightest undergraduate student he had ever encountered, and yes, graduate school not only was worth it, but was required if Feynman wanted to continue a career in science. The die was cast.

  Feynman’s preference was to stay on at MIT. However, wise physics professors generally encourage their students, even their best ones, to pursue their graduate studies at a new institution. It is important for students to get a broad exposure early in their career to the different styles of doing science, and to different focuses of interest, as spending an entire academic career at one institution can be limiting for many people. And so it was that Richard Feynman’s senior dissertation advisor, John Slater, insisted that he go to graduate school elsewhere, telling him, “You should find out what the rest of the world is.”

  Feynman was offered a scholarship to Harvard for graduate school without even applying because he had won the William Lowell Putnam Mathematical Competition in 1939. This is the most prestigious and demanding national mathematics contest open to undergraduates, and was then in its second year. I remember when I was an undergraduate the very best mathematics students would join their university’s team and solve practice problems for months ahead of the examination. No one solves all the problems on the exam, and in many years a significant fraction of the entrants fail to solve a single problem. The mathematics department at MIT had asked Feynman to join MIT’s team for the competition in his senior year, and the gap between Feynman’s score and the scores for all of the other entrants from across the country apparently astounded those grading the exam, so he was offered the Harvard prize scholarship. Feynman would later sometimes feign ignorance of formal mathematics when speaking about physics, but his Putnam score demonstrated that as a mathematician, he could compete with the very best in the world.

  But Feynman turned down Harvard. He had decided he wanted to go to Princeton, I expect for the same reason that so many young physicists wanted to go there: that was where Einstein was. Princeton had accepted him and offered him a job as future Nobel laureate Eugene Wigner’s research assistant. Fortunately for Feynman, he was assigned instead to a young assistant professor, John Archibald Wheeler, a man whose imagination matched Feynman’s mathematical virtuosity.

  In a remembrance of Feynman after his death, Wheeler recalled a discussion among the graduate admissions committee in the spring of 1939, during which one person raved about the fact that no one else applying to the university had math and physics aptitude scores anywhere near as high as Feynman’s (he scored 100 percent in physics), while another member of the committee complained at the same time that they had never let anyone in with scores so low in history and English. Happily for the future of science, physics and math prevailed.

  Interestingly, Wheeler did not describe another key issue, of which he may not have been aware: the so-called Jewish question. The head of the physics department at Princeton had written to Philip Morse about Feynman, asking about his religious affiliation, adding, “We have no definite rule against Jews but have to keep their proportion in our department reasonably small because of the difficulty of placing them.” Ultimately it was decided that Feynman was not sufficiently Jewish “in manner” to get in the way. The fact that Feynman, like many scientists, was essentially uninterested in religion never arose as part of the discussion.

  MORE IMPORTANT THAN all of these external developments, however, was the fact that Feynman had now proceeded to the stage in his education where he could begin to think about the really exciting stuff—namely, the physics that didn’t make sense. Science at the forefront is always on the verge of paradox and inconsistency, and like a bloodhound, great physicists focus precisely on these elements because that is where the true quarry lies.

  The problem that Feynman later said he “fell in love with”
as an undergraduate had been a familiar part of the centerpiece of theoretical physics for almost a century: the classical theory of electromagnetism. Like many deep problems, it can be simply stated. The force between two like charges is repulsive, and therefore it takes work to bring them closer together. The closer they get, the more work it takes. Now imagine a single electron. Think of it as a “ball” of charge with a certain radius. To bring all the charge together at this radius to make up the electron would thus take work. The energy built up by the work bringing the charge together is commonly called the self-energy of the electron.

  The problem is that if we were to shrink the size of the electron down to a single point, the self-energy associated with the electron would go to infinity, because it takes an infinite amount of energy to bring all the charge together at a single point. This problem had been known for some time and various schemes had been put together to solve it, but the simplest was to assume that the electron really wasn’t confined to a single point, but had a finite size.

  By early in the twentieth century this issue took on a different perspective, however. With the development of quantum mechanics, the picture of electrons, and electric and magnetic fields, had completely changed. So-called wave-particle duality, for example, a part of quantum theory, said that both light and matter, in this case electrons, sometimes behaved as if they were particles and sometimes as if they were waves. As our understanding of the quantum universe grew, while the universe also got stranger and stranger, nevertheless some of the key puzzles of classical physics disappeared. But others remained, and the self-energy of the electron was one of them. In order to put this in context, we need to explore the quantum world a little bit.

  Quantum mechanics has two central characteristics, both of which completely defy all of our standard intuition about the world. First, objects that are behaving quantum mechanically are the ultimate multitaskers. They are capable of being in many different configurations at the same time. This includes being in different places and doing different things simultaneously. For example, while an electron behaves almost like a spinning top, it can also act as if it is spinning around in many different directions at the same time.

  If an electron acts as if it is spinning counterclockwise around an axis pointing up from the floor, we say it has spin up. If it is spinning clockwise, we say it has spin down. At any instant the probability that an electron has spin up may be 50 percent, and the probability that it has spin down may be 50 percent. If electrons behaved as our classical intuition would suggest, the implication would be that each electron we measure has either spin up or spin down, and that 50 percent of the electrons will be found to be in one configuration and 50 percent in the other.

  In one sense this is true. If we measure electrons in this way, we will find that 50 percent are spin up and 50 percent are spin down. But, and this is a very important but, it is incorrect to assume that each electron is in one configuration or another before we make the measurement. In the language of quantum mechanics, each electron is in a “superposition of states of spin up and spin down” before the measurement. Put more succinctly, it is spinning both ways.

