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Classical mechanics - Limits of validity | | Classical mechanics - Limits of validity: Encyclopedia II - Classical mechanics - Limits of validity | |
Classical mechanics - The classical approximation to special relativity.
Non-relativistic classical mechanics approximates the relativistic momentum with m0v, so it is only valid when the velocity is much less than the speed of light. For example, the relativistic cyclotron frequency of a cyclotron, gyrotron, or high voltage magnetron is given by , where fc is the classical frequency of an electron (or oth ...
See also:Classical mechanics, Classical mechanics - Description of the theory, Classical mechanics - Position and its derivatives, Classical mechanics - Forces; Newton's second law, Classical mechanics - Energy, Classical mechanics - Beyond Newton's Laws, Classical mechanics - Classical transformations, Classical mechanics - History, Classical mechanics - Limits of validity, Classical mechanics - The classical approximation to special relativity, Classical mechanics - The classical approximation to quantum mechanics, Classical mechanics - Notes | | | Classical mechanics, Classical mechanics - Beyond Newton's Laws, Classical mechanics - Classical transformations, Classical mechanics - Description of the theory, Classical mechanics - Energy, Classical mechanics - Forces; Newton's second law, Classical mechanics - History, Classical mechanics - Limits of validity, Classical mechanics - Notes, Classical mechanics - Position and its derivatives, Classical mechanics - The classical approximation to quantum mechanics, Classical mechanics - The classical approximation to special relativity, Celestial mechanics, List of equations in classical mechanics, List of publications in classical mechanics | | |
| | | Classical mechanics: Encyclopedia II - Classical mechanics - Limits of validity
Classical mechanics - Limits of validity
Classical mechanics - The classical approximation to special relativity
Non-relativistic classical mechanics approximates the relativistic momentum with m0v, so it is only valid when the velocity is much less than the speed of light. For example, the relativistic cyclotron frequency of a cyclotron, gyrotron, or high voltage magnetron is given by , where fc is the classical frequency of an electron (or other charged particle) with kinetic energy T and (rest) mass m0 circling in a magnetic field. The (rest) mass of an electron is 511 KeV. So the frequency correction is 1% for a magnetic vacuum tube with a 5.11 KV. direct current accelerating voltage.
Classical mechanics - The classical approximation to quantum mechanics
The ray approximation of classical mechanics breaks down when the de Broglie wave length is not much smaller than other dimensions of the system. For non-relativistic particles, this wave length is
where is Plank's constant divided by 2π and p is the momentum.
Again, this happens with electrons before it happens with heavier particles. For example, the electrons used by Clinton Davisson and Lester Germer in 1927, accelerated by 54 volts, had a wave length of 0.167 nm, which was long enough to exhibit a single diffraction side lobe when reflecting from the face of a nickel crystal with atomic spacing of 0.215 nm. With a larger vacuum chamber, it would seem relatively easy to increase the angular resolution from around a radian to a milliradian and see quantum diffraction from the periodic patterns of integrated circuit computer memory.
More practical examples of the failure of classical mechanics on an engineering scale are conduction by quantum tunneling in tunnel diodes and very narrow transistor gates in integrated circuits.
Classical mechanics is the same extreme high frequency approximation as geometric optics. It is more often accurate because it describes particles and bodies with rest mass. These have more momentum and therefore shorter De Broglie wave lengths than massless particles, such as light, with the same kinetic energies.
Other related archivesAristotle, Bohr, Brahe, Celestial mechanics, Christiaan Huygens, Clinton Davisson, Einstein, Euclidean geometry, Feynman Lectures on Physics, Galilean relativity, Galilean transformation, Galileo, Galileo Galilei, Gibbs paradox, Gottfried Leibniz, Greeks, Hamiltonian mechanics, History of classical mechanics, Kaons, Kepler, Lagrangian mechanics, Leibniz, Lester Germer, List of equations in classical mechanics, List of publications in classical mechanics, Lorentz force, Maxwell's equations, Newton, Newton's law, Newton's rings, Newton's second law, Newtonian mechanics, Planck, Plank's constant, Sir Isaac Newton, Special Relativity, absolute time, acceleration, angular momentum, angular resolution, baseball, bodies, calculus, center of mass, classical physics, composite, crystal, cyclotron, de Broglie, decays exponentially, degrees of freedom, derivative, diffraction, electrodynamics, electromagnetism, electron, electrons, energy, entropy, force, forces, galaxies, gases, geometric optics, gradient, gravitational force, group transformation, gyrotron, high frequency approximation, inclined plane, integral, integrated, integrated circuit, integrated circuits, kinetic energy, liquids, machinery, magnetron, mass, mechanics, molecules, momentum, negligible, ordinary differential equation, parameters, physical laws, physics, planets, point particles, position, potential energy, projectiles, quantum field theory, quantum mechanics, quantum tunneling, rate of change, reference frames, relativistic, resistive force, rest mass, rocket, side lobe, solids, space, spacecraft, special relativity, speed of light, spin, stars, theory of relativity, thermodynamics, time, tower of Pisa, transistor, tunnel diodes, unit vectors, vacuum chamber, vector, velocity
 Adapted from the Wikipedia article "Limits of validity", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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