(a) (5 pts) Let the mass of the Sun be m S and the mass of the earth be m E . Write down the potential energy function for V ( r ) for the Earth in the Sun’s gravitational field. Substitute this for the Coulomb potential in our analysis of the hydrogen atom to com- pute the Bohr radius a for the Earth-Sun system. Put in the actual numbers to get a value for a in meters. Hint: It will come out to a very small distance. (b) (5 pts) As shown in Townsend (discussion around Eq. 10.50), the most likely radius for an orbiting particle with quantum state n and maximal angular momentum ‘ = n - 1 (hence a nearly circular orbit) is r = n 2 a . By putting in the actual Earth-Sun distance in meters, use this to estimate the principle quantum number n for the Earth-Sun system. Since the orbit of the Earth is described very well by classical mechanics, you should find n 1. Notes for people using the first edition of Townsend • All the references to Townsend in this reading/HW assignment are the same in the first and second editions. 3
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