A small cylindrical tube is drilled through the Earth, passing through the center. A mass m is released essentially at rest at the surface. Assuming the density of the Earth is uniform, show that the mass executes simple harmonic motion and determine the frequency ω. Determine the approximate quantum number n for this state of the mass, using a typical macroscopic value for the magnitude of the mass m. Explain why a single quantum number n is inadequate to specify the state.