  How do we know that the assumption that electrons are in one or another configuration is “incorrect”? It turns out that we can perform experiments whose results depend on what the electron is doing when we are not measuring it, and the results would come out differently if the electron had been behaving sensibly, that is, in one or another specific configuration between measurements.

  The most famous example of this involves shooting electrons at a wall with two slits cut into it. Behind the wall is a scintillating screen, much like the screen on old-fashioned vacuum-tube televisions, that lights up wherever an electron hits it. If we don’t measure the electrons between the time they leave the source and when they hit the screen, so that we cannot tell which slit each electron goes through, we would see a pattern of bright and dark patches emerge on the rear screen—precisely the kind of “interference pattern” that we would see for light or sound waves that traverse a two-slit device, or perhaps more familiarly, the pattern of alternating ripples and calm that often results when two streams of water converge together. Amazingly, this pattern emerges even if we send only a single electron toward the two slits at any time. The pattern thus suggests that somehow the electron “interferes” with itself after going through both slits at the same time.

  At first glance this notion seems like nonsense, so we alter the experiment slightly. We put a nondestructive electron detector by each slit and then send the electrons through. Now we find that for each electron, one and only one detector will signal that an electron has gone through at any time, allowing us to determine that indeed each electron goes through one and only one slit, and moreover we can determine which slit each electron has gone through.

  So far so good, but now comes the quantum kicker. If we examine the pattern on the screen after this seemingly innocent intervention, the new pattern is completely different from the old pattern. It now resembles the pattern we would get if we were shooting bullets at such a screen through the two-slit barrier—namely, there will be a bright spot behind each slit, and the rest will be dark.

  So, like it or not, electrons and other quantum objects can perform classical magic by doing several different things at the same time, at least as long as we do not observe them in the process.

  The other fundamental property at the heart of quantum mechanics involves the so-called Heisenberg uncertainty principle. What this principle says is that there are certain combinations of physical quantities, such as the position of a particle and its momentum (or speed), that we cannot measure at the same instant with absolute accuracy. No matter how good our microscope or measuring device is, multiplying the uncertainty in position by the uncertainty in momentum never results in zero; the product is always bigger than some number, and this number is called Planck’s constant. It is this number that also determines the scale of the spacing between energy levels in atoms. In other words, if we measure the position very accurately so that the uncertainty in position is small, that means our knowledge of the momentum or speed of the particle must be very inaccurate, so that the product of the uncertainty in position and the uncertainty in momentum exceeds Planck’s constant.

  There are other such “Heisenberg pairs,” like energy and time. If we measure the quantum mechanical state of a particle or an atom for a very short time, then there will be a big uncertainty in the measured energy of the particle or atom. In order to measure the energy accurately, we have to measure the object over a long time interval, in which case we cannot say precisely when the energy was being measured.

  If this weren’t bad enough, the quantum world gets even weirder once we add Einstein’s theory of special relativity into the mix, in part because relativity puts mass and energy on the same footing. If we have enough energy available, we can create something with mass.

  So, if we put all of these things together—quantum multiplexing, the Heisenberg uncertainty principle, and relativity—what do we get? We get a picture of electrons that is literally infinitely more confusing than the one presented by the classical theory, which already led to an infinite self-energy for the electron.

  For example, whenever we try to picture an electron, it doesn’t have to be just an electron! To understand this, let’s return back to classical electromagnetism. One of the key features at the heart of this theory is the fact that if we shake an electron, it will emit elecromagnetic radiation, like light, or radio waves. This great discovery resulted from the groundbreaking nineteenth-century experiments of Michael Faraday, Hans Christian Oersted, and others, and the groundbreaking theoretical work of James Clerk Maxwell. Quantum mechanically, this observed phenomenon must still be predicted because if quantum mechanics is to properly describe the world, its predictions had better agree with observations. But the key new feature here is that quantum
mechanics tells us to think of the radiation as being made up of individual quanta, or packets of energy, called photons.

  Now let’s return to the electron. The Heisenberg principle tells us that if we measure the electron for some finite time, there remains some finite uncertainty in knowing its exact energy. But if there is some uncertainty, how do we know we are measuring only the electron? For example, if the electron emits a photon carrying very little energy, the total energy of the system will change, albeit very slightly. But if we don’t know the exact energy of the system, then we cannot say whether it has or hasn’t emitted a low-energy photon. So what we are measuring really could be the energy of the electron plus a photon that it has emitted.

  But why stop there? Perhaps the electron has emitted an infinite number of very-low-energy photons? If we watch the electron for long enough, we can both measure its energy very accurately and put a photon counter nearby to see if there are any photons around. In this case, what will have happened to all the photons that were traveling along with the electron in the interim? Simple: the electron can absorb all those photons before we get a chance to measure them.The kind of photons that an electron can emit and reabsorb on a timescale so short that we cannot measure them are called virtual particles, and as I will describe later, Feynman recognized that when we include the effects of both relativity and quantum mechanics, there is no getting away from the existence of these particles. So when we think of an electron moving around, we now have to think of it as a pretty complicated object, with a cloud of virtual particles surrounding it.

  Virtual particles play another important role in the quantum theory of electromagnetism. They change the way we think of electric and magnetic fields and the forces between particles. For example, say an electron emits a photon. This photon can then in turn interact with another particle, which can absorb it. Depending on the energy of the photon, this will result in a transfer of energy and momentum from one electron to another. But that is what we normally describe as the manifestation of the electromagnetic force between these two charged particles.