with a nucleus at the center and a bunch of circles around it to represent electron orbits, then according to quantum theory, those circles could not be of just any radius; the electron orbits can have only certain fixed allowed radii. This means that in the figure, if the circles drawn correspond to the smallest three allowed orbits, then we cannot draw some other circles in between them to create some intermediate orbits. The situation is just like that for the floors in a multistoried building. Suppose each floor is ten feet high, then people can occupy rooms at ten, twenty, or thirty feet of elevation from the ground (assuming the ground floor is a garage), as shown in , but nobody can be in a room fifteen feet above the ground, because there is no such floor. It is likewise with electrons in their orbits. Electrons in the allowed orbits are in their stationary states, and they would remain there forever, unless disturbed. This striking phenomenon where only specific orbits are allowed is called the quantization of electronic orbits, because the orbital radii can only take discrete or quantized values. The reason this quantization happens is rather surprising, as we will see at the end of this chapter.
. The reverse process can also happen: If a quantum of light with just the right amount of energy comes long, it can be absorbed by an electron to enable it to jump to an outer orbit. And just as the electron states are very specific, all the properties of the packets of light, so absorbed or emitted, are also very specific. Every jump between the same two energy levels will create clones of the exact same quantum of light, which, by the way, are called photons (hence photon torpedoes in Star Trek).
We can visualize the changes in the state of our mind as happening similarly to the little quantum jumps of electrons inside an atom. Our mind remains in a stationary state until stimuli, external ones or internal ones (say due to bodily chemical shifts or memory flashbacks), lead to transitions in our mental state. At every waking moment of our life, there are things happening that influence our mood, with metaphorical quanta of happiness floating in and out: You could have been on your way to work at a job that you hate, and then the car radio confirms that you have won the lottery—that’s a big quantum jolt of happiness—you go from being downright miserable to deliriously happy. Then there are the small quanta that change your mood a bit this way and that all the time: an attractive stranger smiled at you, and that made you a just a bit happier instantaneously. Someone behaved like a jerk for no good reason; your happiness drops a quantum. Most of the time, we simply receive too many stimuli on our mind and senses during our waking hours to distinguish individual “quanta” of happiness, so our change of mood might seem just as fluid as a beam of light composed of countless photons.
of electromagnetic waves (which includes visible light, x-rays, ultra-violet rays, gamma rays, microwaves, and radio-waves) could be explained only by assuming that such waves (including ordinary visible light) actually come in discrete packets of energy that he called quanta. The idea was slow to catch on at first, but when it did, it caught fire and spurred intense research over the next three decades, which ushered in a completely new way of looking at the universe that has come to be known as quantum mechanics. The name underscores the fact that, as with light, many of the things in nature that were thought to exist as a continuum like a fluid actually come in discrete form like grains of sand. But there is a common misconception that everything in quantum mechanics is discrete or “quantized” and, vice versa, that discreteness is a unique feature of quantum physics. The discreteness is not so much about quantum mechanics per se, but is related to the fact that every system we deal with is finite and has boundaries. It is just that in the very small systems where quantum mechanics is most relevant, that discreteness is particularly conspicuous.
But how can boundaries make something discrete? It might seem obvious because all discrete or grainy little things have boundaries, due to their finite size. However, it is more subtle than that, because a river has boundaries, too, and we all think of water as a fluid. The way boundaries lead to discrete behavior in quantum mechanics is rather ironic, because to understand it, we need to look at waves, and waves essentially represent quite the opposite of discreteness—they are associated with continuous media like fluids. Therein lies a lingering mystery of the quantum world—the wave–particle duality: Most quantum entities behave both like waves and like particles depending on how you look at it! Let us now see how waves and boundaries lead to quantization.
. We all know that we won’t get far trying to make music with the strings hanging loose! The strings need to be clamped down at the ends and then tightened to the right tension to tune them. Beyond that, playing the guitar (or any string instrument for that matter) is all about varying the effective length of the strings as you pick on them; your fingers act just like temporary “clamps” where they press down along the neck. Everyone, from rock star wannabes to Eric Clapton does just that when pressing down on the frets along the neck of the guitar; bending the strings simply stretches them a bit more. The effective length of a string, illustrated in , determines the musical note it plays by fixing the wavelength of its vibration. Let’s see how.
. But if the string is not too tight, you would also notice that certain points, spaced out at equal intervals along the string, never move at all! Those points are called nodes, and the interval between adjacent nodes is exactly one half-wavelength of the wave playing on the string. Now here is the crucial point: Since the endpoints are fixed, and all the nodes in between are equally spaced, an allowed vibration of the string will need to fit complete half-wavelengths between the clamped endpoints. Incomplete half-wavelengths will not do, since they would require impossible motion at the fixed endpoints, as illustrated in . Therefore, any string with its ends fixed can vibrate only at very specific wavelengths; the boundaries of the string determine the wavelengths allowed.
. It’s a bit of a Cinderella situation; only a select few fit the “glass slipper”! Those select few waves are the stationary states for the “boundary conditions” of a particular fixed length of string with its ends clamped down. The states are indeed stationary because, if there were no air resistance or friction, they would vibrate forever, and unlike “traveling” waves, such as ripples in the water, these waves do not go anywhere; they remain right there on the string.
. The catch is that higher harmonics require more energy to generate and are therefore harder to reach; for example, normal picking on a guitar usually generates only the lowest harmonics on the strings. This natural bias toward lower harmonics underscores the essential reality of life that we all live with: It is always harder to achieve a happier and contented state of mind than it is to slide into despondency and depression. Hey, nobody ever complained of too much happiness, while antidepressants have become a staple of the modern world—shrinks count upon nature’s proclivity for the lower harmonics!
. Instead of it having clamped boundaries where the boundaries cannot move, here instead we need the ends to match up smoothly on being wrapped around. This is a slightly different boundary condition (called periodic boundary condition), but the argument is exactly the same as for the guitar string. Instead of changing the wavelength of the waves, we just keep on adding more and more half-wavelengths. This means that to squeeze in exactly one more half-wavelength, we need to increase the radius by just the right amount to keep the endpoints matched up.
What is truly amazing is that light and sound—our favorite metaphors for describing the states of the mind—naturally blend into a quantum view of happiness. It is as if somehow we always were in conversation with the universe and understood its language without deciphering the words.
We never see individual packets of light floating around for the same reason that we do not see atoms and molecules. Light quanta, or photons as they are usually called, are very tiny and there are zillions of them in an ordinary beam of light. As a result, we see light as a continuum, the same way we see water as a fluid and never perceive the little molecules it is made of.
Spectrum is like the fingerprint for different sources of electromagnetic waves. Everything in the universe radiates some form of electromagnetic waves (yes, even you—if you have ever seen thermal images of people, that is due to infrared waves emitted by the body). The spectrum of an object maps the intensities of various electromagnetic waves it emits. The spectrum changes with temperature—that is why we say “red hot” or “white hot,” because as an object gets hotter, it not only emits heat (infrared) but visible light, as well, and starts to glow.
The wave–particle duality is one of the strangest features of quantum mechanics and shows that at very small scales all entities behave both as particles and as waves depending on the type of phenomenon in which they are involved. It’s sort of like a Dr. Jekyll and Mr. Hyde situation—the same person but dual characters manifest in different environments, except that with waves and particles, there is no good or bad, they are both equally relevant.
The wavelength of a wave is the distance over which it repeats its shape (see ). The wavelength is a fixed characteristic of a wave; it is the same for any pair of crests of a particular wave. The nodes are a half-wavelength apart because the shape of the wave repeats exactly only at every other node. For more about waves, see .