The Amazing Story of Quantum Mechanics

by Kakalios, James

  We have made use of the intrinsic angular momentum of fundamental subatomic particles when determining which form of quantum statistics—Fermi-Dirac or Bose-Einstein—they would obey. In this way the internal spin is crucial to understanding the nature of metallic or insulating solids but was not employed directly when describing the physics of diodes or transistors. Associated with the spin is a small intrinsic magnetic field that enabled Stern and Gerlach to measure the spin in the first place (Chapter 4), and remotely probing the magnetic field from the spin of nuclear protons enables magnetic imaging.

  We are composed mostly of water—molecules consisting of an oxygen atom bound to two hydrogen atoms. Each hydrogen atom has a single proton in its nucleus. The intrinsic spin of the proton is ±ℏ/2, and there is a small magnetic field associated with each proton. The magnetic field has a north and south pole (Chapter 4, Figure 10), and when we place the water molecule in an external magnetic field, the hydrogen atom’s proton points either in the same direction as the applied field (that is, its north pole points up while the lab magnetic field does likewise) or in the opposite direction (its north pole points down while the external field’s north pole points up). The proton in the nucleus has a series of available quantum levels, just as the electron has its own series of possible states. If the proton’s magnetic field is in the same direction as the external magnetic field, it is in a lower energy state. If the proton’s magnet is opposite to the external magnetic field, then it will have a higher energy, as it takes work to rotate it to the lower-energy, aligned configuration. As indicated in Figure 48, the energy level that is occupied by the single proton can thus be split into two energy values by placing the hydrogen atom between the poles of a strong magnet, with the proton’s energy being lower than what we find with no outside field or the energy being higher, depending on whether the proton’s magnetic field points with or against the external field, respectively.

  Figure 48: Sketch of the energy level of a single proton in the nucleus of a hydrogen atom (a) when no outside magnetic field is applied and (b) when a field is present. In the second case the proton’s energy is lowered if its own intrinsic magnetic field points in the same direction as the outside magnet, and the energy is higher if it points in an opposite direction. In this figure the proton is indicated with its spin aligned with the external magnetic field and thus in the lower energy state. If the spin were opposite to the external field, the proton would reside in the higher energy state.

  Say a proton is aligned in the same direction as the external magnetic field, in the lower-energy split state (Figure 48b). If I provide energy, in the form of a photon, I can promote the proton to the higher-energy state, which corresponds to the proton’s magnetic field being opposite to the outside magnet. This resonant absorption is entirely analogous to the line spectra (Chapter 5, Figure 13) for electronic energy levels in an atom. In essence the photon provides energy to flip the proton’s internal magnet, from pointing up to pointing down (for example). It is as if I had a top that was spinning clockwise, and with an appropriate pulse of energy I caused its direction to reverse, so that it was now rotating counterclockwise.

  The bigger the external magnetic field, the larger the energy splitting. That is, it requires more energy to flip the orientation of the proton’s magnetic field if it is in a large external field than in a weak field. If the hydrogen atom is placed in a magnetic field roughly twenty thousand to sixty thousand times stronger than the Earth’s magnetic field,68 then the separation in energy between when the proton is aligned with and against the external field is less than a millionth of an electron Volt. In comparison, the binding energy holding the hydrogen atom to the oxygen atom in the water molecule is nearly five electron Volts. Recall from Chapter 2 Einstein’s suggestion that the energy of a photon is proportional to its frequency (E = h × f). A photon capable of promoting the proton from one magnetic orientation to the other (as in Figure 48b) is in the radio portion of the electromagnetic spectrum. As this form of electromagnetic radiation penetrates through a person (which is why you can hear your transistor radio even when you place your body between it and the broadcast antenna), this energy region is well suited to probing the proton’s orientation within a person.

  The idea begins to form. Place a person in a large magnetic field, strong enough to generate an appreciable energy splitting for the protons in the water molecules that are in every cell in his or her body. Direct a transmitter of radio waves at the person, and the more photons that are absorbed, promoting a proton from one magnetic orientation to the other, the more water molecules there must be. How do we determine whether a radio wave is absorbed or not? A high exposure will transfer many protons from the lower energy state to the higher energy level. When the radio-frequency light is turned off, the hydrogen atom’s protons flip back down to the lower energy state, emitting photons as they do. In this way the person “glows in the dark,” emitting radio-frequency light that is detected and is the hallmark of the resonant absorption. The number of aligned protons throughout the bulk of the person’s body may thus be measured.69

  How do we obtain spatial resolution throughout a cross section of the person? By varying the strength of the magnetic field. Make the magnetic field very small at the left-hand side of the person and very large on the right, increasing linearly from one side to the other. As the energy spacing depends on the strength of the external magnet, the separation between levels will be small on the left and grow to a larger energy gap on the right. Consequently, the minimum photon energy that will induce a transition will be larger on the right than on the left. By varying the frequency of the radio signal, one can determine the amount of absorption on the left, middle, and right of the person. By using secondary magnets in the cylinder that encloses the person, information on the proton density with full spatial resolution can be obtained. As shown in Figure 49, this imaging of the magnetically induced resonant absorption enables us to probe the inner secrets of a three-pack of chocolate peanut butter cups, confirming, through application of advanced quantum mechanics, that there is indeed delicious peanut butter within the chocolate coating.

