 In 1913, Niels Bohr presented a model of the atom which appealed to concepts from both classical and quantum physics in order to overcome the shortcomings of the planetary model. He started with the idea of electrons held in orbits about the nucleus by electrical forces. Now, we've seen that the two primary problems are radiation collapse and the lack of discrete radiation frequencies. Bohr solved both of these with the radical proposal of stationary states. Why don't orbiting electrons radiate away their energy and crash into the nucleus? Because certain orbits are quote stationary and don't radiate. We can number the orbits one, two, three, and so on. And we'll say that the nth orbit has radius Rn and energy En. And the reason atoms display discrete spectral lines is as follows. An electron in an orbit of energy Ek can transition to a lower energy orbit En by emitting a photon of energy Ek minus En. The frequency of this radiation, new Kn, is fixed by Planck's relation. The energy of the photon is Planck's constant times frequency. Conversely, the electron can transition from the low energy orbit to the high energy orbit by absorbing a photon of the same frequency. The existence of discrete electron energy levels means that only certain discrete frequencies of radiation can be emitted or absorbed. Abstractly, we can think of an electron in an atom as occupying one of a set of possible energy levels. If it has energy E1 and it absorbs a photon of energy E2 minus E1, it'll move to energy level E2. If it's in energy level E4, it can emit a photon of energy E4 minus E3 and move to energy level E3 and so on. Now we look at some details of circular orbits in the hydrogen atom. The next two slides contain a bit of math in the interest of completeness. There's no need to follow them closely. At the end there's a simple takeaway. The nucleus, a proton, with positive charge E is essentially fixed at the center of the atom. The electron with negative charge minus E and mass M orbits at a radius R, moving with velocity V around a circle. We use units in which the magnitude of the electric force is E squared over R squared. Newtonian mechanics tells us that force equals mass times acceleration, and you can show that the acceleration of a circular orbit is velocity squared over radius. The result is an equation MV squared over R equals E squared over R squared. Multiplying both sides by R over 2 converts the left side to the expression for the electron's kinetic energy, one half MV squared, and this equals one half E squared over R. In smaller orbits the electron moves faster and has more kinetic energy. In addition to this kinetic energy, the total energy of the atom also has a contribution from the binding energy between the proton and electron. This potential energy is zero when the two are infinitely far apart. As the electron moves closer to the proton, the potential energy becomes increasingly negative. Negative energy means we have to put energy into the system to pull the electron back to infinite distance. At distance R, potential energy is minus E squared over R. Comparing this to the kinetic energy, we see that potential energy is minus 2 times kinetic energy, so the total energy, kinetic plus potential, is minus the kinetic energy, minus one half E squared over R. Even though the electron's kinetic energy increases as it orbits closer to the proton, this is more than offset by the decrease in potential energy, and therefore the electron is more strongly bound to the atom the closer it gets to the proton. Now Bohr proposes a way to determine which orbits form discrete stationary states that can exist without suffering radiation collapse. We've seen that Planck's constant is at the heart of quantum theory. It has units of energy times time, which are the same units as momentum times distance. This combination of units is important in classical physics and is called action. Since Planck's constant has units of action, maybe action comes in discrete chunks of size H. The action of a circular orbit is the momentum M times V times the distance around the orbit circle to pi R. Now suppose this has to be an integer N times Planck's constant. We call N the quantum number of the orbit, and we say that the orbit is quantized. If we divide by 2 pi, and define H over 2 pi to be H bar, then we have the simpler expression M times V times R equals N times H bar. Now N times V times R is called the angular momentum of the orbit. We say that H bar is the quantum of angular momentum. We propose that only orbits with angular momentum equal to an integer times H bar are stationary. This is Bohr's quantization rule. Previously we saw that Mv squared equals E squared over R. We can manipulate this to find that the quantity MvR squared equals Mre squared and solve for the radius. We find that the nth radius, Rn, equals N squared times a constant. We call this constant A0, the Bohr radius, and we find its value to be 5.3 times 10 to the minus 11th meters. The orbit's energy is minus E squared over 2 Rn, and plugging in values we arrive at the following result. The energy corresponding to orbit with quantum number N equals minus 13.6 electron volts over N squared. An electron volt is a convenient unit of energy. It's equal to what a single electron gains when it passes through a 1 volt battery. The 1 over N squared dependence of orbital energy in Bohr's model explains the 1 over N squared and 1 over k squared terms in the Reidberg formula. This is simply the wavelength corresponding