Question

Problem 2 Consider the Earth-Sun system as a quantum gravitational analog of a hydrogen atom a) What would be, in actual units, the Bohr radius of the system? b) By equating the actual energy of the system to the Bohr formula, estimate the quantum numb the Earth

Answer 1

An amusing little problem is to consider the Earth-Sun system as a giant analogue to the quantum model of the hydrogen atom. Since the grav tational potential is inverse-r, the same as the electromagnetic force, the calculations translate quite easily. The potential is where G is the gravitational constant, G 6.673 x 10-1m3kg 2 The Bohr radius for the Earth can be found by replacing e by mM, and 1/4TEo by G た In MKS, the values are1.0546 x 10-34m2kg/s, mass of the Earth is m = 5.9742 x 1024 kg, and mass of the Sun is M = 1.98892 x 1030 kg Plugging the numbers gives ag-2.349 x 10-138mm (that is, small) From Bohr's formula for the energy levels, as in above, we get doing the same replacemen ts In classical physics, the energy of a planet in a circular orbit of radius ro is the sum of the gravitational potential energy and the kinetic energy. The gravitational potential is ro

Radius is 1.49598 x 10¹¹ m

Quantum number is 2.524 x 10⁷⁴