The fact that the energy of electrons in an atom can have only discrete values, with nothing in between, is indeed bizarre. Imagine the consequences of this discreteness of energy for a car driving down the highway at 50 mph if Planck’s constant h were much larger. The quantum theory tells us that the car could drive at a slower speed of 40 mph, or at a faster speed of 60 mph, but not at any other speeds in between! Even though we can conceive of the car driving at 53 mph, and calculate what its kinetic energy would then be, it would be physically impossible for the car to drive at this speed, according to the principles of quantum physics. If the car absorbed some energy (say, from a gust of wind), it could increase its speed to 60 mph, but only if the energy of the wind could exactly bridge the difference in kinetic energies. For a slightly less energetic gust, the car would ignore the push of the wind as if it were not there and continue along at its original speed. Only if the energy of the wind exactly corresponded to the difference in kinetic energy from 50 to 60 mph, or 50 to 70 mph, would the car “accept” this push and move to a higher speed. The transition to the higher velocity would be almost instantaneous, and the acceleration during this transition would likely do bad things to the car’s occupants. This scenario seems ridiculous when translated to highway traffic, but it accurately describes the situation for electrons in an atom.
Is there any way to understand why the energy of an electron in an atom has only certain discrete values? Yes, actually, but first you must accept one very strange concept. In fact, all of the “weirdness” associated with quantum physics can be reduced to the following statement: There is a wave associated with the motion of any matter, and the greater the momentum of the object, the shorter the wavelength of this wave.
When something moves, it has momentum. The physicist Louis de Broglie suggested in 1924 that associated with this motion is some sort of “matter-wave” connected to the object, and the distance between adjacent peaks or troughs for this wave (its wavelength) depends on the momentum of the object. Physicists refer to an object’s “wave function,” but we’ll stick with “matter-wave” as a reminder that we are referring to a wave associated with the motion of a physical object, whether an electron or a person.
This matter-wave is not a physical wave. Light is a wave of alternating electric and magnetic fields created by an accelerating electric charge. The wind-driven ripples on the surface of a pond or the concentric rings formed when a stone is tossed into the water result from mechanical oscillations of the water’s surface. Sound waves are a series of alternating compressions and expansions of the density of air or some other medium. In contrast, the matter-wave associated with an object’s momentum is not like any of these waves, but in some sense, it is just along for the ride, moving along with the object. It is not an electric or magnetic field, nor can it exist distinct from the object, nor does it need a medium to propagate. Yet this matter-wave has real physical consequences. Matter-waves can interfere when two objects pass near each other, just as when two stones are thrown into a pond a small distance apart and each creates a series of concentric ripple rings on the water’s surface that form a complex pattern where the two rings intersect. If you ask any physicist what this matter-wave actually is, he or she will give a variety of mathematical expressions that always boil down to the same three-word answer: I don’t know. For once, our one-time “miracle exception” applies to the real world, rather than the four-color pages of comic books!
Unless an object is moving at nearly the speed of light, its momentum can be described as the product of its mass and its velocity. A Mack truck has more momentum than a Mini Cooper if they are both traveling at the same speed, since the truck’s mass is much larger. The Mini Cooper could have a larger momentum if it were traveling at a much, much higher speed than the truck. Physicists typically use the letter “p” to represent an object’s momentum, since obviously the p stands for mo-mentum.68 The wavelength of this matter-wave is represented by the Greek letter lambda (λ). The matter-wave’s wavelength was proposed by de Broglie (and experimentally verified in 1926 by Clinton Davisson and Lester Germer) to be related to the object’s momentum by the simple relationship (momentum) times (wavelength) equals a constant, or p × λ= h where h is the same constant that Planck had to introduce in order to account for the glow curve of hot objects.
The fact that the product of an object’s momentum and the matter-wave’s wavelength is a constant means that the bigger the momentum, the smaller the wavelength of the matter-wave. Given that momentum is the product of mass and velocity, large objects such as baseballs or automobiles have very large momenta. A fastball thrown at one hundred mph has a momentum of about 6 kg-meter/second. From the relationship p λ = h, since h is so small, the wavelength (the distance between successive peaks in the wave, for example) of the matter-wave of the baseball is less than a trillion trillionths of the width of an atom. This explains why we have never seen a matter-wave at the ballpark. Obviously there is no way we can ever detect such a tiny wave, and baseballs, for the most part, are well-behaved objects that follow Newton’s laws of classical physics.
On the other hand, an electron’s mass is very small, so it will have a very small momentum. The smaller the momentum, the bigger the matter-wave wavelength will be, since the product is a constant. Inside an atom, the matter-wave wavelength of an electron is about the same size as the atom, and there is no way one can ignore such matter-waves when considering the properties of atoms. When the DC Comics superhero the Atom shrinks down to the size of an atom, he should see some rather strange sights. At this size he is smaller than the wavelength of visible light so, just as we can’t see radio waves, whose wavelength is in the range of several inches to feet, the Atom’s normal vision should be inoperable, and he will be roughly the same size as the matter-waves of the electrons inside the atom. It is suggested in his comic that at this size the Atom’s brain interprets what he sees as a conventional solar system conception of the atom, for he has no other valid frame of reference to decipher the signals sent by his senses.