to a transition from the kth orbit to the nth orbit. For instance, the red line in the Balmer series is produced by a transition from the third to the second orbit. Bohr's model had explained the observed hydrogen spectrum, and this was hailed as a great triumph. But as with the photon concept, it came at a conceptual price. Let's sketch out the first three orbits of hydrogen. An electron in the third orbit can transition to the second orbit and emit red light, or it can jump to the first orbit and emit ultraviolet light. How does this happen? To get from the third orbit to the first orbit, wouldn't the electron somehow have to go through the second orbit, and if it did, wouldn't that create red light? The answer to this is our old friend the quantum jump. The electron jumps from one orbit to another, and a discrete amount of energy at a single wavelength is generated in the process. The jump happens at random with some characteristic probability. There's no deterministic mechanism proposed to explain how this occurs. That an electron starting in the third orbit can generate light of two different frequencies, depending on what orbit it ends up in, shows that it is not the orbital frequency of the third orbit that determines the radiation frequency, yet that is what classical physics unmistakably requires. So we seem to have broken away from classical electromagnetic theory, which is troubling, since that theory has been experimentally validated to a very high degree. But, Bohr pointed out with his correspondence principle, the break is not as sharp as it might seem. Classically, the radiation frequency should equal the orbital frequency. The orbital frequency is the electron velocity divided by the length of the orbit. The radiation frequency, from orbit N to orbit N-1, is just the difference of the orbital energies divided by Planck's constant. It's true that these are different for small values of N, but as N gets very large, the values converge. On this graph we show in red, Bohr's prediction for the radiation frequency, for different quantum numbers, from 1 on the left to 100 on the right. In blue we plot the orbital frequency, and we can see that they converge for large values of the quantum number N, that is, for large orbits. Thus, the quantum theory smoothly branches off from classical theory. The difference becomes prominent at very small scales, that is, for very small orbits, very small quantum numbers. We have the guiding principle that for large quantum numbers, quantum mechanics should correspond to classical mechanics. There were some indications that Bohr's theory might work for other atoms and even molecules. For example, he had some success in explaining how two hydrogen atoms form a hydrogen molecule. The two positive protons repel each other. But if the two electrons circle between them in a quantized Bohr orbit, the proton-electron attractions more than offset the proton-proton repulsion. Bohr showed that a stable molecular configuration results. Here's a page from Bohr's notebook showing how he thought these ideas could be applied to more complicated molecules. Ultimately, however, he had only very limited success. A new breakthrough was needed. That breakthrough came in 1924 when the concept of wave particle duality was applied to the electron by Louis de Broglie. De Broglie's hypothesis was this. Quantum theory had shown that light, which is classically a wave, has both wave and particle properties. Specifically, if the wavelength is lambda, then photons of light have a momentum equal to Planck's constant over lambda. De Broglie has suggested that wave particle duality might be a general principle of nature. Therefore, an electron, which is classically a particle with momentum p equals mass times velocity, should also have an associated wavelength given by Planck's constant over momentum. Now consider the situation when an integer number of electron wavelengths fit into an electron's orbit. The electron wave would repeat, constructively reinforcing itself and forming a stationary state. But, if the orbit is not an integer number of wavelengths, then the electron wave destructively interferes with itself and no stationary state is possible. The circumference of an orbit is 2 pi r, and we want this to equal an integer n times the wavelength. Lambda is h over p, and p is m times v. Multiplying through by mv, and dividing by 2 pi, and recalling our definition of h bar as h over 2 pi, we end up with mvr equals n h bar, which is precisely Bohr's quantization formula. It seems that the discrete radiation spectra of atoms might be a natural consequence of wave particle duality. But is there any evidence for this? Do electrons actually exhibit any wave properties? Well, from as early as 1927, experiments have repeatedly shown that yes, electrons do exhibit wave properties. For example, we've seen that the double slit experiment with light is a powerful demonstration of photon wave particle duality. The same experiment performed with electrons likewise shows a combination of the particle and wave behavior, an electron wave forming an interference pattern. From wave particle duality, it immediately follows that electrons, like photons, must be subject to the uncertainty principle. Therefore, we see that the Bohr model's picture of an electron at a specific place in an orbit moving with a specific momentum cannot be correct. If we want to unlock the secrets of the atom, we're going to need to move even farther away from classical concepts.