## BLACK BODY RADIATION: THE QUANTUM

Max Planck began the revolution, though he never really believed in the consequences of his discovery and, in fact, he was not directly studying atoms. He was instead interested in a problem known as the black body radiation problem, one of the great unsolved problems of physics at the end of the nineteenth century. The problem concerns the spectrum of light emitted from a heated object. We are all familiar with the idea that heated objects give off a color that varies with the temperature. If you have ever stared mesmerized at a campfire as the coals burn down, you are aware of the glowing embers that vary from yellow to red to blue. A black wood stove with coals of a fire burning in it is a crude example of a black body. As described in the theory of global warming in Chapter 1, the earth is also a black body (roughly speaking). The questions for physicists were: What is the spectrum of light emitted from such a black body held at a particular temperature? How much light is emitted at a particular frequency or color?

According to the classical equations developed in the 1860s by James Clerk Maxwell, a Scottish physicist, light is an electromagnetic wave with a wavelength (distance between peaks), frequency (oscillations per second), and velocity. The color of light depends on its frequency f) and its wavelength (X) but they are connected by a relationship that is a constant, namely the speed of light c. The product of the frequency and the wavelength is always equal to the speed of light, which is 3.0 x 108 (300,000,000) meters per second, or 186,000 miles per second. In equation form,

f x X = fX = c

so there is an inverse relationship between frequency and wavelength. Blue light has a higher frequency and shorter wavelength than red light, for example. A spectrum is the amount of light emitted at each frequency or wavelength. Physicists had already measured the light spectrum and knew that it was independent of the size or shape of the black body. It depended only on temperature of the black body.

According to classical physics, the amount of light (the intensity) emitted at a particular frequency from a black body gets smaller and smaller at low frequencies but should approach infinity at high frequencies, leading to what was known as the “ultraviolet catastrophe” (2). This would imply that a black body should emit an infinite amount of energy, and that was clearly absurd. If it were true, you would be fried if you stood in front of your black body wood stove. At the rather old age—for a theoretical physicist—of 42, Max Planck delivered a bombshell in a lecture at the German Physical Society in Berlin on December 14, 1900 (9). He developed an equation that completely described the spectrum of radiation being emitted from a black body based on the statistical thermodynamic distribution of energy in oscillators (what we now call atoms) in the black body. But he made a huge conceptual jump. To avoid the problem of the ultraviolet catastrophe, he postulated that the oscillators could not have all of the infinite possible values of energy allowed by classical physics, but instead the energy had to be a multiple of a new constant he defined as h, which is now known as Planck’s constant. Specifically, the energy of the oscillators could only be multiples of h times f, or hf, where h is Planck’s constant and f is the frequency. This amount of energy is a quantum of energy.

Perhaps this does not seem revolutionary, but it goes against everything that most people think they know about the world. If you made a pendulum by suspending a ball on the end of a string and moved it to one side and then released it, you would expect that it would swing back and forth with a particular frequency (known as resonance), and if you moved it farther to one side, it would swing farther, though still with the same frequency. By moving it aside and up, you are giving it gravitational energy, which is converted to kinetic energy as it swings. But you assume that you can move it anywhere you want before you release it. According to Planck’s theory, however, that is not strictly true. You can only give it energy that is a multiple of hf. In the macroscopic world we live in, you don’t notice the quantum effect because it is extremely small. The value of h is 6.626 x 10-34 joule-seconds, where a joule is a unit of energy.3 But in the atomic world, Planck’s constant rules what can happen.

A much younger fellow German scientist, Albert Einstein, made the critical connection that proved the reality of quantum effects in the atomic world. In his miracle year of 1905, when he was 26 years old, Einstein published five papers that changed the world of physics. One of the papers proved the existence of atoms and molecules based on the Brownian motion of very small particles in a liquid; one gave a calculation of the size of a molecule; one set out the theory of special relativity concerning space and time; a follow-up paper on the theory of relativity set out his most famous equation E = mc2, postulating that mass is just another form of energy; and one proposed the quantum nature of light, for which he received the Nobel Prize (3). That is the one he considered the most revolutionary, and it is directly related to the work of Planck.

Einstein posed the question of why there was an apparent difference in the nature of the material world and light. The natural world of matter was considered to be made up of discrete atoms and molecules, while light was considered to be a continuous wave with a particular frequency and wavelength that are infinitely divisible. Einstein strongly believed in a fundamental beauty of the natural world and thought there should not be these differences between discrete atoms and continuous waves of light. He was aware of Planck’s work on black body radiation, and he derived similar equations for the energy spectrum of black body radiation. But he went further conceptually by considering the light in a black body to be similar to a gas of particles, showing that mathematically they follow the same thermodynamic rules. He concluded that light can be considered as a collection of quantum particles each with energy of hf (9).

Einstein went even further by testing his conclusions against experimental results. A Hungarian physicist, Philip Lenard, had discovered an utterly baffling phenomenon known as the photoelectric effect. He found that if he shone ultraviolet light on a metal, it would emit electrons. This could actually be explained by the wave nature of light. What was especially baffling, though, is that if he increased the intensity of light, it did not increase the energy of the emitted electrons, as it should if light were acting as waves. But if he increased the frequency of the light, the emitted electrons had more energy. Einstein explained the results by postulating that the light consisted not of waves but of quantum particles (later called photons) that had energy hf. The energy of a photon could be given up in a collision with an electron in the atoms of the metal. Increasing the intensity of the light would only produce more electrons of the same energy, but a higher frequency photon would knock out electrons with a higher energy, in agreement with Lenard’s experiments. Thus, Einstein showed that light had properties of a particle, in addition to its well-known wave properties. In fact, light sometimes acts as a particle and sometimes as a continuous wave, but never both at the same time. And even more mysteriously, what we think of as purely matter, such as electrons, sometimes acts as waves. This has come to be known as the wave-particle duality. Welcome to the quantum world, an Alice in Wonderland world full of surprises! We will come back to the photoelectric effect when we consider how radiation causes damage to cells.

The insights ofPlanck and Einstein led to the development of a branch of physics called quantum mechanics, which was unlike anything physicists had contemplated before. Planck discovered the concept of a quantum, but he did not accept that it had a physical reality. It was more of a mathematical convenience to solve a problem. Einstein took this concept and showed that it had a physical reality, namely, that light itself was a quantum particle. But neither Planck nor Einstein could ever fully come to terms with the revolution in physics that they started (3). When quantum mechanics was developed, it described a physical world that Einstein could never believe in. So it was left to others to fully develop the concepts of a quantum atom.