Adobe Digital Edition April 2009 ISBN 978-0-06-192638-9
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*In practice, some of the lines representing transitions between higher energy states do not appear in absorption spectra.
*Accurately speaking, different experimental equipment is required to photograph each series of the hydrogen spectrum. Therefore, most single photographs of the hydrogen spectrum show only about 10 lines. Theoretically, there are an infinite number of lines in each atomic spectrum. In fact, theoretically, there are an infinite number of lines in each series of each spectrum because the lines in the higher frequency range of each series become so closely spaced that, in effect, they form a continuum.
*The dark-adapted eye can detect a single photon. Otherwise, only the effects of subatomic phenomena are available to our senses (a track on a photographic plate, a pointer movement on a meter, etc.).
*At the time of Newton’s discoveries, the power of the church already had been challenged by Martin Luther. Newton himself was a pious person. The specific argument of the church was not with empirical method, but with the theological conclusions that were being developed from Newton’s ideas, conclusions which involved the concept of God as creator and the central position of man in creation.
*Strictly speaking, mass, according to Einstein’s special theory of relativity, is energy and energy is mass. Where there is one, there is the other.
*This was the 5th Solvay Congress at which Bohr and Einstein conducted their now famous debates. The term “Copenhagen Interpretation” reflects the dominant influence of Niels Bohr (from Copenhagen) and his school of thought.
†The Copenhagen Interpretation says that quantum theory is about correlations in our experiences. It is about what will be observed under specified conditions.
*The philosophy of pragmatism was created by the American psychologist William James. Recently, the pragmatic aspects of the Copenhagen Interpretation of Quantum Mechanics have been emphasized by Henry Pierce Stapp, a theoretical physicist at the Lawrence Berkeley Laboratory in Berkeley, California. The Copenhagen Interpretation, in addition to the pragmatic part, has the claim that quantum theory is in some sense complete; that no theory can explain subatomic phenomena in any more detail.
An essential feature of the Copenhagen Interpretation is Bohr’s principle of complementarity (to be discussed later). Some historians practically equate the Copenhagen Interpretation and complementarity. Complementarity is subsumed in a general way in Stapp’s pragmatic interpretation of quantum mechanics, but the special emphasis on complementarity is characteristic of the Copenhagen Interpretation.
*“…the hypothesis of quanta has led to the idea that there are changes in Nature which do not occur continuously but in an explosive manner.”—Max Planck, “Neue Bahnender physikalischen Erkenntnis,” 1913, trans., F. d’Albe, Phil. Mag. vol. 28, 1914.
*h=6.63 × 1027 erg-sec
*Bohr speculated that electronic orbits are arranged by nature at unvarying specific distances from the nucleus of the atom and that, when they absorb energy, the electrons in the atom jump outward from the orbit closest to the nucleus (the “ground state” of the atom) and eventually return to the innermost orbit, in the process emitting energy packets equal to the energy packets that they absorbed in jumping outward. Bohr proposed that when only a little energy is available (low heat), only small energy packets are absorbed by the electrons, and they do not jump out very far. When they return to their lowest energy level, they emit small energy packets, like those of red light. When more energy is available (high heat), larger energy packets are available, the electrons make bigger jumps outward and, on returning, they emit larger energy packets, like those of blue and violet light. Therefore, over low heat, metal glows red, and over high heat, it glows blue-white.
*Each of Einstein’s major 1905 papers dealt with a fundamental physical constant: h, Planck’s constant (the photon hypothesis); k, Boltzmann’s constant (the analysis of Brownian movement); and c, the velocity of light (the special theory of relativity).
*If we assume a particle aspect in the double-slit experiment we will violate the uncertainty relation unless we also assume nonlocality.
*An explanation other than “knowing” might be synchronicity, Jung’s acausal connecting principle.
*According to the complementarity argument, which is at the heart of the Copenhagen Interpretation, the latitude in the choice of possible wave functions exactly corresponds to (or at least includes) the latitude in the set of possible experimental arrangements, so that every possible experimental situation or arrangement is covered by quantum theory.
*Each set of experimental specifications A or B, that can be transcribed into a corresponding theoretical description & A or & B, corresponds to an observable. In the mathematical theory the observable is & A or & B; in the world of our experience the observable is the possible occurrence (coming into our experience) of the satisfied specifications.
