understanding simply the radial distance of the electron from the proton. The other part involves a two-dimensional problem, which governs the angular distribution of the electron “orbits” in the atom. Both of these problems are solved separately and then combined to allow us to classify the total number of states of the hydrogen atom.
Here’s a more recent and more exotic example, along similar lines. Stephen Hawking has become known for his demonstration in 1974 that black holes are not black—that is, they emit radiation with a temperature characteristic of the mass of the black hole. The reason this discovery was so surprising is that black holes were so named because the gravitational field at their surface is so
strong that nothing inside can escape, not even light. So how can they emit radiation? Hawking showed that, in the presence of the strong gravitational field of the black hole, the laws of quantum mechanics allow this result of classical thinking to be evaded. Such evasions of classical “no go” theorems are common in quantum mechanics. For example, in our classical picture of reality, a man resting in a valley between two mountains might never be able to get into a neighboring valley without climbing over one or the other of the mountains. However, quantum mechanics allows an electron in an atom, say, with an energy smaller than that required to escape from the atom, according to classical principles, sometimes to “tunnel” out from inside the electric field binding it, and find itself finally free of its former chains! A standard example of this phenomenon is radioactive decay. Here, the configuration of particles—protons and neutrons—buried deep in the nucleus of an atom can suddenly change. Depending upon the properties of an individual atom or nucleus, quantum mechanics tells us that it is possible for one or more of these particles to escape from the nucleus, even though classically they are all irretrievably bound there. In another example, if I throw a ball at a window, either the ball will have enough energy go through it, or it will bounce off the window and return. If the ball is small enough so that its behavior is governed by quantum-mechanical principles, however, things are different. Electrons, say, impinging on a thin barrier can do both! In a more familiar example, light impinging on the surface of a material like a mirror might normally be reflected. If the mirror is thin enough, however, we find that even though most is reflected, some of the light can “tunnel” through the mirror and appear on the other side! (I shall outline the new “rules” that govern this weird behavior later. For the moment, take it as a given.)
Hawking showed that similar phenomena can occur near the black hole. Particles can tunnel through the gravitational barrier at the surface of the black hole and escape. This demonstration was a tour de force because it was the first time the laws of quantum mechanics had been utilized in connection with general relativity to reveal a new phenomenon. Again, however, it was possible only because, like the hydrogen atom, the quantum-mechanical states of particles around a black hole are “separable”—that is, the three-dimensional calculation can be effectively turned into a one-dimensional problem and an independent two-dimensional problem. If it weren’t for this simplification, we might still be in the dark about black holes.
Interesting as these technical tricks might be, they form only the tip of the iceberg. The real reason we keep repeating ourselves as we discover new laws is not so much due to our character, or lack thereof, as it is due to the character of nature. She keeps repeating herself. It is for this reason that we almost universally check to see whether new physics is really a reinvention of old physics. Newton, in discovering his Universal Law of Gravity, benefited tremendously from the observations and analyses of Galileo, as I have
Here’s a more recent and more exotic example, along similar lines. Stephen Hawking has become known for his demonstration in 1974 that black holes are not black—that is, they emit radiation with a temperature characteristic of the mass of the black hole. The reason this discovery was so surprising is that black holes were so named because the gravitational field at their surface is so
strong that nothing inside can escape, not even light. So how can they emit radiation? Hawking showed that, in the presence of the strong gravitational field of the black hole, the laws of quantum mechanics allow this result of classical thinking to be evaded. Such evasions of classical “no go” theorems are common in quantum mechanics. For example, in our classical picture of reality, a man resting in a valley between two mountains might never be able to get into a neighboring valley without climbing over one or the other of the mountains. However, quantum mechanics allows an electron in an atom, say, with an energy smaller than that required to escape from the atom, according to classical principles, sometimes to “tunnel” out from inside the electric field binding it, and find itself finally free of its former chains! A standard example of this phenomenon is radioactive decay. Here, the configuration of particles—protons and neutrons—buried deep in the nucleus of an atom can suddenly change. Depending upon the properties of an individual atom or nucleus, quantum mechanics tells us that it is possible for one or more of these particles to escape from the nucleus, even though classically they are all irretrievably bound there. In another example, if I throw a ball at a window, either the ball will have enough energy go through it, or it will bounce off the window and return. If the ball is small enough so that its behavior is governed by quantum-mechanical principles, however, things are different. Electrons, say, impinging on a thin barrier can do both! In a more familiar example, light impinging on the surface of a material like a mirror might normally be reflected. If the mirror is thin enough, however, we find that even though most is reflected, some of the light can “tunnel” through the mirror and appear on the other side! (I shall outline the new “rules” that govern this weird behavior later. For the moment, take it as a given.)
Hawking showed that similar phenomena can occur near the black hole. Particles can tunnel through the gravitational barrier at the surface of the black hole and escape. This demonstration was a tour de force because it was the first time the laws of quantum mechanics had been utilized in connection with general relativity to reveal a new phenomenon. Again, however, it was possible only because, like the hydrogen atom, the quantum-mechanical states of particles around a black hole are “separable”—that is, the three-dimensional calculation can be effectively turned into a one-dimensional problem and an independent two-dimensional problem. If it weren’t for this simplification, we might still be in the dark about black holes.
Interesting as these technical tricks might be, they form only the tip of the iceberg. The real reason we keep repeating ourselves as we discover new laws is not so much due to our character, or lack thereof, as it is due to the character of nature. She keeps repeating herself. It is for this reason that we almost universally check to see whether new physics is really a reinvention of old physics. Newton, in discovering his Universal Law of Gravity, benefited tremendously from the observations and analyses of Galileo, as I have
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