The KÝbenhavn Interpretation

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    Suppose you want to make a picture of an electron. The quantum theory tells us that the electron has two aspects B as a particle and as a wave B and that those aspects, incompatible as they are, exist together simultaneously in the electron. It also tells us that those aspects are indeterminate. In this case we don= t select the recording medium (the film), but rather the recording apparatus (the camera) with which we make our picture. And we note that Heisenberg=s indeterminacy principle tells us that no apparatus can exist that will record both aspects of the electron at the same time.

    We can, however, create an apparatus that will respond to the electron=s two aspects one after the other. The apparatus used in the double-slit experiment gives us a perfect example. In that apparatus we have an electron gun projecting electrons through a vacuum toward a thin metal barrier, into which we have cut two narrow, parallel slits. Some distance behind the barrier we have a phosphor-coated screen set up parallel to the barrier. In essence we have a cheapo precursor to the black-and-white television sets of the 1950's.

    With that apparatus in operation the electron gun emits a stream of electrons along a straight line that goes to a point on the barrier midway between the two slits. The barrier will absorb most of the electrons, but some of the electrons will go through the slits. Those electrons hit the screen and, as they get absorbed, they cause the phosphors to emit little flashes of light.

    If we remove the barrier from the apparatus, the electrons will create a small, fuzzy disc of light on the screen, the same kind of luminous dot that we used to see on television sets in the 1950's and 1960's when we turned them off (the electron gun in the picture tube continued emitting electrons in a beam for several minutes after the set was turned off). If we put the barrier back into the apparatus, the fuzzy dot becomes a series of luminous dots arrayed along a straight line that runs parallel to a straight line drawn on the barrier and passing through the centers of the slits.

    As we understand it, the ghostwave associated with each electron diffracts through the slits and then interferes with itself, both constructively and destructively, to create a pattern of greater and lesser probabilities on the screen. More electrons strike the areas of greater probability than strike the areas of lesser probability, thereby creating the pattern of fuzzy dots that we see. That is our knowledge of this phenomenon, but we want more than the Greek temple of mathematics: we want in the theater of our minds to see suitably costumed actors so act out the physics that we can believe that we have gained an appropriately deep understanding of the phenomenon.

    In this case we tend to conceive the electron riding its ghostwave as being like a surfer riding a seawave. That metaphor comes to us, of course, because knowledge of the sport of surfing is part of our culture. It seems to us like a good metaphor to express the wave-particle duality of the quantum theory, but it actually misleads us. The surfer occupies a single, definite location on his seawave, but an electron cannot occupy a single, definite location on its ghostwave.

    According to Born=s theorem, the amplitude of the ghostwave correlates with the probability density of the electron occupying a given point in space. Thus, as long as it rides the ghostwave the electron cannot exist as a point-like particle; rather, it must in some sense have its existence spread over the entire extent of the ghostwave. Only when the ghostwave breaks on some object that will interact with the electron=s particle nature will the electron manifest itself as a point-like object.

    Until it interacts with another object, such as by passing through a slit in a barrier or hitting a phosphor on a screen, an electron exists as an indefinite object. In that state it conforms to the requirements of Heisenberg=s indeterminacy principle and we think of the electron as being not real. Only when we make an observation that compels the electron to reveal its wave nature or its particle nature does the electron become real before slipping back into an indeterminate state. Thus we have the KÝbenhavn interpretation of the quantum theory, so called because it was developed by Niels Bohr and his colleagues while they were working in KÝbenhavn, Denmark.

    That interpretation is founded on six principles:

    1. A quantum system is completely described by a state function that evolves smoothly until it collapses instantaneously, due to a measurement/interaction, to an eigenstate of the measured observable property.

    2. The probability of a given outcome of a measurement follows Born= s theorem and is proportional to the square of the state function.

    3. It is not possible to know the determinate value of all of the properties of a system at the same time (Heisenberg=s indeterminacy principle).

    4. Matter exists in a wave-particle duality (and so does radiation).

    5. Measuring devices are essentially classical devices and can only measure classical properties, such as position and momentum.