*The particle is represented by a wave function which has almost all of the characteristics (when properly squared, to get a probability function) of a probability density function. However, it lacks the crucial feature of a probability density function, namely the property of being positive.
*From the pragmatic point of view, nothing can be said about the world “out there” except via our concepts. However, even within the world of our concepts particles do not seem to have an independent existence. They are represented in theory only by wave functions and the meaning of the wave function lies only in correlations of other (macroscopic) things.
Macroscopic objects, like a “table” or a “chair,” have certain direct experiential meanings, that is, we organize our sensory experiences directly in terms of them. These experiences are such that we can believe that these objects have a persisting existence and well-defined location in space-time that is logically independent of other things. Nonetheless, the concept of independent existence evaporates when we go down to the level of particles. This limitation of the concept of independent entity at the level of particles emphasizes, according to the pragmatic view, that even tables and chairs are, for us, tools for correlating experience.
*What we can predict is the probability corresponding to any specification that can be mapped into a density function. Accurately speaking, we do not calculate probabilities at points, but rather transition probabilities between two states (initial preparation, final detection), each of which is represented by a continuous function of x and p (position and momentum).
*The state of a system containing n particles is represented at each time by a wave function in a 3n dimensional space. If we make an observation on each of the n particles the wave function is reduced to a special form—to a product of n wave functions each of which is in a three-dimensional space. Thus the number of dimensions in the wave function is determined by the number of particles in the system.
*To see the conciseness of mathematical expr
ession, consider that the entire process described in the Theory of Measurement, from photon (system, S) to detectors (measuring device, M) to technician (observer, O) can be represented mathematically by one “sentence”:
†The Theory of Measurement presented here is essentially from John von Neumann’s 1932 discussion.
*The wave function is the physicist’s description of reality. At issue is the interpretation of the wave function and whether it is the best possible description (or simply the only one that fits the language used by physicists).
*The wave function, since it is a tool for our understanding of nature, is something in our thoughts. It represents certain specifications of certain physical systems. Specifications are objective in the sense that scientists and technicians can agree on them. However, specifications do not exist apart from thought. Also, any given physical system satisfies many sets of specifications, and many physical systems can satisfy one set of specifications. All of these characteristics are idea-like and, to that extent, that which is represented by the wave function is idea-like, even though it is objective.
However, these specifications are transcribed into wave functions that develop according to a determined law (the Schrödinger wave equation). This is a matter-like aspect. The thing that develops describes only probabilities. Probabilities can be thought to describe either things that exist apart from thought, or things that exist only within thought. Thus that which the wave function represents has both idea-like and matter-like characteristics.
*“How is one to apply the conventional formulation of quantum mechanics to the space-time geometry itself? The issue becomes especially acute in the case of a closed universe. There is no place to stand outside the system to observe it.”—Hugh Everett III (Reviews of Modern Physics, 29, 3, 1957, 455).
*In practice, it is not clear that a macroscopic object such as a cat actually can be represented by a wave function due to the dominating influence of thermodynamically irreversible processes. Even so, Schrödinger’s cat long has illustrated to physics students the psychedelic aspects of quantum mechanics.
*The dark-adapted eye can detect single photons. All of the other subatomic particles must be detected indirectly.
*Individual events are always particle-like; wave behavior is detected as a statistical pattern, i.e., interference. However, in the words of Paul Dirac (another founder of quantum mechanics) even a single subatomic particle “interferes with itself.” How a single subatomic particle, like an electron, for example, can “interfere with itself” is the basic quantum paradox.
*Planck’s equation: E = hv. Einstein’s equation: E = mc2. De Broglie’s equation: λ = h/mv.
*As you hold this photograph in front of you, the beam of electrons (the “transmitted beam”) is coming directly toward you out of the large white spot in the center. Also located in the white spot is the diffracting material (in this case, the electron beam is being diffracted by small grains of gold, i.e., the beam is being directed through a thin polycrystalline gold foil). The rings on the photograph mark the places where the diffracted electron beams struck the film which was placed on the opposite side of the gold foil from the electron source. The white spot in the center of the photograph was caused by undiffracted electrons in the transmitted beam passing through the gold foil and striking the film directly.