  Figure 49: Magnetic Resonance Image of three packets of peanut butter cup candy. The difference in spin relaxation times provides a basis for contrast between the chocolate coating and the interior filling, confirming the presence of the peanut butter inside the candy without having to bite into the candy (not that we wouldn’t be willing to make such sacrifices for science!). Courtesy of Professor Bruce Hammer at the University of Minnesota.

  But all cells in the body contain water, so there will be strong proton absorption at all points in the body. Where does the contrast come from? There are other elements that exhibit a magnetically induced resonance absorption signal, such as sodium, phosphorus, carbon, oxygen, nitrogen, and calcium. These elements have radio resonances at different frequencies than hydrogen and can thus be distinguished from the single proton signal. However, a more powerful technique involves not the magnetically induced signal, but the manner in which it goes away and then returns.

  When the magnetic field is applied, many of the hydrogen atom’s protons in the water molecules will line up with the external field, so that nearly all of the lower energy states will be occupied, and the higher energy states, corresponding to the proton’s magnetic field opposing the applied field, will be less occupied.70 Under a continuous exposure of radio-frequency light, more and more protons are promoted to the higher energy state, until the situation is reached when the average number in the upper state (with a magnetic field pointing down) is equal to the number in the lower state (with a magnetic field pointing up). At this stage, we have for the collection of protons an equal number with their north poles pointing up as we have with their north poles pointing down. The total net magnetization of the protons will therefore be zero. If we now stop the continuous illumination, the protons in the higher energy state will relax back to the lower energy configuration. The characteris
tic time that this will take is highly sensitive to the local environment in which the particular water molecule resides, as the interaction of the proton’s magnetic field with thermal vibrations of other atoms and with the magnetic field from other elements in its vicinity determines how hard or easy it is to polarize. It was discovered through careful experimentation that the time dependence of the restoration of the net magnetization is different for the various tissues in the body, providing a basis for contrast in the resulting images. One can inspect for blood-vessel blockages, cysts, or growths, and determine whether or not tumors are benign, based on differences in magnetization times. By careful examination of the time dependence of how the protons resume their original magnetization, a thorough diagnosis, which previously would have required X-ray eyes, is possible.

  There are of course a host of complex technical issues that go into generating a three-dimensional image using MRI. I never promised I would tell you how to construct your very own imaging device, only that I would explain the essential quantum mechanics that underlies such a process. Needless to say, all of the above would be useless without high-speed computers utilizing solid-state integrated circuits, to record, store, and analyze the radio-frequency absorption data. So in a sense, quantum mechanics enables MRI machines at two separate levels.

  Dr. James Xavier could have saved himself a great deal of grief with his experimental eye drops by using an MRI to perform diagnoses on patients by peering at their inner organs. Similarly, Professor Charles Xavier (no relation), mutant leader of the X-Men and the world’s most powerful telepath, can read people’s thoughts. While this is not possible, by employing functional magnetic resonance imaging (fMRI), we can determine what regions of the brain a person is using, and from that make inferences as to what they are thinking.71

  All cells in your body have a function and require energy when carrying out their designated tasks. Nerve cells—neurons—process information through the generation and transmission of voltages and ionic currents. When we eat, we ingest molecules originally generated by plants that contain stored chemical energy. The plants utilized the energy in photons from the sun to construct complex sugar molecules. The mitochondria in every cell synthesize adenosine triphosphate (ATP) from these sugars and thereby release some of that stored energy, which the cell can then use to perform various functions. The chemical trigger for the construction of ATP is the incorporation of oxygen molecules and the release of carbon dioxide. Consequently, whenever a cell is actively working, in particular for a prolonged period of time, there is an increase in blood flow to this cell, in order to maintain sufficient oxygen levels for ATP production. By looking where the blood is flowing, we can determine those cells that are most active.

  Neurons do not store glucose, and consequently within a few seconds after you start some heavy thinking, there is an increase in blood flow to the region containing the active neurons. The brain has the same relationship to the body as the United States has to the rest of the world—the brain is roughly 3 percent of the total body mass, while it consumes 20 percent of the energy expended. You use different portions of your brain depending on the task you are performing—sitting while reading this book, walking while reading this book, or showering while reading this book. Thus, by monitoring which regions of the brain are receiving more oxygenated blood, one can ascertain what activity a person is engaged in, without having to look at the person directly. The true power of this technique for determining brain activity from the variations in blood flow involves distinguishing between cerebral tasks—mentally doing arithmetic compared to recalling a pleasant summer memory. In this situation the beatific smile on the person’s face would not betray which task was being performed.

