 Ball was trying to explain the Rydbeg formula for the spectrum of hydrogen and he was starting with the Rutherford model The Rutherford model has a nucleus in the center, a positively charged nucleus and electron going around the outside and hydrogen Just has one electron, which makes it nice and simple If they're going around in circular orbits, then by assuming that the electric force Is equal to the mass times the acceleration and the acceleration if it's going a circle must be the centripetal acceleration then equating those two things Tells us the relationship between the velocity or the speed of rotation of the electron and its radius But Bohr knew that since the light being absorbed or emitted by this atom had to come at very specific frequencies Then it also had to have very specific energies because of the relationship between the photon's energy and its frequency and That energy had to come from the atom itself So therefore when the photon was coming away from the atom with a certain amount of energy The atom had to have changed by that amount of energy And so if only specific energies of photons were being emitted then only certain changes in energy were permitted inside the atom And in much the same way that Planck and Einstein had to restrict the possible energies of photons to integer multiples of certain amounts Bohr found that he could explain the Rydbeg formula If you restricted the product of MV and R Which for a circular orbit is known as the angular momentum to be a multiple of H on 2 pi An H on 2 pi is such a common number used in quantum mechanics that it's now called H bar It's written like this and a completely embarrassing Personal aside I once accidentally wrote on a birthday card habits can be hard to break But like the other pioneers of quantum mechanics Bohr was being quite revolutionary in breaking habits of his Assuming the angular momentum which we normally denote as L can only take particularly discrete values out of the entire continuum Was an extremely revolutionary concept Bohr had no motivation for it Other than he needed to do it in order to explain the emission and absorption lines from hydrogen So let's see how he got the result he wanted Remember that for a circular orbit the angular momentum is just the momentum times the radius and linear momentum is just mass times velocity So that's going to be quantized in units of H bar And what we have is we have one equation here and two unknowns velocity and radius Fortunately, we have another equation over here And if we've got two equations and two unknowns, then we know we can solve it. So let's do it So we simply rearrange the first equation to get velocity in terms of the radius and then substitute that into the other Equations now we just have a single equation for the radius and so we've got the values of the radius And if we cross out the mass and the radius from the top of the bottom line So the radius gets bigger in discrete amounts as n gets larger The very smallest it can be is when n equals 1 and so then the radius would be H bar squared on M ke squared And that's called the Bohr radius now We could solve for the velocities if we wanted to and get them as a function of n as well But what we're really after here is to try and figure out the energies the electron can take around the proton Because it's changes between those energies as possible energies that the electron can have That's giving the photon that's being emitted particular energies and that photon having a particular energy Means that the photon has a particular frequency. So these discrete frequencies are coming out of the energy levels for the electrons And there's two kinds of energy for the electron Kinetic energy and potential energy and the kinetic energy It's just half MV squared just like it normally is when we don't worry about relativity potential energy Well, the force is attractive and is going as one on R squared So the potential energy goes as one on R and because the energy has to go up as they get pulled apart There actually has to be a minus sign there. So to deal with the kinetic energy part We notice that we have an MV squared up in that equation. So that's going to make that easier Then those two terms look almost the same and so you're just going to end up with So the fact that we have quantized radii means we're going to have quantized energy So you just substitute that radius into the energy and we get So the electron has these discrete radii with discrete energies And each possible states characterized by number n So these two states here might have in one this one up here That's a different state the electron can be in that might be in two And so what Bohr said was that his electron can't just emit radiation and decay Because it can only be in these particular levels and so in order to emit a photon it has to make a transition from one level To another level and when it does that it's going to change its energy in a very discrete amount The difference between e n2 and e n1 and when it does that it's going to emit a photon with that energy and because that energy difference is going to be something times 1 on n1 squared minus 1 on n2 squared Then the frequency since the frequencies is proportional to the energy We're going to get the Rydberg formula coming out of that and so that's why Bohr's assumption gave him the Rydberg formula And so Bohr's model worked it explained why people saw discrete lines in spectra And it gave quantitative reasons for why the particular lines were there in terms of fundamental parameters rather than just some Experimentally determined constant but like Planck before him Bohr didn't really change his idea of what electrons were He just invented a crazy rule that got in the right answers And just as Einstein took Planck's ideas about light and really changed our model for light to invent the photon in 1924 de Broglie did the opposite and he took Bohr's rule and he changed our ideas about all of matter