Simple Nature - Light and Matter

dimensional particle in a box, and discuss how this relates to the uncertaintyprinciple.B On a graph of ∆p versus ∆x, sketch the regions that are allowed andforbidden by the Heisenberg uncertainty principle. Interpret the graph:Where does an atom lie on it? An elephant? Can either p or x be measuredwith perfect accuracy if we don’t care about the other?j / An electron in a gentleelectric field gradually shortensits wavelength as it gains energy.k / The wavefunction’s tailsgo where classical physics saysthey shouldn’t.13.3.5 Electrons in electric fieldsSo far the only electron wave patterns we’ve considered havebeen simple sine waves, but whenever an electron finds itself in anelectric field, it must have a more complicated wave pattern. Let’sconsider the example of an electron being accelerated by the electrongun at the back of a TV tube. Newton’s laws are not useful,because they implicitly assume that the path taken by the particle isa meaningful concept. Conservation of energy is still valid in quantumphysics, however. In terms of energy, the electron is movingfrom a region of low voltage into a region of higher voltage. Sinceits charge is negative, it loses electrical energy by moving to a highervoltage, so its kinetic energy increases. As its electrical energy goesdown, its kinetic energy goes up by an equal amount, keeping thetotal energy constant. Increasing kinetic energy implies a growingmomentum, and therefore a shortening wavelength, j.The wavefunction as a whole does not have a single well-definedwavelength, but the wave changes so gradually that if you only lookat a small part of it you can still pick out a wavelength and relateit to the momentum and energy. (The picture actually exaggeratesby many orders of magnitude the rate at which the wavelengthchanges.)But what if the electric field was stronger? The electric field ina TV is only ∼ 10 5 N/C, but the electric field within an atom ismore like 10 12 N/C. In figure l, the wavelength changes so rapidlythat there is nothing that looks like a sine wave at all. We couldget a rough idea of the wavelength in a given region by measuringthe distance between two peaks, but that would only be a roughapproximation. Suppose we want to know the wavelength at pointP . The trick is to construct a sine wave, like the one shown with thedashed line, which matches the curvature of the actual wavefunctionas closely as possible near P . The sine wave that matches as well aspossible is called the “osculating” curve, from a Latin word meaning“to kiss.” The wavelength of the osculating curve is the wavelengththat will relate correctly to conservation of energy.TunnelingWe implicitly assumed that the particle-in-a-box wavefunctionwould cut off abruptly at the sides of the box, k/1, but that would beunphysical. A kink has infinite curvature, and curvature is related866 Chapter 13 Quantum Physics