Imagine an electron orbiting a nucleus, pulled inward by the electrostatic attraction between the positively charged protons in the nucleus and the electron’s negative charge. As the electron travels around the nucleus, only certain wavelengths can fit into a complete cycle. When the electron has returned to its starting point, having completed one full orbit, the matter-wave must be at the same point in the cycle as when it left. As weird as the notion of a matter-wave is, it would be even harder to comprehend if when the wave left it was at a peak (for example), and after having completed one full orbit, was now at a valley. In order to avoid a discontinuous jump from a maximum to a minimum whenever the wave completed a cycle, only certain wavelengths that fit smoothly into a complete orbit are possible for the electron. This is not unlike the situation of a plucked violin string, with only certain possible frequencies of vibration. Because the wavelength of the matter-wave is related to the electron’s momentum, this indicates that the possible momenta for the electron are restricted to only certain definite (discrete) values. The momentum is in turn related to the kinetic energy, so the requirement that the matter-wave not have any discontinuous gaps after finishing an orbit leads us to conclude that the electron can only have certain discrete energy values within the atom.
These finite energies are a direct result of the constraint on the possible wavelengths of the matter-waves, which in turn are due to the fact that the electron is bound within the atom. An electron moving through empty space has no constraints on its momentum, and consequently its matter-wave can have any wavelength it likes.69 A piece of string can have any shape at all when I wiggle one end, provided that the other end is also free to move. But if the string is clamped at both ends, as in the case of a violin string, then the range of motions for the string are severely restricted. When I now pluck the clamped string, it can only vibrate at certain frequencies, determined by the length and width of the string and the tension with wh
ich it is clamped. There is a lowest fundamental frequency for the string and many higher overtones, but the string cannot vibrate at any arbitrary frequency once it is constrained in this manner.
Likewise, the electron is held in an orbit by its electrostatic attraction to the positively charged nucleus. If “plucked” in the right way, a matter-wave for the bound electron can take on a higher energy value. When the electron then relaxes back to its lower fundamental frequency, it must do so by making a discrete jump. Energy is conserved; consequently, the electron can only lower its energy when returning to the lower frequency level by giving off a packet of energy equal to the difference between its higher energy level and the lower level it is relaxing to. Because the energies available to the electron are discrete, well-defined values similar to the overtones possible for a clamped string, this jumping from one energy state to another is termed a “quantum transition” or a “quantum jump.” The discrete packet of energy given off by the electron when making this transition is typically in the form of light, and a quantum of light energy is termed a “photon” (a concept introduced by Albert Einstein, again in 1905—a busy year for him and physics—though the term “photon” was not coined until 1926 by Gilbert Lewis).
If a glass tube is filled with a gas such as neon, and an electrical current is passed through the gas, the energetic electrons of the current will sometimes collide with the neon atoms. When the energy of the energetic electrons is just right, the neon atoms can be excited into a higher energy state. After the collision, the excited neon atoms will relax back to their original lower energy configuration, emitting a photon of light that has the frequency (hence, color) that corresponds to the energy difference between its starting and final states. This is why neon lights have their identifiable color. By changing the type of gas in the tube, different colors of light can be selected. You could do this with any gas, but only certain elements have a transition within the visible portion of the light spectrum. If the atoms suffer highly energetic collisions, such that many higher energy states are excited, then many discrete wavelengths of light will be given off when the different overtones all relax back to the fundamental level. Different elements have differing arrays of overtones and fundamental frequencies, just as different strings on a violin or guitar will have different vibrational modes depending on the length, width, and tension. Two identical violin strings clamped with the same tension will have the same range of possible frequencies when plucked. Similarly, two identical atoms will have the same spectrum of emitted light when they relax from an excited state.
In this way, the spectrum of wavelengths of light emitted by an energetic atom is unique and can be considered the fingerprint of the particular element. The lighter-than-air element helium was discovered by the detection of its characteristic spectrum of light coming from the sun (the word “helium” is derived from Helios, the Greek sun god). By careful comparison with the spectrum of light emitted by hydrogen and other gases, scientists concluded that this array of wavelengths must arise from a new element that at the time had never been found terrestrially. Fortunately for the Macy’s Thanksgiving Day Parade, underground pockets of helium were eventually discovered.
The notion that there is a wave associated with the motion of any object, and that the wavelength of this wave is inversely proportional to its momentum, is weird, but by accepting this mysterious concept, we gain an understanding for the basis of all of chemistry. Bring two atoms close enough together, and they may form a chemical bond, and in so doing create a new basic unit, the molecule. Why would the atoms do this? The negatively charged electrons in the first atom certainly will repel the negatively charged electrons in the second atom. Before quantum mechanics, there was no satisfactory fundamental explanation for why the universe didn’t just consist of isolated elemental atoms.
