The Physics of Superheroes: Spectacular Second Edition (39 page)

The fact that the light from a hot object depends only on its temperature prevents us from getting something for nothing. If two objects made of different materials at the same temperature emitted different radiation spectra, there would be a way to have a net transfer of energy between them and hence useful work, without any heat flow. While this would be a convenient violation of the second law of thermodynamics, it turns out that this does not occur for just that reason. A practical benefit of the fact that the emitted light spectrum depends only on the temperature is that we can use the intensity of emitted light as a function of wavelength to determine the temperature of objects for which normal thermometers are useless. This is how the surface temperature of the sun (roughly 11,000 degrees Fahrenheit) and the background microwave radiation remnants of the Big Bang (3 degrees above absolute zero) are measured—through observations of the spectrum of light they produce.

The second discovery, that the energy emitted by a glowing object is not infinite, was not really a shock to physicists. What they found disturbing was that Maxwell’s electromagnetic theory predicted that the amount of light energy emitted should increase without limit! Calculations using Maxwell’s theory correctly predicted how much light would be emitted at low frequencies, in exact agreement with observations. As the frequency of light emitted by a hot object increased into the ultraviolet portion of the spectrum, the measured light intensity reached a peak, and at higher frequencies decreased again back to a low value, which is what one would expect from both conservation of energy and common sense. However, the curve calculated from Maxwell’s equations and thermodynamics indicated that the intensity would become infinitely high for light above the visible portion of the spectrum. This was labeled the “ultraviolet catastrophe,” though it was only a “catastrophe” for the theorists doing the calculations. Many scientists checked and rechecked the calculations, but they could find nothing wrong with the math. Apparently, there was something wrong—or rather, incomplete—with the physics.

Maxwell’s equations had worked so well in all other cases (they led to the invention of radio in 1895 and would eventually enable the development of television as well as all forms of wireless communication), that it was doubtful that there was something fatally wrong with them. Rather, scientists concluded that the problem must lie with applying Maxwell’s theory to the shaking atoms in a glowing object. Again, many tried to find an alternative approach, some different theory that could account for the observed spectrum of light emitted by a glowing object. Here is where the fact that the spectrum depends only on the temperature of the object becomes important. If the theory of electromagnetism could not account for the behavior of one or two exotic pieces of matter, well, that would be somewhat awkward, but not earthshaking. The inability to explain a property shared by all matter was downright embarrassing, and something had to be done.

In 1900 the theoretical physicist Max Planck, recognizing that desperate times called for desperate measures, did the only thing he could to explain the spectrum of light emitted by a glowing body: He cheated. He first determined the mathematical expression that corresponded to the experimentally obtained glow curve. Once he knew what formula he needed, he then set out to find a physical justification for it. After trying various schemes, the only solution he could come up with that gave him the needed glow-curve formula involved placing restrictions on the energy of the atoms that made up the glowing object. Planck essentially proposed that the electrons in any atom could only have specific energies. From the Latin word for “how much,” this theory was called “quantum physics.” The separation between adjacent energy levels was in practice very small. And I mean really small: If the energy of a well-hit tennis ball is 50 kg-meter
²
/sec
²
, then the separation between adjacent energy levels in an atom is less than a millionth trillionth of a kg-meter
²
/sec
²
. This should provide some perspective the next time you hear a commercial boasting that the latest innovation in an automobile design or a laundry detergent represents a “quantum leap.”

Planck had to introduce a new constant into his calculations, an adjustable parameter that he labeled “h.” He assumed that any change in the energy of an atom could only take on values E = hf, or E = 2hf or E = 3hf, and so on, but nothing in between (so the atom could never have an energy change, say, of E = 1.6hf or 17.9hf), where f is the frequency characteristic to the specific atomic element. This is like saying that a pendulum can swing with a period of one second or ten seconds to complete a cycle, but that it was impossible to make the pendulum swing back and forth in five seconds. Planck himself thought this odd, but found it necessary in order to make his calculations come out right. He intended to let the value of h become zero once he obtained the correct expression for the spectrum for a glowing object. To his dismay, he discovered that when he did this, his mathematical expression went back to the infinite energy result from classical electromagnetism. The only way to avoid this nonsensical infinite result is to say something equally nonsensical (at least to scientists at the time), that the atoms cannot take on any energy value they want, but must always make changes in discrete steps of magnitude E = hf. Since h is very, very small (h = 660 trillion trillion trillionths of a kg-meter
²
/sec), we never notice this “graininess” of energy when we deal with large objects such as baseballs or moving automobiles. For the energy scale of an electron in an atom, it is quite significant and absolutely cannot be ignored.

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 which 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.