If the angular momentum of Earth in its motion around the sun were quantized lik

Question

If the angular momentum of Earth in its motion around the sun were quantized like a hydrogen electron according to Eq.4-17, what would Earth's quantum number be? How much energy would be released in a transition to the next lowest level? Would that energy release (presumably as a gravity wave) be detectable? What would be the radius of that orbot? (the radius of the earth's orbit is 1.50 E 11 m)

Equation 4-17 L = mvr =(nh/2(pie)) = n(h bar)
please be specific where you find any numbers that you use

Explanation / Answer

Louis de Broglie first correctly hypothesized that matter has an associated wavelength.

Neils Bohr extended this theory to suggest the quantization of electron wavelengths in an atom (of different energy levels).

This is the result of Bohr's angular momentum quantization proof:
L = mvr = nh/2(pi)
This shows that orbital angular momentum (of an electron for instance) is quantized in discrete integer packets of h/2pi, or h_bar. Now, if we extend this theory to larger, macroscopic objects such as the Earth, then we can find the principle quantum number (n) of Earth's angular momentum. We recall the mass and tangiential velocity of Earth (about the Sun, it's orbital speed), as well as the radius of orbit (1 Au = 150 Gm), and solving for n: L = mvr = nh/2(pi) n = 2(pi)mvr/h n = 2(pi)(5.97 x 10^24kg)(108 Mm/hr)(150 Gm)/(6.63x10^-34 kg*m^2/s) n = 2(pi)(5.97 x 10^24kg)(108*3,600 Mm/s)(150 Gm)/(6.63x10^-34 kg*m^2/s) n = 3.30 x 10^[8+24+6+9-(-34)] {kg*m^2/s}/{kg*m^2/s} n = 3.30 x 10^81 The Energy release when moving down on integer in principle quantum number would be found using the Rydberg formula: E = (R_ryd)[1/(n_f)^2 - 1/(n_i)^2] but n_f is so close to n_i (n_f = n_i -1 = [3.30 x 10^81] -1), so the difference is negligible.
This implies the Energy release is neglible, or extremely close to zero. Therefore this energy would hardly be detectible, a single photon may be emitted in the form of an ultra-low energy radio wave with very large wavelength. Also, because of the neglible difference in Energy and quantum number, the radius of orbit would remain basically the same, with only a neglible decrease.

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