*Accurately speaking, Schrödinger’s theory does not explain The Jump, either. In fact, Schrödinger did not like the idea of a “jump.”
*These photographs are of mechanical simulations of probability density distributions of different electron states in the hydrogen atom. In other words, they represent where we are most likely to find the point-like electron when we look for it if the atom is in this or that particular state (there are more states than those shown). Initially, Schrödinger pictured electrons as being tenuous clouds actually assuming these patterns.
†A “quantum jump” can be thought of as a transition from one of these pictures to another without anything in between.
*Schrödinger’s early interpretation that electrons literally were standing waves did not stand up to detailed examination and he had to renounce it. Soon, however, the concept of probability based upon a wave function representing an observed system (and developing according to the Schrödinger wave equation) became a fundamental tool in atomic research and Schrödinger’s famous equation became an integral part of quantum theory. Since the Schrödinger wave equation is nonrelativistic, however, it does not work at high energies. Therefore, high-energy particle physicists usually use the S Matrix to calculate transition probabilities. (S Matrix theory is discussed in a later chapter.)
†Until the propagating system interacts with a measuring device. That causes an abrupt, unpredictable transition to another state (a quantum jump).
*If the state is prepared in state ψ(t), the probability that it will be observed to be in state φ(t) is | |2. If it is prepared in state Ψ(t) then the probability that it will be observed in region δ at time t is δ∫d3 × Ψ*(x, t) × Ψ(x, t).
*The Schrödinger wave equation works at lower energies, however, since it is nonrelativistic, it does not work for high energies. Therefore, most particle physicists use the S Matrix together, perhaps, with local relativistic quantum fields to understand quarks and particles.
*Strictly speaking, Newton’s laws do not disappear totally in the subatomic realm: they remain valid as operator equations. Also, in some experiments involving subatomic particles Newton’s laws may be taken as good approximations in the description of what is happening.
*Einstein’s point of departure for the special theory of relativity came from the conflict of classical relativity and Maxwell’s prediction of a light speed, “c.” An often-told story tells how Einstein tried to imagine what it would be like to travel as fast as a light wave. He saw, for example, that the hands on a clock would appear to stand still, since no other light waves from the clock would be able to catch up with him until he slowed down.
*Although we do not experience it directly, the orbital motion of the earth is accelerating.
*The fixed stars provide such a reference frame as far as defining non-rotation.
*In a vacuum. The speed of light changes in matter depending upon the index of refraction of the matter:
*The reverse situation (the source moves and the observer remains stationary) is explainable in terms of prerelativistic physics. In fact, if light is assumed to be a wave phenomenon governed by a wave equation, it is expected that its measured velocity will be independent of the velocity of its source. The velocity of the sound waves reaching us from a jet plane, for example, does not depend upon the velocity of the aircraft. They propagate through a medium (the atmosphere) at a given velocity, from their point of origin, regardless of the motion of the plane (the frequency of the sound shifts as the source moves, e.g., the Doppler effect). Prerelativity theory assumes a medium (like the atmosphere, for sound waves, or the ether, for light waves) through which the waves propagate. The paradox is that the measured velocity of light has been found (the Michelson-Morley experiment) to be independent of the motion of the observer. In other words, assuming a light wave propagating through a medium, how can we move through the same medium toward the approaching wave without increasing its measured velocity?
*Quantum field theory resurrects a new kind of ether, e.g., particles are excited states of the featureless ground state of the field (the vacuum state). The vacuum state is so featureless and has such high symmetry that we cannot assign a velocity to it experimentally.
*It is said that the reasoning process by which Einstein discovered the special theory of relativity did not include the results of the Michelson-Morley experiment. However, the results of this well-publicized experiment were “in the air” for eighteen years prior to Einstein’s paper on special relativity (1905) and they led to the Lorentz transformations which became central to the mathematical formalism of special relativity.
*In a va
cuum. The speed of light changes in matter depending upon the index of refraction of the matter.
*The clocks were flown around the world each way (east and west). Both general relativistic and special relativistic effects were noted. (J. C. Hafele and R. E. Keating, Science, vol. 177, 1972, pp. 168ff.)