(1) omit the necessary steps to solve it - or provide general guidelines based on properties of ellipses - and then present the final result of the energy. Introductory quantum mechanics textbooks that deal with the old quantum theory and present Eq. Our aim is to solve this to find an expression for the energy E. n r is a positive integer, the radial quantum number. Where h is Planck's constant, m is the mass of the electron, - e its charge, and Z the number of protons in the nucleus. Specifically, if the electron orbits the nucleus in an elliptical trajectory, with the radius varying between r max and r min, with constant energy E and angular momentum L, the quantization of the radial action will be given by the integral Inc., London, 1923).– 11 Arnold Sommerfeld, Annalen der Physik 51, 125 (1916).] - fundamental in the old quantum theory - when the electronic orbits are allowed to be elliptical (as opposed to circular), thus allowing for non-zero radial component of the momentum. Inc., London, 1923).] The problem consists in solving an integral resulting from the phase-space quantization conditions Arnold Sommerfeld, Atomic Structure and Spectral Lines (Methuen & Co. Arnold Sommerfeld, Atomic Structure and Spectral Lines (Methuen & Co. Inc., London, 1923).] when deriving the energy levels of a hydrogen-like atom. This article addresses the specific but recurrent inadequate usage of "it is easy to show that" in many quantum mechanics textbooks Robert Eisberg and Robert Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (John Wiley & Sons, New York, 1985).– 6 Arnold Sommerfeld, Atomic Structure and Spectral Lines (Methuen & Co. On the other hand, its inadequate use can make the student lose too much time in a fruitless pursuit of misguided paths, leaving them eventually discouraged when a simple sentence or set of references could point them to the right direction. On the one hand, when well employed, that is, when the steps necessary to reach the final result are a natural consequence of what was explained before, the expression improves the reader's self-confidence by stimulating the use of creativity to fill the logical and mathematical gaps in finding the solution. The quantity T 2/a 3 depends upon the sum of the masses of the Sun and the planet, but since the mass of the Sun is so great, adding the mass of the planet makes very little difference.ĭata from Halliday, Resnick, Walker, Fundamentals of Physics 4th Ed Extended.Few expressions are more frustrating to the student than "it is easy to prove that" followed by a non-trivial result. Table of dataĭata confirming Kepler's Law of Periods comes from measurements of the motion of the planets. In this more rigorous form it is useful for calculation of the orbital period of moons or other binary orbits like those of binary stars. But more precisely the law should be written Kepler's Law of Periods in the above form is an approximation that serves well for the orbits of the planets because the Sun's mass is so dominant. Newton first formulated the law of gravitation from Kepler's 3rd law. This is one of Kepler's laws.This law arises from the law of gravitation. The Law of Periods The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. When the planet is closer to the sun, it moves faster, sweeping through a longer path in a given time. This is one of Kepler's laws.This empirical law discovered by Kepler arises from conservation of angular momentum. The Law of Areas A line that connects a planet to the sun sweeps out equal areas in equal times. Of the planetary orbits, only Pluto has a large eccentricity. Orbit EccentricityThe eccentricity of an ellipse can be defined as the ratio of the distanceīetween the foci to the major axis of the ellipse. The eccentricity of the ellipse is greatly exaggerated here. The elliptical shape of the orbit is a result of the inverse square force of gravity. The Law of Orbits All planets move in elliptical orbits, with the sun at one focus. Kepler's laws were derived for orbits around the sun, but they apply to satellite orbits as well. The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times.ģ. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.Ģ. Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky.ġ.
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