ph2a_final_soln

# ph2a_final_soln - 2 Physics 2a Final Problem Hydrogen Atom...

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David Nichols Physics 2a December 15, 2008 Physics 2a Final Problem – Hydrogen Atom Consider the earth-sun system as a gravitational analog to the hydrogen atom. Let M denote the mass of the sun, m be the mass of the earth, and G be Newton’s gravitational constant. Recall that the gravitational potential energy is - GMm / r . (a) [5 points] What is the “Bohr radius,” a g , for the earth-sun system in terms of M , m , G and ¯ h ? Use the fact that M = 2.0 × 10 30 kg, m = 6.0 × 10 24 kg, G = 6.7 × 10 - 11 m 3 /kg · s 2 and ¯ h = 1.1 × 10 - 34 J · s to compute its numerical value. (b) [5 points] Write down the gravitational “Bohr formula” for the energy E n . Then, set this equal to the classical energy of a planet in a circular orbit of radius r o ( hint: you may recall from Ph1a that this is given by - /2 r o ) to show that the orbital quantum number n = ± r o / a g . Using the fact that that r o ± 1 AU = 1.5 × 10 11 m compute the numerical value of n . (c) [5 points] Finally, suppose that the earth makes a transition from the value of n cal- culated in part (b) to the next lower level n - 1. How much energy, in Joules, would be released? What is the wavelength of the photon (or graviton if you like) produced by this release of energy? (For reference, 1 light-year = 9.5 × 10 15 m.) (a) Given that the gravitational potential energy - / r has the same form as the Coulomb potential - e 2 /4 π± 0 r , one can replace e 2 /4 0 by in the formula for the Bohr radius. Since the Bohr radius is a 0 = ¯ h 2 m 4 0 e 2 , one can immediately see that a g = ¯ h 2 2 ± 2.5 × 10 - 138 m. ± (b) The usual Bohr energies are given by E n = - e 2 8 0 a 0 1 n 2 , so the gravitational analog will be E n = - 2 a g 1 n 2 .

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ph2a_final_soln - 2 Physics 2a Final Problem Hydrogen Atom...

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