Once Before Time - Martin Bojowald [25]
Besides the possibility of superpositions, quantum theory shows several important differences from the principles taken for granted in prequantum, classical physics. In contrast to a classical particle, objects described by a wave function cannot be assigned distinct positions or velocities. Measurements of position or velocity are always subject to an inherent uncertainty that cannot be eliminated by improving the measurement apparatus. In a single position measurement, one could, in principle, find any value within the region where the wave function has nonzero height. Performing many measurements, one will find a different value in that region each time; a large number of measurements leads to a distribution of results in which the frequency of a specific position measured is given by the height (or rather, its norm squared) of the wave function at this place. Certainly, one would not expect to localize a wave function far away from its center except in rare cases; but since the height there can still be nonzero, one may sometimes take a place far from the center as the actual position of the wave. In contrast to a water wave, which, by daylight, can be seen in its full extension, measurements of a quantum theoretical wave function are random tests. One checks out the wave function just a few times as if trying to feel its shape at night. And before one can measure the wave often enough to find its precise position, it will have moved away or—just by having been touched—changed its form.
The height of a wave function at some point, so says quantum theory, determines how often one would take this place as the actual position of a particle. In mathematical terms, this distribution is a probability: the likelihood of the result “position x” is given by the height of the wave function there. If this height at some place is larger than it is somewhere else, attempts to feel the wave would more often make one take this place for the position of the particle and one is more likely to identify this place as the actual position. Similarly, probabilities can be found for measurements of velocity or any other quantity of interest. One would think that one could then simply determine the whole wave function by measuring the position sufficiently often, deriving from the individual results for all points in space the shape of the wave function. When each position measurement is plotted in a diagram, darker regions would be seen where the particle was found more often, surrounded by lighter shades where the particle would enter only rarely. Indistinct but noticeable contours would result that one could interpret as the shape of an object measured.
But once again this is too classical a picture of a wave to be applicable in quantum theory. Such measurements would be so sensitive that their process itself, as a physical procedure, must influence the state described by a wave function. After, say, the position has been measured, a wave function is different than before. With most measurements, this goes so far that the wave function is said to “collapse” due to the measurement process—similarly to Gawain’s “quantum bed,” the Lit marveile:
He [Gawain] entered. Its pavement shone smooth and clear as glass. Upon it stood that fabulous Bed: Lit marveile! … He went at a peradventure. And as often as he made a step, the Bed moved from where it was.… “How shall I get at you?” he