  When an MRI scan is taken of a person’s brain, spatial resolution on the order of millimeters and time resolution of one to four seconds are possible. When red blood cells are carrying oxygen, they are diamagnetic—which means that their internal magnets orient opposite to the direction of an external magnetic field. Conversely, deoxygenated hemoglobin is paramagnetic; that is, it will align in the same direction as an external field but will have no net magnetization in the absence of an applied magnetic field. Using magnetic resonance imaging, one can examine a region deep within the cerebral cortex and determine its rate of blood flow. By measuring the time dependence of the variation in magnetization when the saturating radio-frequency radiation is removed, one can distinguish which regions of the brain are in high need of additional oxygen and energy.

  Alfred Bester wrote of the difficulty that Ben Reich faced plotting and carrying out an undetected murder in a society where nearly everyone, particularly the police, is telepathic in his 1953 novel The Demolished Man. The ability to read minds, to know what another person is silently thinking, has long been a hallmark of science fiction stories, predating the pulps and continuing into the present. The cooperative behavior exhibited by the characters in Theodore Sturgeon’s novels More Than Human and The Cosmic Rape, and the Hammer film Village of the Damned, presumes the ability to remotely link neuronal activity for various individual agents. Certainly, the apparatus for a functional MRI device is considerably larger than the compact helmets for “mind reading” frequently depicted in science fiction magazines and comic books, and this technique provides information only about blood flow in the brain. While many remain unconvinced, some believe that this technique may someday serve as an accurate lie detector, enabling us to directly discern a person’s thoughts and intentions. Quantum mechanics brought us a world unimagined by the science fiction stories of fifty years ago, and it may now actually start bringing aspects of the science fictional world into reality.

  SECTION 6

  THE WORLD OF TOMORROW

  CHAPTER TWENTY

  Coming Attractions

  I have described the basic concepts underlying quantum mechanics and have discussed how these principles account for the properties of single atoms, nuclei, and many-body systems, such as metals and semiconductors. By elucidating the physics of the laser, the diode, the transistor, and disc drives, we now have an understanding of the basic building blocks of such modern technology as laptops, DVDs, and cell phones, which, for many, are part of everyday life in the twenty-first century. All of us routinely make use of devices and applications that would not be possible without the understanding of nature provided by quantum mechanics.

  We have obviously not gone into any detailed descriptions as to how consumer products, such as a computer, operate. By combining the electrical-current inputs in a series of transistors and diodes in ingenious ways, one can arrange it so that two high currents cancel each other out and lead to either a low current (two “ones” combine to form a “zero”) or a high current (two “ones” yield another “one”). Similarly, if one current level is high and the other is low, then a circuit can be constructed so that the output is either high or low, depending on the required logic operation. In this way the “ones” and “zeros” in the computer can be manipulated. A full discussion of Boolean mathematics, logic gates, and data storage and processing employed in a computer would be fascinating (in my nerdy opinion), but it would involve no new principles or applications of quantum mechanics.

  Nevertheless, I would like to note in passing that sometimes the physics from Section 3 (radioactivity) interferes with the physics of Section 5 (solid state devices). Few have not experienced the frustration of having a computer program freeze or crash for no particular reason, solved only by a rebooting of the operating system. Sometimes the source of the problem turns out to be thorium, a radioactive element that is a contaminant (thorium is as common as lead) in the circuitry packaging. When the thorium nucleus decays it emits an alpha particle and the high-energy helium nucleus can disrupt the current in an unlucky transistor. The loss of information in the middle of a calculation often requires that the entire program be restarted from scratch. What quantum mechanics giveth, quantum mechanics taketh away.

  Similarly,
the scheme by which mobile phones send and receive electromagnetic signals to radio towers that then connect to a land-based call router is rather clever. Present models use very sophisticated protocols to determine the optimal region, or “cell,” with which to transmit the call, but the basic quantum mechanics in the cell phone itself essentially involves applications of transistors and diodes. The heart of the cordless phone is the analog-to-digital converter, which takes a variable voltage generated when spoken sound waves are transformed into electrical signals (as occurs in a conventional landline phone) and, through what is effectively an operational amplifier (a series of transistors and resistors), converts this variable voltage to either a “one” or a “zero.” Obviously, devices must also exist that operate in reverse, so that a digital-to-analog conversion can transform the received signal into the variable sound we hear.

  One really cool feature in the latest generations of cell phones is the touch screen. While many simple touch screens, as in kiosk information displays or automatic teller machines, detect the alteration in electric field caused by the electrical conductance and capacitance of your finger, the newest multitouch versions send an LED-generated infrared light beam skimming along the inner surface of the screen. Touching the screen causes some of the infrared light to be scattered, and the location of your finger is ascertained by determining which photodetector behind the screen collects the scattered light. Here again, all the quantum physics in a touch screen is represented in the infrared light-emitting diodes and photodetectors.