En = -m / 2 hbar2 ( e2/4 Pi e0)2 1/n2 for n = 1,2,3....

This equation describes the allowed energies of a single electron when bound to a hydrogen atom nucleus (i.e., a proton). It was discovered by Niels Bohr in 1913, using a mix of classical and then-nascent quantum physics, and was one of the finds really kicked off the development of quantum mechanics.

Bohr found this equation by assuming that the angular momentum of an electron was quantized, in other words, it could only have specific values, and nothing in between those. From this restriction, he could work out what 'orbits' the electrons would be allowed to inhabit around the nucleus, and what the energies of those orbits should be.

While the equation is completely correct according to modern quantum theory, his derivation has several faults (completely understandable. Things like the Schodinger Equation did not arrive until a few years later). Electrons do not orbit around nuclei in any way; they simply have a probability distribution which places them near the nucleus if they do not have sufficient energy to scatter away.

Bohr's formula, also, only works for atoms with only a single electron. (Ionized helium, etc). Any more electrons, and there was no way to patch the formula to result in answers that were consistent with observation.

A modern derivation of this formula can be done using the 3-dimensional Schrodinger equation, and the potential energy equation for electric forces. That, and solving a few differential equations, doing a lot of substitutions, simplifications, and clever math tricks.

Of course, the benefit of the modern way is that one can apply it to any atom.

Oh, the mess of constants in front of the 1/n2 term turns out to be roughly 13.6 eV.

As quantumet notes, this derivation has several errors (although it still comes up with an equation which works under certain special circumstances). Mainly, Bohr assumes (in Step 1) that the electron is just an ordinary Newtonian object in uniform circular motion. But here, for your viewing pleasure, is the actual derivation. You'll need to know a fair amount of basic physics and a few formulas from very basic quantum mechanics.

1. F = ke*ze / (r2) = m*v2 / r
2. mv2 = k*z*e2 / r
3. KE = 0.5mv2 = k*z*e2 / (2*r)
4. U = - k*z*e2 / r
5. E = KE + U = - k*z*e2 / (2*r)
6. L = I*omega (Angular Momentum = Moment of Inertia * Angular Velocity)
7. I = mr2
8. w = v / r
9. L = (mr2)*(v / r) = mvr
10. mvr = n*h/(2*pi)
11. v = nh / (2*pi*m*r)
12. m*(nh / 2*pi*m*r)2 = k*z*e2 / r
13. r = (n2 / z)*(h2epsilon0 / (pi*m*e2))
14. E = -kze2 / (2*r) = -kze2 / (2*(n2 / z)*(h2epsilon0 / (pi*m*e2)))
15. E = -(z / n)2 * ((e4m) / (8*(epsilon0h)2))
16. E = (R/h)*(1/n)2
17. h*c/lambda = (R/h)*(1 / nf2 - 1 / ni2)
18. 1 / lambda = R*(1 / nf2 - 1 / ni2)
Notes:
• Step 1: For a single electron orbiting around a nucleus, where e is the basic electrical charge, k is a constant, r is the radius, z is the number of protons, and m is the mass of an electron.
• Step 10: This was a big leap of faith for Bohr. As de Broglie later proved in 1923, this step works because the electron moves as a standing particle wave around the nucleus. For more information, see de Broglie's explanation of Bohr's assumption about angular momentum
• Step 12: Plugging in the results for v to step 2
• Step 13: Since k = 1 / (4*pi*epsilon0)
• Step 14: Plugging in the results for r to step 6
• Step 16: Assuming that this is a hydrogen atom, and therefore z = 1. Also, the whole mess of constants out in front simplifies down to Rydberg's constant, R (actually R divided by h, but the h kindly disappears in the next step).

For further reference:
Cutnell, John D. and Kenneth W. Johnson, Physics. New York: John Wiley & Sons, 1998 (4th edition),
or any good physics book.

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