    6. The quantum-mechanical description will closely approximate the classical description (the correspondence principle).

    If we apply those principles to an electron in a television picture tube after the set has been turned off, we get:

    1. As the electron goes from the electron gun to the phosphor-coated screen we associate it with a state function that consists of an imaginary exponential (cosine plus imaginary sine) whose argument consists of the algebraic description of a single point moving in a straight line from the tip of the electron gun to the center of the screen at a uniform speed. The state function has a value at each and every point within the picture tube and it evolves smoothly and continuously with the elapse of time from the emission of the electron to the electron=s absorption on the screen. Because the state function has the continuous nature of a forcefield, we expect that the material structure of the picture tube will provide the boundary conditions that give the field its unique shape.

    2. If we multiply the state function at any point at any instant by its complex conjugate, we get the product as a probability density of finding the electron and its properties at that point at that instant. Integrating that probability density over a given volume inside the picture tube gives us a number telling us the probability of finding the electron somewhere in that volume at the given instant. Using that probability, we can calculate expectation values for the electron=s properties; what we can reasonably expect the electron=s location to be at the given instant, what we can reasonably expect the electron=s linear momentum to be, and so on.

    3. The electron certainly conforms to Heisenberg=s indeterminacy principle. If multiplying together the mathematical descriptions of two of the electron=s properties yields a number with the units of action, then the product of multiplying together the indeterminacies in those properties must be greater than or equal to Planck=s constant. Thus we cannot know both the electron=s position and its linear momentum to arbitrary precision. The same holds true to Reality for the energy and location in time associated with the electron, also the angular displacement and the angular momentum, and so on.

    4. In passing from the electron gun to the screen the electron exists as a wave-particle duality; that is, it consists of the nature of a wave and the nature of a particle intermingled. It exists as such until it undergoes an interaction that causes a discontinuous change in its state function. We call such an interaction an observation or a measurement and it reveals something about either the wave nature of the electron or the particle nature of the electron, but not both at once.

    5. Although the electron exists as a purely quantum-mechanical object, it can, because of its wave nature, interact with objects large enough to conform to the laws of classical physics. Thus we can make observations or measurements, but only of properties defined in Newtonian mechanics, properties such as location, linear momentum, energy, etc. In the present example the electron strikes an atom in the phosphor coating of the picture tube and induces it to emit a photon. By seeing whence the photon came we can determine the location occupied by the electron at the instant in which it struck the atom. We have thus used a classical-scale device to measure a property of a quantum-scale object.

    6. That the electron= s behavior conforms to the correspondence principle is not so clear. In Relativity we know that, if the relative velocity between two observers tends toward zero, the special effects of the Lorentz Transformation go away and the two observers end up sharing a common Newtonian physics. The relationship between quantum mechanics and classical physics is harder to see because we use two different mathematical formalisms to represent them (and we don=t have two observers trying to translate one observer=s measurements into the other observer=s frame of reference). Nonetheless, we want to prove and verify the proposition that if the simplest actions of an object or a system become very much larger than Planck=s constant, that object or system will conform its behavior to the laws of classical physics.

    In quantum mechanics we represent the electron and its properties with a state function, whose evolution in time describes the electron=s behavior. That state function consists of a kind of wave function whose amplitude, if plotted on a graph, resembles a smoothly-curved, sharp-pointed mountain peak. If we could somehow make the electron=s mass increase without doing anything else to the electron (such as making it move at a speed close to the speed of light), then that peak would become taller and narrower until it comes to closely resemble an infinitely-tall, infinitely-narrow spike, the Dirac delta. At that point our electron would be obeying the rules of classical physics: it would have gone from being a squishy, fluffy entity in the indeterminate realm of quantum mechanics to being a point-like, billiard-ball hard object in the determinate realm of classical dynamics. From using the state function and Born=s theorem to calculate probabilities and expectation values, we have gone to using differential equations involving energy and momentum to lay out equations of motion, although, instead of using Isaac Newton=s version of dynamics, we end up using the Hamiltonian version (which is, of course, fully equivalent to the Newtonian version). And we have made that transition so smoothly that we cannot draw a line and say that quantum mechanics works only on one side and classical dynamics on the other. The correspondence principle tells us that we should expect that result.

    Thus we have the KÝbenhavn interpretation of the quantum theory.

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