The driving force underlying the bonding between atoms is the interactions of the matter-waves of the electrons from the different atoms. When the two atoms are held very far apart, the matter-waves of the atomic electrons do not overlap. When the atoms are brought close enough together so that the electron clouds around each atom intersect, the electronic matter-waves from each atom begin to interfere, forming a new wave pattern, just as two stones tossed into a pond create an intricate pattern of ripples that is very different from the pattern that would be created by each stone separately. In most cases this new pattern is a high-energy, discordant mess, similar to the sound resulting from a clarinet and violin played simultaneously by novices with no musical training or talent. In these cases, the two atoms do not form a chemical bond and do not chemically interact. In a few special cases, the two matter-waves interact harmoniously, creating a new wave pattern that has a lower-energy configuration than the two separate matter-waves. In these special cases, the two atoms can lower their total energy by allowing the matter-waves to interact in this manner, and once in a lower-energy state, it requires the addition of energy (called the “binding energy” or “bonding energy”) to physically separate them. In this way, despite the considerable repulsion between the negatively charged electrons, the two atoms are held together in a chemical bond, owing to the wave-like nature of the electrons.
These arguments about discrete energy levels in an atom arising from those particular orbits that correspond to an integral number of wavelengths of the electron matter-wave seem so reasonable that it’s a shame that they’re not really correct. The electron cannot literally be considered to move in a circular or elliptical orbit around the positively charged nucleus, despite the appealing analogy to our solar system. For one thing, the electron would be constantly accelerated as it bends onto a curved path. As argued in the previous chapter, an accelerating electric charge emits electromagnetic waves that carry energy, so if the electron continuously radiates light in its orbit, it would lose kinetic energy. Eventually, the electron should spiral into the nucleus in a little less than a trillionth of a second, so no elements should be stable, and hence there would be no chemistry and no anything if the electrons really moved in curved orbits.
Nevertheless, the notion that only certain wavelengths are allowed, with corresponding discrete energy levels, is still a valid one, even if the picture we employed to get there can only be considered a useful metaphor and not a literal description. Instead of thinking of the electron as a point particle that moves in a circular orbit with a particular matter-wave associated with it, the full quantum theory of Heisenberg and Schrödinger, to be discussed in the next chapter, tells us that there is a “wave function” for the electron. For a plucked violin string, it makes no sense to ask at what position on the string the wave is. Similarly for the electron in an atom, its matter-wave extends over the atom, and we cannot specify the electron’s position or trajectory more accurately than this. Electrons only emit or absorb light when moving from one wave pattern to another in the atom. As we’ll see in the next chapter, these matter-waves are also responsible for a crisis on infinite Earths!
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NOT A DREAM! NOT A HOAX! NOT AN IMAGINARY TALE!—QUANTUM MECHANICS
THE ORIGIN TALE in Showcase # 4 describing how Barry Allen gained his superspeed powers and became the Silver Age Flash featured a symbolic passing of the torch from the earlier Golden Age of superheroes. Just prior to being struck by the lightning bolt that simultaneously doused him with exotic chemicals, and that naturally would enable him to run at the speed of light, police scientist Allen relaxed with a milk-and-pie break in his lab while reading Flash Comics # 13, featuring the Golden Age Flash on its cover. After the freak accident gave Barry his superpowers, his immediate thoughts turned to how he could use his superspeed to help humanity. Taking his inspiration from the Flash comic book he had been reading before he was struck by lightning, he donned a red-and-yellow costume and began his crime-fighting career as the Silver Age Flash (though he referred to himself simply as the Flash, not realizing that he was himself a comic-book character in the dawning Silver Age of superheroes). In a
turn of events that nowadays would be described as “postmodern,” and back then was simply considered “pretty neat,” it was established in the Flash comic books of the 1960s that the Flash character of the 1940s (who wore a different costume and gained his superspeed in a different, though equally implausible, chemical accident) was a comic-book character in Barry Allen’s “reality.”
The Golden Age Flash (whose secret identity was Jay Garrick) was considered fictional, as far as the Silver Age Flash was concerned, until September 1961. That month saw the publication of the classic story “Flash of Two Worlds” in The Flash # 123 (fig. 30), where it was revealed that the Silver Age Flash and the Golden Age Flash both existed, but on parallel Earths, separated by a “vibrational barrier.” In this story, the Silver Age (Barry Allen) Flash accidentally vibrated at superspeed at the exact frequency necessary to cross over to the Earth on which his idol, the Golden Age (Jay Garrick) Flash, lived. Once he realized that he was in the world of the Golden Age heroes, Barry met Jay and introduced himself. “As you know,” the police scientist explained, “two objects can occupy the same space and time—if they vibrate at different speeds!” Apparently Barry Allen was a better forensic scientist than he was a theoretical physicist. Regardless of their vibrational frequency (and as we saw in Section 2, atoms in a solid do vibrate simply because they have some finite temperature), there is no way that two objects can be in the same place in space and time (unless we are discussing massless quantities such as light photons).
Fig. 30. The cover to The Flash # 123, giving comic-book fans their first hint that there were at least two worlds beyond their own.
The Physics of Superheroes: Spectacular Second Edition Page 28
